Counting Words in the Alphabet, {1, 2}, That Avoid A Certain set of , 3, consecutive subwords of length, 3, For all the possibilities, According to their Number of Good Neighbors By Shalosh B. Ekhad ------------------------------------------------ "Theorem Number 1" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [1, 2, 1], [2, 1, 1]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = / | / | ----- | ----- | m = 0 \ n = 0 / 4 3 4 2 3 3 3 2 2 2 2 x y - 2 x y + 3 x y - 3 x y - 2 x y - x y - 1 - --------------------------------------------------------- 5 2 5 4 2 4 3 x y - x y + x y - x y - x y - x y + 1 and in Maple notation -(2*x^4*y^3-2*x^4*y^2+3*x^3*y^3-3*x^3*y^2-2*x^2*y^2-x*y-1)/(x^5*y^2-x^5*y+x^4*y ^2-x^4*y-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 4 2 2 (a + 2 a - 2 a - 1) - ------------------------- 2 2 (3 a + 1) (2 a + a + 1) 3 where a is the root of the polynomial, x + x - 1, and in decimals this is, 0.3885080076 BTW the ratio for words with, 500, letters is, 0.3895060905 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2], [1, 2, 1], [2, 1, 1]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 4 8 3 8 2 6 4 8 ) | ) C(m, n) x y | = (3 x y - 5 x y + x y - 3 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 2 6 3 7 5 3 6 5 2 4 3 5 - x y + 2 x y + x y + 3 x y + x y - 3 x y - 3 x y + x y 4 2 3 3 4 3 2 3 2 2 3 2 + x y + 3 x y + 2 x y + 2 x y - 2 x y - 2 x y - x + x y - x y / 5 2 5 4 2 4 3 + x - 1) / ((x - 1) (x y - x y + x y - x y - x y - x y + 1)) / and in Maple notation (3*x^8*y^4-5*x^8*y^3+x^8*y^2-3*x^6*y^4+x^8*y-x^7*y^2+2*x^6*y^3+x^7*y+3*x^5*y^3+ x^6*y-3*x^5*y^2-3*x^4*y^3+x^5*y+x^4*y^2+3*x^3*y^3+2*x^4*y+2*x^3*y^2-2*x^3*y-2*x ^2*y^2-x^3+x^2*y-x*y+x-1)/(x-1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 5 4 3 2 3 2 2 (2 a + 4 a - 4 a + 2 a - 4 a + 2 a + 1) (4 a - 3 a + 2 a - 2) ---------------------------------------------------------------------- 2 2 5 3 2 (3 a + 1) (a + 2 a - a - 1) 3 where a is the root of the polynomial, (x - 1) (x + x - 1), and in decimals this is, 0.3885080076 BTW the ratio for words with, 500, letters is, 0.3886363662 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2], [1, 2, 1], [2, 1, 2]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 3 5 2 4 3 5 4 2 ) | ) C(m, n) x y | = (x y - 2 x y + x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 3 2 3 2 2 2 2 - 3 x y - 5 x y + 4 x y + 4 x y - 4 x y + x + 2 x y - 2 x + 1) / 2 / (x - 2 x + 1) / and in Maple notation (x^5*y^3-2*x^5*y^2+x^4*y^3+x^5*y+3*x^4*y^2-3*x^4*y-5*x^3*y^2+4*x^3*y+4*x^2*y^2-\ 4*x^2*y+x^2+2*x*y-2*x+1)/(x^2-2*x+1) ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [2, 1, 1], [2, 2, 1]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 3 5 2 4 3 5 ) | ) C(m, n) x y | = - (x y - 2 x y - 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 2 3 3 4 3 2 3 2 2 3 2 2 + 4 x y - x y - 2 x y + x y + x y - 4 x y - x + 2 x y + x / 2 - 2 x y + x - 1) / ((x - 1) (x - 1)) / and in Maple notation -(x^5*y^3-2*x^5*y^2-2*x^4*y^3+x^5*y+4*x^4*y^2-x^3*y^3-2*x^4*y+x^3*y^2+x^3*y-4*x ^2*y^2-x^3+2*x^2*y+x^2-2*x*y+x-1)/(x-1)/(x^2-1) ------------------------------------------------ "Theorem Number 5" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [1, 2, 1], [2, 2, 1]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = / | / | ----- | ----- | m = 0 \ n = 0 / 3 3 3 2 3 2 2 2 x y - x y + x y + 4 x y - 2 x y + 2 x y - x + 1 - ------------------------------------------------------- x - 1 and in Maple notation -(x^3*y^3-x^3*y^2+x^3*y+4*x^2*y^2-2*x^2*y+2*x*y-x+1)/(x-1) ------------------------------------------------ "Theorem Number 6" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2], [1, 2, 2], [2, 1, 2]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 4 3 4 2 3 3 4 3 2 ) | ) C(m, n) x y | = - (x y - 2 x y - x y + x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 2 2 2 2 / 2 - 3 x y - 4 x y + 4 x y - x - 2 x y + 2 x - 1) / (x - 2 x + 1) / and in Maple notation -(x^4*y^3-2*x^4*y^2-x^3*y^3+x^4*y+5*x^3*y^2-3*x^3*y-4*x^2*y^2+4*x^2*y-x^2-2*x*y +2*x-1)/(x^2-2*x+1) ------------------------------------------------ "Theorem Number 7" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [1, 2, 1], [2, 1, 2]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 4 9 3 9 2 8 3 ) | ) C(m, n) x y | = (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 2 8 7 2 6 3 5 4 7 6 2 7 - 4 x y + 2 x y + x y - 2 x y - x y - 2 x y + 5 x y + x 6 5 2 4 3 6 5 4 2 5 3 2 3 - 4 x y - 3 x y - 2 x y + x + 4 x y + x y - x + x y - 2 x y 2 2 2 / 4 - 4 x y + 2 x y - 2 x y + x - 1) / (x + x - 1) / and in Maple notation (x^9*y^4-2*x^9*y^3+x^9*y^2+2*x^8*y^3-4*x^8*y^2+2*x^8*y+x^7*y^2-2*x^6*y^3-x^5*y^ 4-2*x^7*y+5*x^6*y^2+x^7-4*x^6*y-3*x^5*y^2-2*x^4*y^3+x^6+4*x^5*y+x^4*y^2-x^5+x^3 *y^2-2*x^3*y-4*x^2*y^2+2*x^2*y-2*x*y+x-1)/(x^4+x-1) ------------------------------------------------ "Theorem Number 8" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [1, 2, 2], [2, 2, 2]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 3 7 2 6 3 7 6 2 ) | ) C(m, n) x y | = (x y - 2 x y - x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 5 2 4 3 5 4 2 4 3 2 2 2 3 - x y + 2 x y - x y - 2 x y + x y - x y - 3 x y - 4 x y + x 2 / 3 2 + x - 2 x y - 1) / (x + x - 1) / and in Maple notation (x^7*y^3-2*x^7*y^2-x^6*y^3+x^7*y+2*x^6*y^2-x^6*y+2*x^5*y^2-x^4*y^3-2*x^5*y+x^4* y^2-x^4*y-3*x^3*y^2-4*x^2*y^2+x^3+x^2-2*x*y-1)/(x^3+x^2-1) ------------------------------------------------ "Theorem Number 9" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [1, 2, 1], [2, 2, 2]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 2 9 9 7 2 7 ) | ) C(m, n) x y | = (x y - 2 x y + x - 2 x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 2 7 6 5 2 6 5 4 2 3 2 + 3 x y - 2 x - 2 x y + 3 x y - x - 4 x y - 3 x y - 3 x y 3 2 2 3 / 4 3 - 2 x y - 4 x y + x - 2 x y - 1) / (x + x - 1) / and in Maple notation (x^9*y^2-2*x^9*y+x^9-2*x^7*y^2+4*x^7*y+3*x^6*y^2-2*x^7-2*x^6*y+3*x^5*y^2-x^6-4* x^5*y-3*x^4*y^2-3*x^3*y^2-2*x^3*y-4*x^2*y^2+x^3-2*x*y-1)/(x^4+x^3-1) ------------------------------------------------ "Theorem Number 10" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1], [2, 1, 2], [2, 2, 1]}, and having m neighbors (i.e. Hamming d\ istance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 2 5 3 6 5 2 4 3 ) | ) C(m, n) x y | = (x y + x y - 2 x y - 3 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 5 4 2 3 3 5 4 3 2 3 2 2 + x + 3 x y + 6 x y + x y - x - 4 x y - 8 x y + 5 x y + 4 x y 3 2 2 / 2 + x - 4 x y + x + 2 x y - 2 x + 1) / (x - 1) / and in Maple notation (x^6*y^2+x^5*y^3-2*x^6*y-3*x^5*y^2-2*x^4*y^3+x^6+3*x^5*y+6*x^4*y^2+x^3*y^3-x^5-\ 4*x^4*y-8*x^3*y^2+5*x^3*y+4*x^2*y^2+x^3-4*x^2*y+x^2+2*x*y-2*x+1)/(x-1)^2 ---------------------------------- This ends this paper that took, 0.930, seconds to produce ------------------------------ Counting Words in the Alphabet, {1, 2}, That Avoid A Certain set of , 3, consecutive subwords of length, 4, For all the possibilities, According to their Number of Good Neighbors By Shalosh B. Ekhad ------------------------------------------------ "Theorem Number 1" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [1, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 12 17 11 16 12 ) | ) C(m, n) x y | = - (x y - 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 10 16 11 15 12 17 9 16 10 15 11 + 6 x y - 3 x y + 2 x y - 4 x y + 2 x y - 7 x y 14 12 17 8 16 9 15 10 14 11 16 8 + 2 x y + x y + 2 x y + 8 x y - 8 x y - 3 x y 15 9 14 10 13 11 16 7 15 8 14 9 - 2 x y + 11 x y - 2 x y + x y - 2 x y - 5 x y 13 10 12 11 15 7 14 8 13 9 12 10 14 7 + 5 x y - 2 x y + x y - x y - 3 x y + 6 x y + x y 13 8 12 9 13 7 12 8 11 9 12 7 11 8 - x y - 5 x y + x y - 2 x y + 2 x y + 5 x y - 11 x y 10 9 12 6 11 7 10 8 11 6 10 7 9 8 - x y - 2 x y + 16 x y - 4 x y - 7 x y + 9 x y - 6 x y 10 6 9 7 8 8 10 5 9 6 8 7 10 4 - x y + 16 x y + x y - 4 x y - 13 x y + 2 x y + x y 9 5 8 6 7 7 9 4 8 5 7 6 8 4 + 2 x y - 3 x y + 4 x y + x y - 4 x y - 6 x y + 4 x y 7 5 6 6 7 4 6 5 6 4 5 5 6 3 5 4 - x y + x y + 3 x y - 4 x y + 8 x y + x y - 4 x y + 5 x y 5 3 4 4 5 2 4 3 4 2 3 3 3 2 3 - 3 x y + 3 x y - x y - 4 x y + x y - 4 x y + x y + x y 2 2 2 / 18 12 18 11 18 10 17 11 - 2 x y + x y - x y - 1) / (x y - 4 x y + 6 x y + x y / 16 12 18 9 17 10 16 11 18 8 17 9 + 2 x y - 4 x y - 4 x y - 8 x y + x y + 6 x y 16 10 17 8 16 9 15 10 14 11 17 7 + 12 x y - 4 x y - 8 x y - x y - 2 x y + x y 16 8 15 9 14 10 15 8 14 9 15 7 + 2 x y + 3 x y + 6 x y - 3 x y - 6 x y + x y 14 8 13 9 13 8 13 7 12 8 11 9 13 6 + 2 x y + x y - 4 x y + 5 x y - 3 x y - x y - 2 x y 12 7 11 8 12 6 11 7 12 5 11 6 10 7 + 5 x y - 2 x y - x y + 8 x y - x y - 6 x y + 2 x y 9 8 11 5 10 6 9 7 9 6 10 4 9 5 9 4 + x y + x y - 3 x y + x y - 3 x y + x y - x y + 2 x y 8 5 8 4 7 5 7 4 6 5 7 3 6 3 5 4 - x y + x y + x y + 2 x y + 2 x y - 2 x y - x y - x y 5 3 4 4 5 2 3 2 3 2 - x y - x y + x y + x y - x y - x y - x y + 1) and in Maple notation -(x^17*y^12-4*x^17*y^11+x^16*y^12+6*x^17*y^10-3*x^16*y^11+2*x^15*y^12-4*x^17*y^ 9+2*x^16*y^10-7*x^15*y^11+2*x^14*y^12+x^17*y^8+2*x^16*y^9+8*x^15*y^10-8*x^14*y^ 11-3*x^16*y^8-2*x^15*y^9+11*x^14*y^10-2*x^13*y^11+x^16*y^7-2*x^15*y^8-5*x^14*y^ 9+5*x^13*y^10-2*x^12*y^11+x^15*y^7-x^14*y^8-3*x^13*y^9+6*x^12*y^10+x^14*y^7-x^ 13*y^8-5*x^12*y^9+x^13*y^7-2*x^12*y^8+2*x^11*y^9+5*x^12*y^7-11*x^11*y^8-x^10*y^ 9-2*x^12*y^6+16*x^11*y^7-4*x^10*y^8-7*x^11*y^6+9*x^10*y^7-6*x^9*y^8-x^10*y^6+16 *x^9*y^7+x^8*y^8-4*x^10*y^5-13*x^9*y^6+2*x^8*y^7+x^10*y^4+2*x^9*y^5-3*x^8*y^6+4 *x^7*y^7+x^9*y^4-4*x^8*y^5-6*x^7*y^6+4*x^8*y^4-x^7*y^5+x^6*y^6+3*x^7*y^4-4*x^6* y^5+8*x^6*y^4+x^5*y^5-4*x^6*y^3+5*x^5*y^4-3*x^5*y^3+3*x^4*y^4-x^5*y^2-4*x^4*y^3 +x^4*y^2-4*x^3*y^3+x^3*y^2+x^3*y-2*x^2*y^2+x^2*y-x*y-1)/(x^18*y^12-4*x^18*y^11+ 6*x^18*y^10+x^17*y^11+2*x^16*y^12-4*x^18*y^9-4*x^17*y^10-8*x^16*y^11+x^18*y^8+6 *x^17*y^9+12*x^16*y^10-4*x^17*y^8-8*x^16*y^9-x^15*y^10-2*x^14*y^11+x^17*y^7+2*x ^16*y^8+3*x^15*y^9+6*x^14*y^10-3*x^15*y^8-6*x^14*y^9+x^15*y^7+2*x^14*y^8+x^13*y ^9-4*x^13*y^8+5*x^13*y^7-3*x^12*y^8-x^11*y^9-2*x^13*y^6+5*x^12*y^7-2*x^11*y^8-x ^12*y^6+8*x^11*y^7-x^12*y^5-6*x^11*y^6+2*x^10*y^7+x^9*y^8+x^11*y^5-3*x^10*y^6+x ^9*y^7-3*x^9*y^6+x^10*y^4-x^9*y^5+2*x^9*y^4-x^8*y^5+x^8*y^4+x^7*y^5+2*x^7*y^4+2 *x^6*y^5-2*x^7*y^3-x^6*y^3-x^5*y^4-x^5*y^3-x^4*y^4+x^5*y^2+x^3*y^2-x^3*y-x^2*y- x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 2 (3 a + 2 a - 6 a - 4 a + 3 a + 2 a + 1) --------------------------------------------------------------------- 6 5 4 3 6 5 3 2 (7 a + 6 a - 5 a - 4 a - 2 a - 1) (a + 2 a - 2 a - a - a - 1) 7 6 5 4 2 where a is the root of the polynomial, x + x - x - x - x - x + 1, and in decimals this is, 0.6677422162 BTW the ratio for words with, 500, letters is, 0.6677622203 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 7 7 6 7 5 7 4 ) | ) C(m, n) x y | = - (3 x y - 7 x y + 5 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 5 6 4 6 3 5 4 5 3 4 4 5 2 + 3 x y - 4 x y + x y + 2 x y - 3 x y + 3 x y + x y 4 3 3 3 3 2 2 2 2 / 8 6 - 3 x y - 4 x y + 2 x y - 2 x y + x y - x y - 1) / (x y / 8 5 8 4 7 4 6 5 7 3 6 4 6 3 5 3 - 2 x y + x y + x y + x y - x y - 2 x y + x y + 2 x y 4 4 5 2 4 3 4 2 2 - x y - 2 x y + x y - x y - x y - x y + 1) and in Maple notation -(3*x^7*y^7-7*x^7*y^6+5*x^7*y^5-x^7*y^4+3*x^6*y^5-4*x^6*y^4+x^6*y^3+2*x^5*y^4-3 *x^5*y^3+3*x^4*y^4+x^5*y^2-3*x^4*y^3-4*x^3*y^3+2*x^3*y^2-2*x^2*y^2+x^2*y-x*y-1) /(x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4+x^6*y^5-x^7*y^3-2*x^6*y^4+x^6*y^3+2*x^5*y^3 -x^4*y^4-2*x^5*y^2+x^4*y^3-x^4*y^2-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 4 3 2 2 (a + 2 a + 4 a - 3 a - 2 a - 1) - ------------------------------------- 3 3 2 (4 a + 2 a + 1) (2 a + a + a + 1) 4 2 where a is the root of the polynomial, x + x + x - 1, and in decimals this is, 0.6527572904 BTW the ratio for words with, 500, letters is, 0.6530856433 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 9 11 8 11 7 10 8 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 6 10 7 9 8 11 5 10 6 9 7 10 5 - 4 x y - 7 x y + 3 x y + x y + 9 x y - 8 x y - 5 x y 9 6 8 7 10 4 9 5 8 6 7 7 8 5 + 7 x y - x y + x y - 2 x y + 6 x y - 3 x y - 8 x y 7 6 8 4 6 6 7 4 6 5 7 3 6 4 + 5 x y + 3 x y - x y - 3 x y + 3 x y + x y - 4 x y 6 3 5 4 5 3 4 4 5 2 4 3 3 3 + 2 x y - 8 x y + 9 x y - 3 x y - 2 x y + 3 x y + 4 x y 3 2 2 2 2 / 8 6 8 5 8 4 7 4 - 2 x y + 2 x y - x y + x y + 1) / (x y - 2 x y + x y + x y / 6 5 7 3 6 4 6 3 5 3 4 4 5 2 4 3 + x y - x y - 2 x y + x y + 2 x y - x y - 2 x y + x y 4 2 2 - x y - x y - x y + 1) and in Maple notation (x^11*y^9-4*x^11*y^8+6*x^11*y^7+2*x^10*y^8-4*x^11*y^6-7*x^10*y^7+3*x^9*y^8+x^11 *y^5+9*x^10*y^6-8*x^9*y^7-5*x^10*y^5+7*x^9*y^6-x^8*y^7+x^10*y^4-2*x^9*y^5+6*x^8 *y^6-3*x^7*y^7-8*x^8*y^5+5*x^7*y^6+3*x^8*y^4-x^6*y^6-3*x^7*y^4+3*x^6*y^5+x^7*y^ 3-4*x^6*y^4+2*x^6*y^3-8*x^5*y^4+9*x^5*y^3-3*x^4*y^4-2*x^5*y^2+3*x^4*y^3+4*x^3*y ^3-2*x^3*y^2+2*x^2*y^2-x^2*y+x*y+1)/(x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4+x^6*y^5- x^7*y^3-2*x^6*y^4+x^6*y^3+2*x^5*y^3-x^4*y^4-2*x^5*y^2+x^4*y^3-x^4*y^2-x^2*y-x*y +1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 3 2 (2 a - 4 a + 8 a - 8 a + 2 a + 1) (4 a + 2 a + 1) - --------------------------------------------------------- 2 2 5 3 2 (3 a - 2 a + 2) (a - 2 a - a - a - 1) 4 2 where a is the root of the polynomial, x + x + x - 1, and in decimals this is, 0.6527572908 BTW the ratio for words with, 500, letters is, 0.6528944886 ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 10 13 9 12 10 13 8 ) | ) C(m, n) x y | = (x y - 3 x y - x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 9 12 8 11 9 13 6 12 7 11 8 + 7 x y - 13 x y - 3 x y - 3 x y + 8 x y + 7 x y 10 9 13 5 11 7 10 8 13 4 11 6 10 7 - x y + 3 x y - x y + 7 x y - x y - 5 x y - 16 x y 9 8 12 4 11 5 10 6 9 7 8 8 12 3 - 5 x y - 2 x y - x y + 13 x y + 11 x y + x y + x y 11 4 10 5 8 7 11 3 10 4 9 5 7 7 + 4 x y - 2 x y - 6 x y - x y + x y - 13 x y + 4 x y 10 3 9 4 8 5 7 6 10 2 8 4 7 5 - 4 x y + 6 x y + 9 x y - 7 x y + 2 x y - 5 x y + 6 x y 6 6 9 2 8 3 7 4 6 5 7 3 6 4 5 5 + x y + x y + x y - 3 x y - 2 x y + x y - 4 x y + x y 7 2 6 3 5 4 7 6 2 5 3 4 4 6 - x y + 5 x y + 2 x y + x y - x y - 6 x y + 3 x y + x y 5 2 4 3 5 4 2 3 3 5 4 3 2 + 4 x y + 4 x y + x y - 4 x y - 4 x y - x - x y + 4 x y 3 2 2 2 / 8 6 8 5 8 4 - x y - 2 x y + 2 x y - x y + x - 1) / ((x y - 2 x y + x y / 7 4 6 5 7 3 6 4 6 3 5 3 4 4 5 2 + x y + x y - x y - 2 x y + x y + 2 x y - x y - 2 x y 4 3 4 2 2 + x y - x y - x y - x y + 1) (x - 1)) and in Maple notation (x^13*y^10-3*x^13*y^9-x^12*y^10+3*x^13*y^8+7*x^12*y^9-13*x^12*y^8-3*x^11*y^9-3* x^13*y^6+8*x^12*y^7+7*x^11*y^8-x^10*y^9+3*x^13*y^5-x^11*y^7+7*x^10*y^8-x^13*y^4 -5*x^11*y^6-16*x^10*y^7-5*x^9*y^8-2*x^12*y^4-x^11*y^5+13*x^10*y^6+11*x^9*y^7+x^ 8*y^8+x^12*y^3+4*x^11*y^4-2*x^10*y^5-6*x^8*y^7-x^11*y^3+x^10*y^4-13*x^9*y^5+4*x ^7*y^7-4*x^10*y^3+6*x^9*y^4+9*x^8*y^5-7*x^7*y^6+2*x^10*y^2-5*x^8*y^4+6*x^7*y^5+ x^6*y^6+x^9*y^2+x^8*y^3-3*x^7*y^4-2*x^6*y^5+x^7*y^3-4*x^6*y^4+x^5*y^5-x^7*y^2+5 *x^6*y^3+2*x^5*y^4+x^7*y-x^6*y^2-6*x^5*y^3+3*x^4*y^4+x^6*y+4*x^5*y^2+4*x^4*y^3+ x^5*y-4*x^4*y^2-4*x^3*y^3-x^5-x^4*y+4*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/ (x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4+x^6*y^5-x^7*y^3-2*x^6*y^4+x^6*y^3+2*x^5*y^3- x^4*y^4-2*x^5*y^2+x^4*y^3-x^4*y^2-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.6525849339 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 5" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 10 13 9 12 10 13 8 ) | ) C(m, n) x y | = (x y - 3 x y - x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 9 12 8 11 9 13 6 12 7 11 8 + 7 x y - 13 x y - 3 x y - 3 x y + 8 x y + 7 x y 10 9 13 5 11 7 10 8 13 4 11 6 10 7 - x y + 3 x y - x y + 7 x y - x y - 5 x y - 16 x y 9 8 12 4 11 5 10 6 9 7 8 8 12 3 - 5 x y - 2 x y - x y + 13 x y + 11 x y + x y + x y 11 4 10 5 8 7 11 3 10 4 9 5 7 7 + 4 x y - 2 x y - 6 x y - x y + x y - 13 x y + 4 x y 10 3 9 4 8 5 7 6 10 2 8 4 7 5 - 4 x y + 6 x y + 9 x y - 7 x y + 2 x y - 5 x y + 6 x y 6 6 9 2 8 3 7 4 6 5 7 3 6 4 5 5 + x y + x y + x y - 3 x y - 2 x y + x y - 4 x y + x y 7 2 6 3 5 4 7 6 2 5 3 4 4 6 - x y + 5 x y + 2 x y + x y - x y - 6 x y + 3 x y + x y 5 2 4 3 5 4 2 3 3 5 4 3 2 + 4 x y + 4 x y + x y - 4 x y - 4 x y - x - x y + 4 x y 3 2 2 2 / 8 6 8 5 8 4 - x y - 2 x y + 2 x y - x y + x - 1) / ((x y - 2 x y + x y / 7 4 6 5 7 3 6 4 6 3 5 3 4 4 5 2 + x y + x y - x y - 2 x y + x y + 2 x y - x y - 2 x y 4 3 4 2 2 + x y - x y - x y - x y + 1) (x - 1)) and in Maple notation (x^13*y^10-3*x^13*y^9-x^12*y^10+3*x^13*y^8+7*x^12*y^9-13*x^12*y^8-3*x^11*y^9-3* x^13*y^6+8*x^12*y^7+7*x^11*y^8-x^10*y^9+3*x^13*y^5-x^11*y^7+7*x^10*y^8-x^13*y^4 -5*x^11*y^6-16*x^10*y^7-5*x^9*y^8-2*x^12*y^4-x^11*y^5+13*x^10*y^6+11*x^9*y^7+x^ 8*y^8+x^12*y^3+4*x^11*y^4-2*x^10*y^5-6*x^8*y^7-x^11*y^3+x^10*y^4-13*x^9*y^5+4*x ^7*y^7-4*x^10*y^3+6*x^9*y^4+9*x^8*y^5-7*x^7*y^6+2*x^10*y^2-5*x^8*y^4+6*x^7*y^5+ x^6*y^6+x^9*y^2+x^8*y^3-3*x^7*y^4-2*x^6*y^5+x^7*y^3-4*x^6*y^4+x^5*y^5-x^7*y^2+5 *x^6*y^3+2*x^5*y^4+x^7*y-x^6*y^2-6*x^5*y^3+3*x^4*y^4+x^6*y+4*x^5*y^2+4*x^4*y^3+ x^5*y-4*x^4*y^2-4*x^3*y^3-x^5-x^4*y+4*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/ (x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4+x^6*y^5-x^7*y^3-2*x^6*y^4+x^6*y^3+2*x^5*y^3- x^4*y^4-2*x^5*y^2+x^4*y^3-x^4*y^2-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.6525849339 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 6" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 10 14 9 13 10 14 8 ) | ) C(m, n) x y | = - (x y - 2 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 9 14 7 13 8 12 9 14 6 13 7 12 8 + 3 x y + 5 x y - 3 x y - x y - 4 x y + x y + 3 x y 11 9 14 5 12 7 10 9 12 6 11 7 10 8 + x y + x y - 3 x y - x y + x y - 3 x y + 4 x y 11 6 10 7 9 8 11 5 10 6 9 7 8 8 + x y - 2 x y - 5 x y + 2 x y - 6 x y + 5 x y + x y 11 4 10 5 9 6 8 7 9 5 8 6 7 7 - x y + 6 x y + 7 x y - 5 x y - 9 x y + 3 x y + 4 x y 10 3 9 4 8 5 7 6 8 4 7 5 6 6 - x y + 2 x y + 8 x y - 5 x y - 10 x y - x y + x y 8 3 7 4 6 5 5 5 6 3 5 4 6 2 5 3 + 3 x y + 2 x y - 5 x y + x y + 2 x y + x y + x y - x y 4 4 4 3 4 2 3 3 4 3 2 2 2 + 3 x y + 3 x y - 2 x y - 4 x y - 2 x y + 3 x y - 2 x y 2 / 14 8 14 7 14 6 + 2 x y - x y + x - 1) / ((x - 1) (x y - 3 x y + 3 x y / 14 5 11 7 11 6 10 7 11 5 11 4 10 5 9 6 - x y + x y - x y + x y - x y + x y - 3 x y - 2 x y 10 4 9 5 8 6 9 3 8 4 8 3 7 4 6 5 + 2 x y + 3 x y - x y - x y + 3 x y - 2 x y + x y - x y 7 3 6 3 6 2 5 3 4 4 5 2 3 2 3 2 - x y - x y + x y - x y + x y + x y - x y + x y + x y + x y - 1)) and in Maple notation -(x^14*y^10-2*x^14*y^9-x^13*y^10-x^14*y^8+3*x^13*y^9+5*x^14*y^7-3*x^13*y^8-x^12 *y^9-4*x^14*y^6+x^13*y^7+3*x^12*y^8+x^11*y^9+x^14*y^5-3*x^12*y^7-x^10*y^9+x^12* y^6-3*x^11*y^7+4*x^10*y^8+x^11*y^6-2*x^10*y^7-5*x^9*y^8+2*x^11*y^5-6*x^10*y^6+5 *x^9*y^7+x^8*y^8-x^11*y^4+6*x^10*y^5+7*x^9*y^6-5*x^8*y^7-9*x^9*y^5+3*x^8*y^6+4* x^7*y^7-x^10*y^3+2*x^9*y^4+8*x^8*y^5-5*x^7*y^6-10*x^8*y^4-x^7*y^5+x^6*y^6+3*x^8 *y^3+2*x^7*y^4-5*x^6*y^5+x^5*y^5+2*x^6*y^3+x^5*y^4+x^6*y^2-x^5*y^3+3*x^4*y^4+3* x^4*y^3-2*x^4*y^2-4*x^3*y^3-2*x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/ (x^14*y^8-3*x^14*y^7+3*x^14*y^6-x^14*y^5+x^11*y^7-x^11*y^6+x^10*y^7-x^11*y^5+x^ 11*y^4-3*x^10*y^5-2*x^9*y^6+2*x^10*y^4+3*x^9*y^5-x^8*y^6-x^9*y^3+3*x^8*y^4-2*x^ 8*y^3+x^7*y^4-x^6*y^5-x^7*y^3-x^6*y^3+x^6*y^2-x^5*y^3+x^4*y^4+x^5*y^2-x^3*y^2+x ^3*y+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.6085499744 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 7" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 10 14 9 13 10 14 8 ) | ) C(m, n) x y | = - (x y - 2 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 9 14 7 13 8 12 9 14 6 13 7 12 8 + 3 x y + 5 x y - 3 x y - x y - 4 x y + x y + 3 x y 11 9 14 5 12 7 10 9 12 6 11 7 10 8 + x y + x y - 3 x y - x y + x y - 3 x y + 4 x y 11 6 10 7 9 8 11 5 10 6 9 7 8 8 + x y - 2 x y - 5 x y + 2 x y - 6 x y + 5 x y + x y 11 4 10 5 9 6 8 7 9 5 8 6 7 7 - x y + 6 x y + 7 x y - 5 x y - 9 x y + 3 x y + 4 x y 10 3 9 4 8 5 7 6 8 4 7 5 6 6 - x y + 2 x y + 8 x y - 5 x y - 10 x y - x y + x y 8 3 7 4 6 5 5 5 6 3 5 4 6 2 5 3 + 3 x y + 2 x y - 5 x y + x y + 2 x y + x y + x y - x y 4 4 4 3 4 2 3 3 4 3 2 2 2 + 3 x y + 3 x y - 2 x y - 4 x y - 2 x y + 3 x y - 2 x y 2 / 14 8 14 7 14 6 + 2 x y - x y + x - 1) / ((x - 1) (x y - 3 x y + 3 x y / 14 5 11 7 11 6 10 7 11 5 11 4 10 5 9 6 - x y + x y - x y + x y - x y + x y - 3 x y - 2 x y 10 4 9 5 8 6 9 3 8 4 8 3 7 4 6 5 + 2 x y + 3 x y - x y - x y + 3 x y - 2 x y + x y - x y 7 3 6 3 6 2 5 3 4 4 5 2 3 2 3 2 - x y - x y + x y - x y + x y + x y - x y + x y + x y + x y - 1)) and in Maple notation -(x^14*y^10-2*x^14*y^9-x^13*y^10-x^14*y^8+3*x^13*y^9+5*x^14*y^7-3*x^13*y^8-x^12 *y^9-4*x^14*y^6+x^13*y^7+3*x^12*y^8+x^11*y^9+x^14*y^5-3*x^12*y^7-x^10*y^9+x^12* y^6-3*x^11*y^7+4*x^10*y^8+x^11*y^6-2*x^10*y^7-5*x^9*y^8+2*x^11*y^5-6*x^10*y^6+5 *x^9*y^7+x^8*y^8-x^11*y^4+6*x^10*y^5+7*x^9*y^6-5*x^8*y^7-9*x^9*y^5+3*x^8*y^6+4* x^7*y^7-x^10*y^3+2*x^9*y^4+8*x^8*y^5-5*x^7*y^6-10*x^8*y^4-x^7*y^5+x^6*y^6+3*x^8 *y^3+2*x^7*y^4-5*x^6*y^5+x^5*y^5+2*x^6*y^3+x^5*y^4+x^6*y^2-x^5*y^3+3*x^4*y^4+3* x^4*y^3-2*x^4*y^2-4*x^3*y^3-2*x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/ (x^14*y^8-3*x^14*y^7+3*x^14*y^6-x^14*y^5+x^11*y^7-x^11*y^6+x^10*y^7-x^11*y^5+x^ 11*y^4-3*x^10*y^5-2*x^9*y^6+2*x^10*y^4+3*x^9*y^5-x^8*y^6-x^9*y^3+3*x^8*y^4-2*x^ 8*y^3+x^7*y^4-x^6*y^5-x^7*y^3-x^6*y^3+x^6*y^2-x^5*y^3+x^4*y^4+x^5*y^2-x^3*y^2+x ^3*y+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.6085499744 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 8" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 7 7 6 7 5 6 6 ) | ) C(m, n) x y | = - (x y - 3 x y + 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 4 6 5 7 3 6 4 5 5 6 3 5 4 5 3 - 3 x y - 5 x y + x y + 7 x y - x y - 3 x y + x y + x y 4 4 5 2 4 3 4 2 3 3 3 2 3 2 2 + 2 x y - x y - 3 x y + x y - 4 x y + 2 x y + x y - 2 x y 2 / + x y - x y - 1) / ( / 6 5 6 4 6 3 5 4 5 3 4 3 3 2 x y - 2 x y + x y - x y + x y + x y - x y - x y - x y + 1) and in Maple notation -(x^7*y^7-3*x^7*y^6+4*x^7*y^5+x^6*y^6-3*x^7*y^4-5*x^6*y^5+x^7*y^3+7*x^6*y^4-x^5 *y^5-3*x^6*y^3+x^5*y^4+x^5*y^3+2*x^4*y^4-x^5*y^2-3*x^4*y^3+x^4*y^2-4*x^3*y^3+2* x^3*y^2+x^3*y-2*x^2*y^2+x^2*y-x*y-1)/(x^6*y^5-2*x^6*y^4+x^6*y^3-x^5*y^4+x^5*y^3 +x^4*y^3-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 2 (3 a - 2 a - 4 a - 6 a + 3 a + 2 a + 1) - ---------------------------------------------- 6 5 4 3 2 4 a + a - a - 2 a - 6 a - 3 a - 1 4 3 2 where a is the root of the polynomial, x - x - x - x + 1, and in decimals this is, 0.6043178092 BTW the ratio for words with, 500, letters is, 0.6052158767 ------------------------------------------------ "Theorem Number 9" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 8 9 7 9 6 8 7 ) | ) C(m, n) x y | = (3 x y - 9 x y + 9 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 5 8 6 8 5 7 6 7 5 6 6 6 5 - 3 x y + 6 x y - 3 x y + 3 x y - 3 x y - x y + 3 x y 6 4 5 5 6 3 5 4 5 3 4 4 5 2 4 3 - 7 x y - x y + 4 x y - 2 x y + x y - 3 x y + x y + 3 x y 3 3 3 2 3 2 2 2 / + 4 x y - 2 x y - x y + 2 x y - x y + x y + 1) / ( / 6 5 6 4 6 3 5 4 5 3 4 3 3 2 x y - 2 x y + x y - x y + x y + x y - x y - x y - x y + 1) and in Maple notation (3*x^9*y^8-9*x^9*y^7+9*x^9*y^6-3*x^8*y^7-3*x^9*y^5+6*x^8*y^6-3*x^8*y^5+3*x^7*y^ 6-3*x^7*y^5-x^6*y^6+3*x^6*y^5-7*x^6*y^4-x^5*y^5+4*x^6*y^3-2*x^5*y^4+x^5*y^3-3*x ^4*y^4+x^5*y^2+3*x^4*y^3+4*x^3*y^3-2*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y+1)/(x^6* y^5-2*x^6*y^4+x^6*y^3-x^5*y^4+x^5*y^3+x^4*y^3-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 10 9 8 7 6 5 4 3 2 2 (a - 4 a - a + 4 a + 12 a - a - 7 a - 7 a + 3 a + 2 a + 1) ---------------------------------------------------------------------- 3 2 5 2 (a + 1) (4 a - 3 a - 2 a - 1) (a - a - 1) 4 3 2 where a is the root of the polynomial, x - x - x - x + 1, and in decimals this is, 0.6043178092 BTW the ratio for words with, 500, letters is, 0.6046864205 ------------------------------------------------ "Theorem Number 10" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [1, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 10 15 9 14 10 15 8 ) | ) C(m, n) x y | = - (x y - 3 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 9 15 7 14 8 13 9 12 10 15 6 - 4 x y + 2 x y + 6 x y + x y + 2 x y - 3 x y 14 7 13 8 12 9 15 5 14 6 13 7 12 8 - 4 x y - 3 x y - 6 x y + x y + x y + 3 x y + 5 x y 13 6 11 8 11 7 12 5 11 6 10 7 9 8 - x y - x y + 4 x y - 2 x y - 6 x y + x y - x y 12 4 11 5 10 6 9 7 11 4 10 5 9 6 8 7 + x y + 4 x y - 2 x y + 2 x y - x y + x y + x y - x y 9 5 8 6 9 4 8 5 8 4 8 3 7 4 - 5 x y + 2 x y + 3 x y - 2 x y - x y + 2 x y - 5 x y 6 5 7 3 5 5 6 3 5 4 4 4 5 2 - 2 x y + 5 x y - x y + 3 x y + x y - 3 x y + 2 x y 4 3 4 2 3 3 3 2 2 2 / 16 10 + 4 x y + x y + 4 x y - x y + 2 x y + x y + 1) / (x y / 16 9 16 8 16 7 14 9 16 6 14 8 16 5 - 3 x y + 2 x y + 2 x y + x y - 3 x y - 3 x y + x y 14 7 14 6 13 7 13 6 13 5 13 4 11 6 + 3 x y - x y - x y + 3 x y - 3 x y + x y + x y 11 5 9 7 11 4 10 5 9 6 10 4 9 5 9 4 - 2 x y - x y + x y - x y + 2 x y + x y - x y - 2 x y 9 3 8 4 8 3 7 3 6 3 6 2 5 2 4 2 + 2 x y - 3 x y + 3 x y + x y - x y + 2 x y + x y + x y 3 2 + x y + x y - 1) and in Maple notation -(x^15*y^10-3*x^15*y^9+x^14*y^10+2*x^15*y^8-4*x^14*y^9+2*x^15*y^7+6*x^14*y^8+x^ 13*y^9+2*x^12*y^10-3*x^15*y^6-4*x^14*y^7-3*x^13*y^8-6*x^12*y^9+x^15*y^5+x^14*y^ 6+3*x^13*y^7+5*x^12*y^8-x^13*y^6-x^11*y^8+4*x^11*y^7-2*x^12*y^5-6*x^11*y^6+x^10 *y^7-x^9*y^8+x^12*y^4+4*x^11*y^5-2*x^10*y^6+2*x^9*y^7-x^11*y^4+x^10*y^5+x^9*y^6 -x^8*y^7-5*x^9*y^5+2*x^8*y^6+3*x^9*y^4-2*x^8*y^5-x^8*y^4+2*x^8*y^3-5*x^7*y^4-2* x^6*y^5+5*x^7*y^3-x^5*y^5+3*x^6*y^3+x^5*y^4-3*x^4*y^4+2*x^5*y^2+4*x^4*y^3+x^4*y ^2+4*x^3*y^3-x^3*y^2+2*x^2*y^2+x*y+1)/(x^16*y^10-3*x^16*y^9+2*x^16*y^8+2*x^16*y ^7+x^14*y^9-3*x^16*y^6-3*x^14*y^8+x^16*y^5+3*x^14*y^7-x^14*y^6-x^13*y^7+3*x^13* y^6-3*x^13*y^5+x^13*y^4+x^11*y^6-2*x^11*y^5-x^9*y^7+x^11*y^4-x^10*y^5+2*x^9*y^6 +x^10*y^4-x^9*y^5-2*x^9*y^4+2*x^9*y^3-3*x^8*y^4+3*x^8*y^3+x^7*y^3-x^6*y^3+2*x^6 *y^2+x^5*y^2+x^4*y^2+x^3*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 2 8 7 6 5 4 3 2 / (a + a + a + 2 a - 3 a + a - 5 a - 2 a - 1) / ( / 6 5 4 3 2 (7 a + 6 a + 5 a + 4 a + 3 a + 1) 6 5 4 3 2 (a + 2 a + 2 a + 3 a + 2 a + a + 1)) 7 6 5 4 3 where a is the root of the polynomial, x + x + x + x + x + x - 1, and in decimals this is, 0.5936629974 BTW the ratio for words with, 500, letters is, 0.5941597278 ------------------------------------------------ "Theorem Number 11" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 10 15 9 14 10 15 8 ) | ) C(m, n) x y | = - (x y - 3 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 9 15 7 14 8 13 9 12 10 15 6 - 4 x y + 2 x y + 6 x y + x y + 2 x y - 3 x y 14 7 13 8 12 9 15 5 14 6 13 7 12 8 - 4 x y - 3 x y - 6 x y + x y + x y + 3 x y + 5 x y 13 6 11 8 11 7 12 5 11 6 10 7 9 8 - x y - x y + 4 x y - 2 x y - 6 x y + x y - x y 12 4 11 5 10 6 9 7 11 4 10 5 9 6 8 7 + x y + 4 x y - 2 x y + 2 x y - x y + x y + x y - x y 9 5 8 6 9 4 8 5 8 4 8 3 7 4 - 5 x y + 2 x y + 3 x y - 2 x y - x y + 2 x y - 5 x y 6 5 7 3 5 5 6 3 5 4 4 4 5 2 - 2 x y + 5 x y - x y + 3 x y + x y - 3 x y + 2 x y 4 3 4 2 3 3 3 2 2 2 / 16 10 + 4 x y + x y + 4 x y - x y + 2 x y + x y + 1) / (x y / 16 9 16 8 16 7 14 9 16 6 14 8 16 5 - 3 x y + 2 x y + 2 x y + x y - 3 x y - 3 x y + x y 14 7 14 6 13 7 13 6 13 5 13 4 11 6 + 3 x y - x y - x y + 3 x y - 3 x y + x y + x y 11 5 9 7 11 4 10 5 9 6 10 4 9 5 9 4 - 2 x y - x y + x y - x y + 2 x y + x y - x y - 2 x y 9 3 8 4 8 3 7 3 6 3 6 2 5 2 4 2 + 2 x y - 3 x y + 3 x y + x y - x y + 2 x y + x y + x y 3 2 + x y + x y - 1) and in Maple notation -(x^15*y^10-3*x^15*y^9+x^14*y^10+2*x^15*y^8-4*x^14*y^9+2*x^15*y^7+6*x^14*y^8+x^ 13*y^9+2*x^12*y^10-3*x^15*y^6-4*x^14*y^7-3*x^13*y^8-6*x^12*y^9+x^15*y^5+x^14*y^ 6+3*x^13*y^7+5*x^12*y^8-x^13*y^6-x^11*y^8+4*x^11*y^7-2*x^12*y^5-6*x^11*y^6+x^10 *y^7-x^9*y^8+x^12*y^4+4*x^11*y^5-2*x^10*y^6+2*x^9*y^7-x^11*y^4+x^10*y^5+x^9*y^6 -x^8*y^7-5*x^9*y^5+2*x^8*y^6+3*x^9*y^4-2*x^8*y^5-x^8*y^4+2*x^8*y^3-5*x^7*y^4-2* x^6*y^5+5*x^7*y^3-x^5*y^5+3*x^6*y^3+x^5*y^4-3*x^4*y^4+2*x^5*y^2+4*x^4*y^3+x^4*y ^2+4*x^3*y^3-x^3*y^2+2*x^2*y^2+x*y+1)/(x^16*y^10-3*x^16*y^9+2*x^16*y^8+2*x^16*y ^7+x^14*y^9-3*x^16*y^6-3*x^14*y^8+x^16*y^5+3*x^14*y^7-x^14*y^6-x^13*y^7+3*x^13* y^6-3*x^13*y^5+x^13*y^4+x^11*y^6-2*x^11*y^5-x^9*y^7+x^11*y^4-x^10*y^5+2*x^9*y^6 +x^10*y^4-x^9*y^5-2*x^9*y^4+2*x^9*y^3-3*x^8*y^4+3*x^8*y^3+x^7*y^3-x^6*y^3+2*x^6 *y^2+x^5*y^2+x^4*y^2+x^3*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 2 8 7 6 5 4 3 2 / (a + a + a + 2 a - 3 a + a - 5 a - 2 a - 1) / ( / 6 5 4 3 2 (7 a + 6 a + 5 a + 4 a + 3 a + 1) 6 5 4 3 2 (a + 2 a + 2 a + 3 a + 2 a + a + 1)) 7 6 5 4 3 where a is the root of the polynomial, x + x + x + x + x + x - 1, and in decimals this is, 0.5936629974 BTW the ratio for words with, 500, letters is, 0.5941597278 ------------------------------------------------ "Theorem Number 12" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 13 17 12 17 11 ) | ) C(m, n) x y | = (2 x y - 7 x y + 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 12 17 10 16 11 17 9 16 10 15 11 + 3 x y + 2 x y - 13 x y - 8 x y + 16 x y - x y 17 8 16 9 15 10 17 7 16 8 15 9 + 5 x y + 4 x y - 4 x y - x y - 21 x y + 22 x y 14 10 16 7 15 8 14 9 13 10 12 11 - 4 x y + 11 x y - 29 x y + 22 x y + x y + 2 x y 16 6 15 7 14 8 13 9 12 10 16 5 + 2 x y + 10 x y - 38 x y - 2 x y - 5 x y - 2 x y 15 6 14 7 12 9 11 10 15 5 14 6 + 5 x y + 20 x y + 3 x y - 3 x y - 3 x y + 8 x y 13 7 12 8 11 9 14 5 13 6 12 7 11 8 - 2 x y - x y + 5 x y - 10 x y + 10 x y + x y - 6 x y 14 4 13 5 12 6 11 7 10 8 13 4 + 2 x y - 9 x y + 4 x y + 22 x y - 4 x y + x y 12 5 11 6 10 7 9 8 13 3 11 5 - 5 x y - 28 x y + 10 x y - 2 x y + x y + 5 x y 10 6 9 7 8 8 12 3 11 4 10 5 9 6 - 13 x y + 3 x y - 3 x y + x y + 7 x y + 5 x y + 5 x y 8 7 11 3 10 4 9 5 8 6 7 7 10 3 + 2 x y - 2 x y + 5 x y - 15 x y + 3 x y + 4 x y - 2 x y 9 4 8 5 7 6 10 2 9 3 8 4 7 5 + 8 x y + 2 x y + 2 x y - x y + 2 x y - 4 x y - 4 x y 6 6 9 2 8 3 7 4 6 5 7 3 6 4 5 5 + 2 x y - x y + x y - 7 x y - 2 x y + 4 x y + 6 x y + x y 7 2 6 3 5 4 5 3 4 4 6 5 2 4 3 + 2 x y - 3 x y - 3 x y + x y + 4 x y - x y + x y - 3 x y 4 2 3 3 4 3 2 2 3 / 14 10 - 2 x y - 4 x y + x y + x y - 2 x y + x - x y - 1) / (x y / 14 9 14 8 14 7 13 7 13 6 12 7 13 5 - 3 x y + 3 x y - x y - 2 x y + 5 x y - 2 x y - 3 x y 12 6 13 4 12 5 11 6 10 7 9 8 13 3 + 5 x y - x y - 3 x y + 2 x y + x y + x y + x y 12 4 11 5 10 6 9 7 12 3 10 5 9 6 - x y - 3 x y - 2 x y - 2 x y + x y - x y + 2 x y 8 7 11 3 10 4 9 5 8 6 9 4 8 5 10 2 - x y + x y + 3 x y - 3 x y + x y + 3 x y - x y - x y 8 4 9 2 8 3 6 5 8 2 7 3 6 4 5 5 + 4 x y - x y - 2 x y - x y - x y - 3 x y + x y - x y 7 2 6 3 6 2 4 4 6 5 2 4 3 4 3 + 2 x y - x y + x y + x y - x y + x y + x y - x y + x y 3 + x + x y - 1) and in Maple notation (2*x^17*y^13-7*x^17*y^12+7*x^17*y^11+3*x^16*y^12+2*x^17*y^10-13*x^16*y^11-8*x^ 17*y^9+16*x^16*y^10-x^15*y^11+5*x^17*y^8+4*x^16*y^9-4*x^15*y^10-x^17*y^7-21*x^ 16*y^8+22*x^15*y^9-4*x^14*y^10+11*x^16*y^7-29*x^15*y^8+22*x^14*y^9+x^13*y^10+2* x^12*y^11+2*x^16*y^6+10*x^15*y^7-38*x^14*y^8-2*x^13*y^9-5*x^12*y^10-2*x^16*y^5+ 5*x^15*y^6+20*x^14*y^7+3*x^12*y^9-3*x^11*y^10-3*x^15*y^5+8*x^14*y^6-2*x^13*y^7- x^12*y^8+5*x^11*y^9-10*x^14*y^5+10*x^13*y^6+x^12*y^7-6*x^11*y^8+2*x^14*y^4-9*x^ 13*y^5+4*x^12*y^6+22*x^11*y^7-4*x^10*y^8+x^13*y^4-5*x^12*y^5-28*x^11*y^6+10*x^ 10*y^7-2*x^9*y^8+x^13*y^3+5*x^11*y^5-13*x^10*y^6+3*x^9*y^7-3*x^8*y^8+x^12*y^3+7 *x^11*y^4+5*x^10*y^5+5*x^9*y^6+2*x^8*y^7-2*x^11*y^3+5*x^10*y^4-15*x^9*y^5+3*x^8 *y^6+4*x^7*y^7-2*x^10*y^3+8*x^9*y^4+2*x^8*y^5+2*x^7*y^6-x^10*y^2+2*x^9*y^3-4*x^ 8*y^4-4*x^7*y^5+2*x^6*y^6-x^9*y^2+x^8*y^3-7*x^7*y^4-2*x^6*y^5+4*x^7*y^3+6*x^6*y ^4+x^5*y^5+2*x^7*y^2-3*x^6*y^3-3*x^5*y^4+x^5*y^3+4*x^4*y^4-x^6*y+x^5*y^2-3*x^4* y^3-2*x^4*y^2-4*x^3*y^3+x^4*y+x^3*y-2*x^2*y^2+x^3-x*y-1)/(x^14*y^10-3*x^14*y^9+ 3*x^14*y^8-x^14*y^7-2*x^13*y^7+5*x^13*y^6-2*x^12*y^7-3*x^13*y^5+5*x^12*y^6-x^13 *y^4-3*x^12*y^5+2*x^11*y^6+x^10*y^7+x^9*y^8+x^13*y^3-x^12*y^4-3*x^11*y^5-2*x^10 *y^6-2*x^9*y^7+x^12*y^3-x^10*y^5+2*x^9*y^6-x^8*y^7+x^11*y^3+3*x^10*y^4-3*x^9*y^ 5+x^8*y^6+3*x^9*y^4-x^8*y^5-x^10*y^2+4*x^8*y^4-x^9*y^2-2*x^8*y^3-x^6*y^5-x^8*y^ 2-3*x^7*y^3+x^6*y^4-x^5*y^5+2*x^7*y^2-x^6*y^3+x^6*y^2+x^4*y^4-x^6*y+x^5*y^2+x^4 *y^3-x^4*y+x^3*y+x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 13 bors of a random word of length n tends to n times, 2 (a + 2 a - 6 a 12 11 10 9 8 7 6 5 4 3 - 9 a - 6 a + 9 a + 10 a + 19 a - 4 a + a - 18 a + a - 4 a 2 / 6 5 3 2 + 5 a + 2 a + 1) / ((7 a + 6 a - 4 a - 6 a - 1) / 8 7 6 3 2 (a + a + 2 a - 2 a - 2 a - a - 1)) 7 6 4 3 where a is the root of the polynomial, x + x - x - 2 x - x + 1, and in decimals this is, 0.5876210596 BTW the ratio for words with, 500, letters is, 0.5880043953 ------------------------------------------------ "Theorem Number 13" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 9 10 8 10 7 10 6 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 7 10 5 9 6 8 7 9 5 8 6 7 7 9 4 - 2 x y + x y + 6 x y + x y - 6 x y - 4 x y + x y + 2 x y 8 5 7 6 8 4 7 5 6 6 8 3 7 4 6 5 + 6 x y - 2 x y - 4 x y + 2 x y + x y + x y - x y - 3 x y 6 4 5 5 6 3 5 4 5 3 4 4 5 2 4 3 + 5 x y + x y - 3 x y - x y - 2 x y + 2 x y + x y - 4 x y 4 2 3 3 3 2 3 2 2 / 8 6 8 5 + x y - 4 x y + x y + x y - 2 x y - x y - 1) / (x y - 3 x y / 8 4 7 5 8 3 7 4 7 3 5 4 5 2 3 2 3 + 3 x y - x y - x y + 2 x y - x y + x y - x y - x y - x y - x y + 1) and in Maple notation -(x^10*y^9-4*x^10*y^8+6*x^10*y^7-4*x^10*y^6-2*x^9*y^7+x^10*y^5+6*x^9*y^6+x^8*y^ 7-6*x^9*y^5-4*x^8*y^6+x^7*y^7+2*x^9*y^4+6*x^8*y^5-2*x^7*y^6-4*x^8*y^4+2*x^7*y^5 +x^6*y^6+x^8*y^3-x^7*y^4-3*x^6*y^5+5*x^6*y^4+x^5*y^5-3*x^6*y^3-x^5*y^4-2*x^5*y^ 3+2*x^4*y^4+x^5*y^2-4*x^4*y^3+x^4*y^2-4*x^3*y^3+x^3*y^2+x^3*y-2*x^2*y^2-x*y-1)/ (x^8*y^6-3*x^8*y^5+3*x^8*y^4-x^7*y^5-x^8*y^3+2*x^7*y^4-x^7*y^3+x^5*y^4-x^5*y^2- x^3*y^2-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (a - 3 a - 2 a - 9 a - a - 2 a + 5 a + 2 a + 1) -------------------------------------------------------- 2 2 2 (a + 1) (6 a + 1) (a + 1) 3 where a is the root of the polynomial, 2 x + x - 1, and in decimals this is, 0.5835981154 BTW the ratio for words with, 500, letters is, 0.5845350575 ------------------------------------------------ "Theorem Number 14" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 9 11 8 10 9 11 7 ) | ) C(m, n) x y | = (2 x y - 8 x y - x y + 12 x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 8 11 6 11 5 10 6 10 5 9 6 8 7 + 2 x y - 8 x y + 2 x y - 2 x y + x y - x y + 3 x y 9 5 8 6 9 4 8 5 7 6 8 4 7 5 6 6 + 2 x y - x y - x y - 7 x y - x y + 6 x y + 3 x y - 2 x y 8 3 7 4 7 3 6 4 5 5 6 3 5 4 4 4 - x y - 3 x y + x y - 3 x y - x y + 4 x y + 2 x y - 3 x y 4 3 3 3 3 2 3 2 2 / 8 6 + 4 x y + 4 x y - x y - x y + 2 x y + x y + 1) / (x y / 8 5 8 4 7 5 8 3 7 4 7 3 5 4 5 2 - 3 x y + 3 x y - x y - x y + 2 x y - x y + x y - x y 3 2 3 - x y - x y - x y + 1) and in Maple notation (2*x^11*y^9-8*x^11*y^8-x^10*y^9+12*x^11*y^7+2*x^10*y^8-8*x^11*y^6+2*x^11*y^5-2* x^10*y^6+x^10*y^5-x^9*y^6+3*x^8*y^7+2*x^9*y^5-x^8*y^6-x^9*y^4-7*x^8*y^5-x^7*y^6 +6*x^8*y^4+3*x^7*y^5-2*x^6*y^6-x^8*y^3-3*x^7*y^4+x^7*y^3-3*x^6*y^4-x^5*y^5+4*x^ 6*y^3+2*x^5*y^4-3*x^4*y^4+4*x^4*y^3+4*x^3*y^3-x^3*y^2-x^3*y+2*x^2*y^2+x*y+1)/(x ^8*y^6-3*x^8*y^5+3*x^8*y^4-x^7*y^5-x^8*y^3+2*x^7*y^4-x^7*y^3+x^5*y^4-x^5*y^2-x^ 3*y^2-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 9 8 7 6 5 3 2 2 (a - 9 a + 3 a - 5 a + 14 a + 3 a - 5 a - 2 a - 1) ----------------------------------------------------------- 2 6 5 4 3 2 (6 a + 1) (a - a - a - 2 a - 2 a - a - 1) 3 where a is the root of the polynomial, 2 x + x - 1, and in decimals this is, 0.5835981154 BTW the ratio for words with, 500, letters is, 0.5842197485 ------------------------------------------------ "Theorem Number 15" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 8 11 7 11 6 10 7 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 8 11 5 10 6 9 7 11 4 10 5 9 6 + x y - 5 x y - 5 x y - 3 x y + x y + 8 x y + x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 8 4 - 5 x y + 3 x y - x y + x y - 2 x y + x y - x y + x y 7 5 6 6 8 3 7 4 6 5 7 3 6 4 5 5 + 2 x y - x y - x y - 3 x y + x y + 3 x y - 2 x y - 3 x y 7 2 6 3 5 4 4 4 4 3 4 2 3 3 3 - x y + 2 x y + 4 x y - 2 x y + 5 x y - x y + 4 x y - x y 2 2 / 11 7 11 6 11 5 11 4 10 5 + 2 x y + x y + 1) / (x y - 3 x y + 3 x y - x y - x y / 10 4 9 5 10 3 9 4 9 3 6 5 7 3 6 4 + 2 x y - x y - x y + 2 x y - x y - x y - x y + x y 7 2 6 3 6 2 5 2 4 3 3 + x y - x y + x y + x y + x y + x y + x y - 1) and in Maple notation -(2*x^11*y^8-7*x^11*y^7+9*x^11*y^6+x^10*y^7+x^9*y^8-5*x^11*y^5-5*x^10*y^6-3*x^9 *y^7+x^11*y^4+8*x^10*y^5+x^9*y^6-5*x^10*y^4+3*x^9*y^5-x^8*y^6+x^10*y^3-2*x^9*y^ 4+x^8*y^5-x^7*y^6+x^8*y^4+2*x^7*y^5-x^6*y^6-x^8*y^3-3*x^7*y^4+x^6*y^5+3*x^7*y^3 -2*x^6*y^4-3*x^5*y^5-x^7*y^2+2*x^6*y^3+4*x^5*y^4-2*x^4*y^4+5*x^4*y^3-x^4*y^2+4* x^3*y^3-x^3*y+2*x^2*y^2+x*y+1)/(x^11*y^7-3*x^11*y^6+3*x^11*y^5-x^11*y^4-x^10*y^ 5+2*x^10*y^4-x^9*y^5-x^10*y^3+2*x^9*y^4-x^9*y^3-x^6*y^5-x^7*y^3+x^6*y^4+x^7*y^2 -x^6*y^3+x^6*y^2+x^5*y^2+x^4*y^3+x^3*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 9 8 7 6 5 2 2 (a + 2 a + 4 a + 3 a + 5 a - 5 a - 2 a - 1) - ---------------------------------------------------------- 4 3 2 5 4 3 2 (5 a + 4 a + 3 a + 1) (a + 2 a + 3 a + 2 a + a + 1) 5 4 3 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.5797292632 BTW the ratio for words with, 500, letters is, 0.5805512886 ------------------------------------------------ "Theorem Number 16" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [1, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 14 10 16 7 14 9 ) | ) C(m, n) x y | = (x y - x y - 4 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 10 16 6 14 8 13 9 16 5 14 7 13 8 + x y + 6 x y + x y - 5 x y - 4 x y - 4 x y + 9 x y 12 9 16 4 14 6 13 7 12 8 14 5 13 6 + 3 x y + x y + x y - 6 x y - 11 x y + 2 x y - 2 x y 12 7 11 8 14 4 13 5 12 6 11 7 + 17 x y + 3 x y - x y + 6 x y - 15 x y - 7 x y 10 8 13 4 12 5 11 6 10 7 13 3 12 4 - 2 x y - 4 x y + 8 x y + 3 x y + 7 x y + x y - 2 x y 11 5 10 6 9 7 10 5 9 6 8 7 11 3 + 2 x y - 9 x y - 7 x y + 3 x y + 16 x y + x y - x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 + 3 x y - 12 x y - 2 x y - 2 x y + 5 x y + 4 x y + 3 x y 9 3 8 4 7 5 9 2 7 4 6 5 8 2 - 3 x y - 4 x y - 6 x y + x y + 6 x y + 2 x y + x y 7 3 5 5 6 3 5 4 5 3 4 4 5 2 4 3 - 3 x y - x y - 2 x y - 3 x y + 3 x y - 3 x y - x y - x y 5 4 2 3 3 3 2 3 2 2 2 + x y + 3 x y + 4 x y - 4 x y + x y + 2 x y - 2 x y + x y - x / 11 5 10 6 11 4 10 5 9 6 11 3 9 5 + 1) / (x y - x y - 2 x y + x y + x y + x y - 2 x y / 9 4 9 3 8 3 7 4 8 2 7 3 6 4 6 3 + 2 x y - x y - x y + 2 x y + x y - x y - 2 x y + x y 4 3 4 2 3 + x y - x y + x y - x y - x + 1) and in Maple notation (x^16*y^8-x^14*y^10-4*x^16*y^7+2*x^14*y^9+x^13*y^10+6*x^16*y^6+x^14*y^8-5*x^13* y^9-4*x^16*y^5-4*x^14*y^7+9*x^13*y^8+3*x^12*y^9+x^16*y^4+x^14*y^6-6*x^13*y^7-11 *x^12*y^8+2*x^14*y^5-2*x^13*y^6+17*x^12*y^7+3*x^11*y^8-x^14*y^4+6*x^13*y^5-15*x ^12*y^6-7*x^11*y^7-2*x^10*y^8-4*x^13*y^4+8*x^12*y^5+3*x^11*y^6+7*x^10*y^7+x^13* y^3-2*x^12*y^4+2*x^11*y^5-9*x^10*y^6-7*x^9*y^7+3*x^10*y^5+16*x^9*y^6+x^8*y^7-x^ 11*y^3+3*x^10*y^4-12*x^9*y^5-2*x^8*y^6-2*x^10*y^3+5*x^9*y^4+4*x^8*y^5+3*x^7*y^6 -3*x^9*y^3-4*x^8*y^4-6*x^7*y^5+x^9*y^2+6*x^7*y^4+2*x^6*y^5+x^8*y^2-3*x^7*y^3-x^ 5*y^5-2*x^6*y^3-3*x^5*y^4+3*x^5*y^3-3*x^4*y^4-x^5*y^2-x^4*y^3+x^5*y+3*x^4*y^2+4 *x^3*y^3-4*x^3*y^2+x^3*y+2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^11*y^5-x^10*y^6-2*x^11*y ^4+x^10*y^5+x^9*y^6+x^11*y^3-2*x^9*y^5+2*x^9*y^4-x^9*y^3-x^8*y^3+2*x^7*y^4+x^8* y^2-x^7*y^3-2*x^6*y^4+x^6*y^3+x^4*y^3-x^4*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5656419657 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 17" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 1, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 14 10 16 7 14 9 ) | ) C(m, n) x y | = (x y - x y - 4 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 10 16 6 14 8 13 9 16 5 14 7 13 8 + x y + 6 x y + x y - 5 x y - 4 x y - 4 x y + 9 x y 12 9 16 4 14 6 13 7 12 8 14 5 13 6 + 3 x y + x y + x y - 6 x y - 11 x y + 2 x y - 2 x y 12 7 11 8 14 4 13 5 12 6 11 7 + 17 x y + 3 x y - x y + 6 x y - 15 x y - 7 x y 10 8 13 4 12 5 11 6 10 7 13 3 12 4 - 2 x y - 4 x y + 8 x y + 3 x y + 7 x y + x y - 2 x y 11 5 10 6 9 7 10 5 9 6 8 7 11 3 + 2 x y - 9 x y - 7 x y + 3 x y + 16 x y + x y - x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 + 3 x y - 12 x y - 2 x y - 2 x y + 5 x y + 4 x y + 3 x y 9 3 8 4 7 5 9 2 7 4 6 5 8 2 - 3 x y - 4 x y - 6 x y + x y + 6 x y + 2 x y + x y 7 3 5 5 6 3 5 4 5 3 4 4 5 2 4 3 - 3 x y - x y - 2 x y - 3 x y + 3 x y - 3 x y - x y - x y 5 4 2 3 3 3 2 3 2 2 2 + x y + 3 x y + 4 x y - 4 x y + x y + 2 x y - 2 x y + x y - x / 10 5 9 6 10 4 9 5 10 3 9 4 9 3 + 1) / ((x y - x y - 2 x y + 2 x y + x y - 2 x y + x y / 7 3 6 4 7 2 6 3 6 2 5 3 5 2 4 3 - x y + 2 x y + x y - 2 x y + x y - x y + x y - x y 4 2 2 + x y + x y + x y - 1) (x - 1)) and in Maple notation (x^16*y^8-x^14*y^10-4*x^16*y^7+2*x^14*y^9+x^13*y^10+6*x^16*y^6+x^14*y^8-5*x^13* y^9-4*x^16*y^5-4*x^14*y^7+9*x^13*y^8+3*x^12*y^9+x^16*y^4+x^14*y^6-6*x^13*y^7-11 *x^12*y^8+2*x^14*y^5-2*x^13*y^6+17*x^12*y^7+3*x^11*y^8-x^14*y^4+6*x^13*y^5-15*x ^12*y^6-7*x^11*y^7-2*x^10*y^8-4*x^13*y^4+8*x^12*y^5+3*x^11*y^6+7*x^10*y^7+x^13* y^3-2*x^12*y^4+2*x^11*y^5-9*x^10*y^6-7*x^9*y^7+3*x^10*y^5+16*x^9*y^6+x^8*y^7-x^ 11*y^3+3*x^10*y^4-12*x^9*y^5-2*x^8*y^6-2*x^10*y^3+5*x^9*y^4+4*x^8*y^5+3*x^7*y^6 -3*x^9*y^3-4*x^8*y^4-6*x^7*y^5+x^9*y^2+6*x^7*y^4+2*x^6*y^5+x^8*y^2-3*x^7*y^3-x^ 5*y^5-2*x^6*y^3-3*x^5*y^4+3*x^5*y^3-3*x^4*y^4-x^5*y^2-x^4*y^3+x^5*y+3*x^4*y^2+4 *x^3*y^3-4*x^3*y^2+x^3*y+2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^10*y^5-x^9*y^6-2*x^10*y^ 4+2*x^9*y^5+x^10*y^3-2*x^9*y^4+x^9*y^3-x^7*y^3+2*x^6*y^4+x^7*y^2-2*x^6*y^3+x^6* y^2-x^5*y^3+x^5*y^2-x^4*y^3+x^4*y^2+x^2*y+x*y-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5656419657 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 18" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 16 10 17 8 16 9 ) | ) C(m, n) x y | = - (x y - x y - 5 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 7 16 8 17 6 16 7 15 8 13 10 + 10 x y - 16 x y - 10 x y + 24 x y - 2 x y + x y 17 5 16 6 15 7 14 8 13 9 17 4 16 5 + 5 x y - 21 x y + 8 x y - x y - 3 x y - x y + 10 x y 15 6 14 7 13 8 12 9 16 4 15 5 - 12 x y + 4 x y + 8 x y - 3 x y - 2 x y + 8 x y 14 6 13 7 12 8 15 4 14 5 13 6 - 5 x y - 19 x y + 10 x y - 2 x y + x y + 26 x y 12 7 11 8 14 4 13 5 12 6 11 7 - 14 x y - x y + 2 x y - 20 x y + 13 x y + 2 x y 10 8 14 3 13 4 12 5 11 6 13 3 12 4 - x y - x y + 9 x y - 12 x y + x y - 2 x y + 9 x y 11 5 10 6 9 7 12 3 11 4 10 5 - 6 x y + 2 x y + 7 x y - 3 x y + 6 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 10 3 9 4 - 13 x y - 2 x y - 7 x y + 5 x y - x y + 3 x y + x y 7 6 9 3 8 4 7 5 9 2 8 3 7 4 6 5 - 4 x y + x y + x y + 6 x y - x y + x y - 5 x y - x y 8 2 7 3 7 2 6 3 5 4 6 2 5 3 4 4 - x y + 4 x y - x y + 3 x y + 3 x y - x y - x y + 5 x y 5 2 4 3 4 2 3 3 3 2 3 2 2 2 - x y - x y - 3 x y - 4 x y + 4 x y - x y - 2 x y + 2 x y / 10 5 9 6 10 4 9 5 10 3 - x y + x - 1) / ((x - 1) (x y - x y - 2 x y + 2 x y + x y / 9 4 9 3 7 3 6 4 7 2 6 3 6 2 5 3 - 2 x y + x y - x y + 2 x y + x y - 2 x y + x y - x y 5 2 4 3 4 2 2 + x y - x y + x y + x y + x y - 1)) and in Maple notation -(x^17*y^9-x^16*y^10-5*x^17*y^8+6*x^16*y^9+10*x^17*y^7-16*x^16*y^8-10*x^17*y^6+ 24*x^16*y^7-2*x^15*y^8+x^13*y^10+5*x^17*y^5-21*x^16*y^6+8*x^15*y^7-x^14*y^8-3*x ^13*y^9-x^17*y^4+10*x^16*y^5-12*x^15*y^6+4*x^14*y^7+8*x^13*y^8-3*x^12*y^9-2*x^ 16*y^4+8*x^15*y^5-5*x^14*y^6-19*x^13*y^7+10*x^12*y^8-2*x^15*y^4+x^14*y^5+26*x^ 13*y^6-14*x^12*y^7-x^11*y^8+2*x^14*y^4-20*x^13*y^5+13*x^12*y^6+2*x^11*y^7-x^10* y^8-x^14*y^3+9*x^13*y^4-12*x^12*y^5+x^11*y^6-2*x^13*y^3+9*x^12*y^4-6*x^11*y^5+2 *x^10*y^6+7*x^9*y^7-3*x^12*y^3+6*x^11*y^4+3*x^10*y^5-13*x^9*y^6-2*x^11*y^3-7*x^ 10*y^4+5*x^9*y^5-x^8*y^6+3*x^10*y^3+x^9*y^4-4*x^7*y^6+x^9*y^3+x^8*y^4+6*x^7*y^5 -x^9*y^2+x^8*y^3-5*x^7*y^4-x^6*y^5-x^8*y^2+4*x^7*y^3-x^7*y^2+3*x^6*y^3+3*x^5*y^ 4-x^6*y^2-x^5*y^3+5*x^4*y^4-x^5*y^2-x^4*y^3-3*x^4*y^2-4*x^3*y^3+4*x^3*y^2-x^3*y -2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/(x^10*y^5-x^9*y^6-2*x^10*y^4+2*x^9*y^5+x^10*y ^3-2*x^9*y^4+x^9*y^3-x^7*y^3+2*x^6*y^4+x^7*y^2-2*x^6*y^3+x^6*y^2-x^5*y^3+x^5*y^ 2-x^4*y^3+x^4*y^2+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5653104178 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 19" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 16 10 17 8 16 9 ) | ) C(m, n) x y | = - (x y - x y - 5 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 7 16 8 17 6 16 7 15 8 13 10 + 10 x y - 16 x y - 10 x y + 24 x y - 2 x y + x y 17 5 16 6 15 7 14 8 13 9 17 4 16 5 + 5 x y - 21 x y + 8 x y - x y - 3 x y - x y + 10 x y 15 6 14 7 13 8 12 9 16 4 15 5 - 12 x y + 4 x y + 8 x y - 3 x y - 2 x y + 8 x y 14 6 13 7 12 8 15 4 14 5 13 6 - 5 x y - 19 x y + 10 x y - 2 x y + x y + 26 x y 12 7 11 8 14 4 13 5 12 6 11 7 - 14 x y - x y + 2 x y - 20 x y + 13 x y + 2 x y 10 8 14 3 13 4 12 5 11 6 13 3 12 4 - x y - x y + 9 x y - 12 x y + x y - 2 x y + 9 x y 11 5 10 6 9 7 12 3 11 4 10 5 - 6 x y + 2 x y + 7 x y - 3 x y + 6 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 10 3 9 4 - 13 x y - 2 x y - 7 x y + 5 x y - x y + 3 x y + x y 7 6 9 3 8 4 7 5 9 2 8 3 7 4 6 5 - 4 x y + x y + x y + 6 x y - x y + x y - 5 x y - x y 8 2 7 3 7 2 6 3 5 4 6 2 5 3 4 4 - x y + 4 x y - x y + 3 x y + 3 x y - x y - x y + 5 x y 5 2 4 3 4 2 3 3 3 2 3 2 2 2 - x y - x y - 3 x y - 4 x y + 4 x y - x y - 2 x y + 2 x y / 10 5 9 6 10 4 9 5 10 3 - x y + x - 1) / ((x - 1) (x y - x y - 2 x y + 2 x y + x y / 9 4 9 3 7 3 6 4 7 2 6 3 6 2 5 3 - 2 x y + x y - x y + 2 x y + x y - 2 x y + x y - x y 5 2 4 3 4 2 2 + x y - x y + x y + x y + x y - 1)) and in Maple notation -(x^17*y^9-x^16*y^10-5*x^17*y^8+6*x^16*y^9+10*x^17*y^7-16*x^16*y^8-10*x^17*y^6+ 24*x^16*y^7-2*x^15*y^8+x^13*y^10+5*x^17*y^5-21*x^16*y^6+8*x^15*y^7-x^14*y^8-3*x ^13*y^9-x^17*y^4+10*x^16*y^5-12*x^15*y^6+4*x^14*y^7+8*x^13*y^8-3*x^12*y^9-2*x^ 16*y^4+8*x^15*y^5-5*x^14*y^6-19*x^13*y^7+10*x^12*y^8-2*x^15*y^4+x^14*y^5+26*x^ 13*y^6-14*x^12*y^7-x^11*y^8+2*x^14*y^4-20*x^13*y^5+13*x^12*y^6+2*x^11*y^7-x^10* y^8-x^14*y^3+9*x^13*y^4-12*x^12*y^5+x^11*y^6-2*x^13*y^3+9*x^12*y^4-6*x^11*y^5+2 *x^10*y^6+7*x^9*y^7-3*x^12*y^3+6*x^11*y^4+3*x^10*y^5-13*x^9*y^6-2*x^11*y^3-7*x^ 10*y^4+5*x^9*y^5-x^8*y^6+3*x^10*y^3+x^9*y^4-4*x^7*y^6+x^9*y^3+x^8*y^4+6*x^7*y^5 -x^9*y^2+x^8*y^3-5*x^7*y^4-x^6*y^5-x^8*y^2+4*x^7*y^3-x^7*y^2+3*x^6*y^3+3*x^5*y^ 4-x^6*y^2-x^5*y^3+5*x^4*y^4-x^5*y^2-x^4*y^3-3*x^4*y^2-4*x^3*y^3+4*x^3*y^2-x^3*y -2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/(x^10*y^5-x^9*y^6-2*x^10*y^4+2*x^9*y^5+x^10*y ^3-2*x^9*y^4+x^9*y^3-x^7*y^3+2*x^6*y^4+x^7*y^2-2*x^6*y^3+x^6*y^2-x^5*y^3+x^5*y^ 2-x^4*y^3+x^4*y^2+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5653104178 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 20" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 10 15 9 14 10 ) | ) C(m, n) x y | = - (x y - 4 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 14 9 13 10 12 11 15 7 14 8 + 6 x y - 8 x y + 2 x y + x y - 4 x y + 12 x y 13 9 12 10 15 6 14 7 13 8 12 9 - 10 x y - 4 x y + x y - 8 x y + 17 x y + x y 14 6 13 7 12 8 11 9 13 6 12 7 + 2 x y - 11 x y + 11 x y - 4 x y + x y - 15 x y 11 8 10 9 13 5 12 6 11 7 10 8 + 12 x y - 2 x y + x y + 8 x y - 9 x y + 7 x y 12 5 11 6 10 7 9 8 12 4 11 5 10 6 - 3 x y - 3 x y - 6 x y + x y + x y + 5 x y - 3 x y 9 7 11 4 10 5 9 6 8 7 10 4 9 5 - x y - x y + 6 x y - 3 x y - x y - 2 x y + 4 x y 8 5 9 3 8 4 6 6 8 3 7 4 6 5 7 3 + 3 x y - x y - x y + 3 x y - x y + 5 x y - 4 x y - 6 x y 6 4 5 5 7 2 6 3 5 3 4 4 4 3 + 7 x y + 2 x y + x y - 6 x y - 3 x y + 2 x y - 4 x y 3 3 3 2 2 2 2 / 16 10 16 9 - 4 x y + x y - 2 x y + x y - x y - 1) / (x y - 4 x y / 15 10 16 8 15 9 16 7 15 8 14 9 16 6 + x y + 6 x y - 4 x y - 4 x y + 6 x y - 2 x y + x y 15 7 14 8 13 9 15 6 14 7 13 8 14 6 - 4 x y + 5 x y - 2 x y + x y - 3 x y + 5 x y - x y 13 7 12 8 14 5 13 6 12 7 13 5 12 6 - 3 x y + 2 x y + x y - x y - 3 x y + x y - x y 12 5 11 6 10 7 12 4 11 5 10 6 11 4 + 3 x y - x y - x y - x y + 2 x y + x y - x y 10 5 10 4 8 6 9 4 7 6 9 3 8 3 6 5 6 4 + x y - x y + x y + x y + x y - x y - x y - x y - x y 7 2 6 3 5 2 4 2 3 2 2 - x y + x y - x y - x y + x y - x y - x y + 1) and in Maple notation -(x^15*y^10-4*x^15*y^9+2*x^14*y^10+6*x^15*y^8-8*x^14*y^9+2*x^13*y^10+x^12*y^11-\ 4*x^15*y^7+12*x^14*y^8-10*x^13*y^9-4*x^12*y^10+x^15*y^6-8*x^14*y^7+17*x^13*y^8+ x^12*y^9+2*x^14*y^6-11*x^13*y^7+11*x^12*y^8-4*x^11*y^9+x^13*y^6-15*x^12*y^7+12* x^11*y^8-2*x^10*y^9+x^13*y^5+8*x^12*y^6-9*x^11*y^7+7*x^10*y^8-3*x^12*y^5-3*x^11 *y^6-6*x^10*y^7+x^9*y^8+x^12*y^4+5*x^11*y^5-3*x^10*y^6-x^9*y^7-x^11*y^4+6*x^10* y^5-3*x^9*y^6-x^8*y^7-2*x^10*y^4+4*x^9*y^5+3*x^8*y^5-x^9*y^3-x^8*y^4+3*x^6*y^6- x^8*y^3+5*x^7*y^4-4*x^6*y^5-6*x^7*y^3+7*x^6*y^4+2*x^5*y^5+x^7*y^2-6*x^6*y^3-3*x ^5*y^3+2*x^4*y^4-4*x^4*y^3-4*x^3*y^3+x^3*y^2-2*x^2*y^2+x^2*y-x*y-1)/(x^16*y^10-\ 4*x^16*y^9+x^15*y^10+6*x^16*y^8-4*x^15*y^9-4*x^16*y^7+6*x^15*y^8-2*x^14*y^9+x^ 16*y^6-4*x^15*y^7+5*x^14*y^8-2*x^13*y^9+x^15*y^6-3*x^14*y^7+5*x^13*y^8-x^14*y^6 -3*x^13*y^7+2*x^12*y^8+x^14*y^5-x^13*y^6-3*x^12*y^7+x^13*y^5-x^12*y^6+3*x^12*y^ 5-x^11*y^6-x^10*y^7-x^12*y^4+2*x^11*y^5+x^10*y^6-x^11*y^4+x^10*y^5-x^10*y^4+x^8 *y^6+x^9*y^4+x^7*y^6-x^9*y^3-x^8*y^3-x^6*y^5-x^6*y^4-x^7*y^2+x^6*y^3-x^5*y^2-x^ 4*y^2+x^3*y^2-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 2 (a + a - 3 a - 2 a + 3 a + 2 a + 1) ---------------------------------------------------------------------- 10 9 8 7 6 5 4 3 6 a + 17 a + 32 a + 26 a + 19 a + 11 a + 14 a + 6 a + 3 a + 1 6 5 4 3 2 where a is the root of the polynomial, x + x + x - x + x + x - 1, and in decimals this is, 0.5643024420 BTW the ratio for words with, 500, letters is, 0.5650518446 ------------------------------------------------ "Theorem Number 21" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 10 15 9 14 10 ) | ) C(m, n) x y | = - (x y - 4 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 14 9 13 10 12 11 15 7 14 8 + 6 x y - 8 x y + 2 x y + x y - 4 x y + 12 x y 13 9 12 10 15 6 14 7 13 8 12 9 - 10 x y - 4 x y + x y - 8 x y + 17 x y + x y 14 6 13 7 12 8 11 9 13 6 12 7 + 2 x y - 11 x y + 11 x y - 4 x y + x y - 15 x y 11 8 10 9 13 5 12 6 11 7 10 8 + 12 x y - 2 x y + x y + 8 x y - 9 x y + 7 x y 12 5 11 6 10 7 9 8 12 4 11 5 10 6 - 3 x y - 3 x y - 6 x y + x y + x y + 5 x y - 3 x y 9 7 11 4 10 5 9 6 8 7 10 4 9 5 - x y - x y + 6 x y - 3 x y - x y - 2 x y + 4 x y 8 5 9 3 8 4 6 6 8 3 7 4 6 5 7 3 + 3 x y - x y - x y + 3 x y - x y + 5 x y - 4 x y - 6 x y 6 4 5 5 7 2 6 3 5 3 4 4 4 3 + 7 x y + 2 x y + x y - 6 x y - 3 x y + 2 x y - 4 x y 3 3 3 2 2 2 2 / 16 10 16 9 - 4 x y + x y - 2 x y + x y - x y - 1) / (x y - 4 x y / 15 10 16 8 15 9 16 7 15 8 14 9 16 6 + x y + 6 x y - 4 x y - 4 x y + 6 x y - 2 x y + x y 15 7 14 8 13 9 15 6 14 7 13 8 14 6 - 4 x y + 5 x y - 2 x y + x y - 3 x y + 5 x y - x y 13 7 12 8 14 5 13 6 12 7 13 5 12 6 - 3 x y + 2 x y + x y - x y - 3 x y + x y - x y 12 5 11 6 10 7 12 4 11 5 10 6 11 4 + 3 x y - x y - x y - x y + 2 x y + x y - x y 10 5 10 4 8 6 9 4 7 6 9 3 8 3 6 5 6 4 + x y - x y + x y + x y + x y - x y - x y - x y - x y 7 2 6 3 5 2 4 2 3 2 2 - x y + x y - x y - x y + x y - x y - x y + 1) and in Maple notation -(x^15*y^10-4*x^15*y^9+2*x^14*y^10+6*x^15*y^8-8*x^14*y^9+2*x^13*y^10+x^12*y^11-\ 4*x^15*y^7+12*x^14*y^8-10*x^13*y^9-4*x^12*y^10+x^15*y^6-8*x^14*y^7+17*x^13*y^8+ x^12*y^9+2*x^14*y^6-11*x^13*y^7+11*x^12*y^8-4*x^11*y^9+x^13*y^6-15*x^12*y^7+12* x^11*y^8-2*x^10*y^9+x^13*y^5+8*x^12*y^6-9*x^11*y^7+7*x^10*y^8-3*x^12*y^5-3*x^11 *y^6-6*x^10*y^7+x^9*y^8+x^12*y^4+5*x^11*y^5-3*x^10*y^6-x^9*y^7-x^11*y^4+6*x^10* y^5-3*x^9*y^6-x^8*y^7-2*x^10*y^4+4*x^9*y^5+3*x^8*y^5-x^9*y^3-x^8*y^4+3*x^6*y^6- x^8*y^3+5*x^7*y^4-4*x^6*y^5-6*x^7*y^3+7*x^6*y^4+2*x^5*y^5+x^7*y^2-6*x^6*y^3-3*x ^5*y^3+2*x^4*y^4-4*x^4*y^3-4*x^3*y^3+x^3*y^2-2*x^2*y^2+x^2*y-x*y-1)/(x^16*y^10-\ 4*x^16*y^9+x^15*y^10+6*x^16*y^8-4*x^15*y^9-4*x^16*y^7+6*x^15*y^8-2*x^14*y^9+x^ 16*y^6-4*x^15*y^7+5*x^14*y^8-2*x^13*y^9+x^15*y^6-3*x^14*y^7+5*x^13*y^8-x^14*y^6 -3*x^13*y^7+2*x^12*y^8+x^14*y^5-x^13*y^6-3*x^12*y^7+x^13*y^5-x^12*y^6+3*x^12*y^ 5-x^11*y^6-x^10*y^7-x^12*y^4+2*x^11*y^5+x^10*y^6-x^11*y^4+x^10*y^5-x^10*y^4+x^8 *y^6+x^9*y^4+x^7*y^6-x^9*y^3-x^8*y^3-x^6*y^5-x^6*y^4-x^7*y^2+x^6*y^3-x^5*y^2-x^ 4*y^2+x^3*y^2-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 2 (a + a - 3 a - 2 a + 3 a + 2 a + 1) ---------------------------------------------------------------------- 10 9 8 7 6 5 4 3 6 a + 17 a + 32 a + 26 a + 19 a + 11 a + 14 a + 6 a + 3 a + 1 6 5 4 3 2 where a is the root of the polynomial, x + x + x - x + x + x - 1, and in decimals this is, 0.5643024420 BTW the ratio for words with, 500, letters is, 0.5650518446 ------------------------------------------------ "Theorem Number 22" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 12 16 10 15 11 16 9 ) | ) C(m, n) x y | = (x y + x y - 3 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 10 14 11 16 8 15 9 14 10 16 7 + 2 x y + 3 x y + 10 x y - x y - 13 x y - 10 x y 15 8 14 9 13 10 16 6 15 7 14 8 + 9 x y + 21 x y + 2 x y + 5 x y - 17 x y - 14 x y 13 9 12 10 16 5 15 6 14 7 13 8 12 9 - 8 x y - x y - x y + 12 x y + x y + 12 x y + 5 x y 15 5 14 6 13 7 12 8 11 9 14 5 - 3 x y + 3 x y - 8 x y - 11 x y - 4 x y - x y 13 6 12 7 11 8 12 6 11 7 10 8 + 2 x y + 14 x y + 13 x y - 11 x y - 17 x y - 3 x y 12 5 11 6 10 7 12 4 11 5 10 6 9 7 + 5 x y + 13 x y + 6 x y - x y - 7 x y - 3 x y - 6 x y 11 4 10 5 9 6 8 7 10 4 9 5 10 3 + 2 x y + 3 x y + 16 x y + x y - 6 x y - 10 x y + 3 x y 9 4 8 5 7 6 9 3 8 4 7 5 8 3 - 4 x y + x y + 3 x y + 4 x y - 6 x y - 4 x y + 4 x y 7 4 6 5 7 3 6 4 5 5 7 2 6 3 5 4 - x y + x y + x y - 4 x y - x y + x y + x y - 4 x y 6 2 5 3 4 4 5 2 4 3 4 2 3 3 + 2 x y + 3 x y - 3 x y + 2 x y + 3 x y + x y + 4 x y 3 2 2 2 2 / 11 8 11 7 11 6 - 2 x y + 2 x y - x y + x y + 1) / (x y - 3 x y + 3 x y / 11 5 11 4 9 6 11 3 10 4 9 5 8 6 10 3 - 2 x y + 2 x y + x y - x y + x y - 2 x y - x y - x y 9 4 8 5 9 3 8 4 8 3 7 4 8 2 7 3 + 2 x y + 2 x y - x y - x y + x y - x y - x y + x y 6 4 7 2 6 3 6 2 5 3 5 2 4 3 4 2 2 - 2 x y - x y + 2 x y - x y + x y - x y + x y - x y - x y - x y + 1) and in Maple notation (x^15*y^12+x^16*y^10-3*x^15*y^11-5*x^16*y^9+2*x^15*y^10+3*x^14*y^11+10*x^16*y^8 -x^15*y^9-13*x^14*y^10-10*x^16*y^7+9*x^15*y^8+21*x^14*y^9+2*x^13*y^10+5*x^16*y^ 6-17*x^15*y^7-14*x^14*y^8-8*x^13*y^9-x^12*y^10-x^16*y^5+12*x^15*y^6+x^14*y^7+12 *x^13*y^8+5*x^12*y^9-3*x^15*y^5+3*x^14*y^6-8*x^13*y^7-11*x^12*y^8-4*x^11*y^9-x^ 14*y^5+2*x^13*y^6+14*x^12*y^7+13*x^11*y^8-11*x^12*y^6-17*x^11*y^7-3*x^10*y^8+5* x^12*y^5+13*x^11*y^6+6*x^10*y^7-x^12*y^4-7*x^11*y^5-3*x^10*y^6-6*x^9*y^7+2*x^11 *y^4+3*x^10*y^5+16*x^9*y^6+x^8*y^7-6*x^10*y^4-10*x^9*y^5+3*x^10*y^3-4*x^9*y^4+x ^8*y^5+3*x^7*y^6+4*x^9*y^3-6*x^8*y^4-4*x^7*y^5+4*x^8*y^3-x^7*y^4+x^6*y^5+x^7*y^ 3-4*x^6*y^4-x^5*y^5+x^7*y^2+x^6*y^3-4*x^5*y^4+2*x^6*y^2+3*x^5*y^3-3*x^4*y^4+2*x ^5*y^2+3*x^4*y^3+x^4*y^2+4*x^3*y^3-2*x^3*y^2+2*x^2*y^2-x^2*y+x*y+1)/(x^11*y^8-3 *x^11*y^7+3*x^11*y^6-2*x^11*y^5+2*x^11*y^4+x^9*y^6-x^11*y^3+x^10*y^4-2*x^9*y^5- x^8*y^6-x^10*y^3+2*x^9*y^4+2*x^8*y^5-x^9*y^3-x^8*y^4+x^8*y^3-x^7*y^4-x^8*y^2+x^ 7*y^3-2*x^6*y^4-x^7*y^2+2*x^6*y^3-x^6*y^2+x^5*y^3-x^5*y^2+x^4*y^3-x^4*y^2-x^2*y -x*y+1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 10 bors of a random word of length n tends to n times, 2 (a + 4 a + 4 a 8 7 6 5 4 3 2 / - a + 2 a + 7 a + 4 a - 7 a - 5 a + 3 a + 2 a + 1) / ( / 6 5 4 3 2 (7 a + 6 a + 2 a + 1) (a + 2 a + a + a + 1)) 7 6 2 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.5620809050 BTW the ratio for words with, 500, letters is, 0.5625922012 ------------------------------------------------ "Theorem Number 23" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [1, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 12 16 10 15 11 16 9 ) | ) C(m, n) x y | = (x y + x y - 3 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 10 14 11 16 8 15 9 14 10 16 7 + 2 x y + 3 x y + 10 x y - x y - 13 x y - 10 x y 15 8 14 9 13 10 16 6 15 7 14 8 + 9 x y + 21 x y + 2 x y + 5 x y - 17 x y - 14 x y 13 9 12 10 16 5 15 6 14 7 13 8 12 9 - 8 x y - x y - x y + 12 x y + x y + 12 x y + 5 x y 15 5 14 6 13 7 12 8 11 9 14 5 - 3 x y + 3 x y - 8 x y - 11 x y - 4 x y - x y 13 6 12 7 11 8 12 6 11 7 10 8 + 2 x y + 14 x y + 13 x y - 11 x y - 17 x y - 3 x y 12 5 11 6 10 7 12 4 11 5 10 6 9 7 + 5 x y + 13 x y + 6 x y - x y - 7 x y - 3 x y - 6 x y 11 4 10 5 9 6 8 7 10 4 9 5 10 3 + 2 x y + 3 x y + 16 x y + x y - 6 x y - 10 x y + 3 x y 9 4 8 5 7 6 9 3 8 4 7 5 8 3 - 4 x y + x y + 3 x y + 4 x y - 6 x y - 4 x y + 4 x y 7 4 6 5 7 3 6 4 5 5 7 2 6 3 5 4 - x y + x y + x y - 4 x y - x y + x y + x y - 4 x y 6 2 5 3 4 4 5 2 4 3 4 2 3 3 + 2 x y + 3 x y - 3 x y + 2 x y + 3 x y + x y + 4 x y 3 2 2 2 2 / 11 8 11 7 11 6 - 2 x y + 2 x y - x y + x y + 1) / (x y - 3 x y + 3 x y / 11 5 11 4 9 6 11 3 10 4 9 5 8 6 10 3 - 2 x y + 2 x y + x y - x y + x y - 2 x y - x y - x y 9 4 8 5 9 3 8 4 8 3 7 4 8 2 7 3 + 2 x y + 2 x y - x y - x y + x y - x y - x y + x y 6 4 7 2 6 3 6 2 5 3 5 2 4 3 4 2 2 - 2 x y - x y + 2 x y - x y + x y - x y + x y - x y - x y - x y + 1) and in Maple notation (x^15*y^12+x^16*y^10-3*x^15*y^11-5*x^16*y^9+2*x^15*y^10+3*x^14*y^11+10*x^16*y^8 -x^15*y^9-13*x^14*y^10-10*x^16*y^7+9*x^15*y^8+21*x^14*y^9+2*x^13*y^10+5*x^16*y^ 6-17*x^15*y^7-14*x^14*y^8-8*x^13*y^9-x^12*y^10-x^16*y^5+12*x^15*y^6+x^14*y^7+12 *x^13*y^8+5*x^12*y^9-3*x^15*y^5+3*x^14*y^6-8*x^13*y^7-11*x^12*y^8-4*x^11*y^9-x^ 14*y^5+2*x^13*y^6+14*x^12*y^7+13*x^11*y^8-11*x^12*y^6-17*x^11*y^7-3*x^10*y^8+5* x^12*y^5+13*x^11*y^6+6*x^10*y^7-x^12*y^4-7*x^11*y^5-3*x^10*y^6-6*x^9*y^7+2*x^11 *y^4+3*x^10*y^5+16*x^9*y^6+x^8*y^7-6*x^10*y^4-10*x^9*y^5+3*x^10*y^3-4*x^9*y^4+x ^8*y^5+3*x^7*y^6+4*x^9*y^3-6*x^8*y^4-4*x^7*y^5+4*x^8*y^3-x^7*y^4+x^6*y^5+x^7*y^ 3-4*x^6*y^4-x^5*y^5+x^7*y^2+x^6*y^3-4*x^5*y^4+2*x^6*y^2+3*x^5*y^3-3*x^4*y^4+2*x ^5*y^2+3*x^4*y^3+x^4*y^2+4*x^3*y^3-2*x^3*y^2+2*x^2*y^2-x^2*y+x*y+1)/(x^11*y^8-3 *x^11*y^7+3*x^11*y^6-2*x^11*y^5+2*x^11*y^4+x^9*y^6-x^11*y^3+x^10*y^4-2*x^9*y^5- x^8*y^6-x^10*y^3+2*x^9*y^4+2*x^8*y^5-x^9*y^3-x^8*y^4+x^8*y^3-x^7*y^4-x^8*y^2+x^ 7*y^3-2*x^6*y^4-x^7*y^2+2*x^6*y^3-x^6*y^2+x^5*y^3-x^5*y^2+x^4*y^3-x^4*y^2-x^2*y -x*y+1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 10 bors of a random word of length n tends to n times, 2 (a + 4 a + 4 a 8 7 6 5 4 3 2 / - a + 2 a + 7 a + 4 a - 7 a - 5 a + 3 a + 2 a + 1) / ( / 6 5 4 3 2 (7 a + 6 a + 2 a + 1) (a + 2 a + a + a + 1)) 7 6 2 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.5620809050 BTW the ratio for words with, 500, letters is, 0.5625922012 ------------------------------------------------ "Theorem Number 24" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 8 11 7 10 8 11 6 ) | ) C(m, n) x y | = - (x y - 2 x y - x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 10 6 9 7 9 6 8 7 9 5 8 6 + 2 x y - x y + x y - 3 x y - 2 x y + 2 x y + 5 x y 8 5 8 4 7 5 7 4 6 5 5 5 6 3 5 4 - 4 x y + x y - x y + x y + 2 x y + x y - x y - 4 x y 5 3 4 4 4 2 3 3 3 2 2 2 2 + 2 x y + 3 x y - 2 x y - 4 x y + 3 x y - 2 x y + x y - x y / 11 6 11 5 11 4 9 6 9 5 + x - 1) / ((x - 1) (x y - 2 x y + x y + x y - 2 x y / 9 3 8 3 7 4 8 2 7 2 5 4 6 2 5 2 4 2 + x y - x y - x y + x y + x y - x y + x y + x y + x y 3 2 + x y + x y - 1)) and in Maple notation -(x^11*y^8-2*x^11*y^7-x^10*y^8+x^11*y^6+2*x^10*y^7-x^10*y^6+x^9*y^7-3*x^9*y^6-2 *x^8*y^7+2*x^9*y^5+5*x^8*y^6-4*x^8*y^5+x^8*y^4-x^7*y^5+x^7*y^4+2*x^6*y^5+x^5*y^ 5-x^6*y^3-4*x^5*y^4+2*x^5*y^3+3*x^4*y^4-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-2*x^2*y^2 +x^2*y-x*y+x-1)/(x-1)/(x^11*y^6-2*x^11*y^5+x^11*y^4+x^9*y^6-2*x^9*y^5+x^9*y^3-x ^8*y^3-x^7*y^4+x^8*y^2+x^7*y^2-x^5*y^4+x^6*y^2+x^5*y^2+x^4*y^2+x^3*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5580967063 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 25" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 8 11 7 10 8 11 6 ) | ) C(m, n) x y | = - (x y - 2 x y - x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 10 6 9 7 9 6 8 7 9 5 8 6 + 2 x y - x y + x y - 3 x y - 2 x y + 2 x y + 5 x y 8 5 8 4 7 5 7 4 6 5 5 5 6 3 5 4 - 4 x y + x y - x y + x y + 2 x y + x y - x y - 4 x y 5 3 4 4 4 2 3 3 3 2 2 2 2 + 2 x y + 3 x y - 2 x y - 4 x y + 3 x y - 2 x y + x y - x y / 11 6 11 5 11 4 9 6 9 5 + x - 1) / ((x - 1) (x y - 2 x y + x y + x y - 2 x y / 9 3 8 3 7 4 8 2 7 2 5 4 6 2 5 2 4 2 + x y - x y - x y + x y + x y - x y + x y + x y + x y 3 2 + x y + x y - 1)) and in Maple notation -(x^11*y^8-2*x^11*y^7-x^10*y^8+x^11*y^6+2*x^10*y^7-x^10*y^6+x^9*y^7-3*x^9*y^6-2 *x^8*y^7+2*x^9*y^5+5*x^8*y^6-4*x^8*y^5+x^8*y^4-x^7*y^5+x^7*y^4+2*x^6*y^5+x^5*y^ 5-x^6*y^3-4*x^5*y^4+2*x^5*y^3+3*x^4*y^4-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-2*x^2*y^2 +x^2*y-x*y+x-1)/(x-1)/(x^11*y^6-2*x^11*y^5+x^11*y^4+x^9*y^6-2*x^9*y^5+x^9*y^3-x ^8*y^3-x^7*y^4+x^8*y^2+x^7*y^2-x^5*y^4+x^6*y^2+x^5*y^2+x^4*y^2+x^3*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5580967063 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 26" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 10 18 9 17 10 18 8 ) | ) C(m, n) x y | = - (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 9 16 10 18 7 17 8 16 9 15 10 - 5 x y + x y - 10 x y + 10 x y - 4 x y - x y 18 6 17 7 16 8 15 9 14 10 18 5 17 6 + 5 x y - 10 x y + 6 x y + 2 x y - x y - x y + 5 x y 16 7 15 8 14 9 13 10 17 5 16 6 15 7 - 4 x y + 2 x y + 6 x y - x y - x y + x y - 8 x y 14 8 15 6 14 7 13 8 12 9 15 5 - 14 x y + 7 x y + 17 x y + 4 x y + 3 x y - 2 x y 14 6 13 7 12 8 14 5 13 6 12 7 - 12 x y - 2 x y - 11 x y + 5 x y - 4 x y + 13 x y 11 8 14 4 13 5 12 6 11 7 10 8 13 4 + 2 x y - x y + 4 x y - 5 x y - x y + 2 x y - x y 11 6 10 7 9 8 11 5 10 6 9 7 11 4 - 3 x y - 5 x y + x y + x y + 6 x y - x y + 2 x y 10 5 9 6 8 7 11 3 10 4 9 5 8 6 - 5 x y + 4 x y - 4 x y - x y + 3 x y - 6 x y + 2 x y 10 3 9 4 8 5 7 6 9 3 7 5 7 4 6 5 - x y + x y + x y - 2 x y + x y + 4 x y - 5 x y + 2 x y 7 3 6 4 5 5 7 2 5 4 5 3 4 4 4 3 + 4 x y - x y + x y - x y - 3 x y + 3 x y + 5 x y - 2 x y 5 4 2 3 3 3 2 2 2 2 / - x y - 2 x y - 4 x y + 3 x y - 2 x y + x y - x y + x - 1) / ( / 12 6 12 5 11 6 12 4 11 5 10 6 11 4 10 5 x y - 2 x y - x y + x y + 2 x y + x y - x y - 2 x y 9 6 9 5 10 3 9 3 8 4 9 2 8 3 7 4 - x y + 2 x y + x y - 2 x y - x y + x y + x y + x y 6 4 5 4 3 2 2 - x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^18*y^10-5*x^18*y^9+x^17*y^10+10*x^18*y^8-5*x^17*y^9+x^16*y^10-10*x^18*y^7+ 10*x^17*y^8-4*x^16*y^9-x^15*y^10+5*x^18*y^6-10*x^17*y^7+6*x^16*y^8+2*x^15*y^9-x ^14*y^10-x^18*y^5+5*x^17*y^6-4*x^16*y^7+2*x^15*y^8+6*x^14*y^9-x^13*y^10-x^17*y^ 5+x^16*y^6-8*x^15*y^7-14*x^14*y^8+7*x^15*y^6+17*x^14*y^7+4*x^13*y^8+3*x^12*y^9-\ 2*x^15*y^5-12*x^14*y^6-2*x^13*y^7-11*x^12*y^8+5*x^14*y^5-4*x^13*y^6+13*x^12*y^7 +2*x^11*y^8-x^14*y^4+4*x^13*y^5-5*x^12*y^6-x^11*y^7+2*x^10*y^8-x^13*y^4-3*x^11* y^6-5*x^10*y^7+x^9*y^8+x^11*y^5+6*x^10*y^6-x^9*y^7+2*x^11*y^4-5*x^10*y^5+4*x^9* y^6-4*x^8*y^7-x^11*y^3+3*x^10*y^4-6*x^9*y^5+2*x^8*y^6-x^10*y^3+x^9*y^4+x^8*y^5-\ 2*x^7*y^6+x^9*y^3+4*x^7*y^5-5*x^7*y^4+2*x^6*y^5+4*x^7*y^3-x^6*y^4+x^5*y^5-x^7*y ^2-3*x^5*y^4+3*x^5*y^3+5*x^4*y^4-2*x^4*y^3-x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-\ 2*x^2*y^2+x^2*y-x*y+x-1)/(x^12*y^6-2*x^12*y^5-x^11*y^6+x^12*y^4+2*x^11*y^5+x^10 *y^6-x^11*y^4-2*x^10*y^5-x^9*y^6+2*x^9*y^5+x^10*y^3-2*x^9*y^3-x^8*y^4+x^9*y^2+x ^8*y^3+x^7*y^4-x^6*y^4+x^5*y^4-x^3*y^2+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5576661576 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 27" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 10 18 9 17 10 18 8 ) | ) C(m, n) x y | = - (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 9 16 10 18 7 17 8 16 9 15 10 - 5 x y + x y - 10 x y + 10 x y - 4 x y - x y 18 6 17 7 16 8 15 9 14 10 18 5 17 6 + 5 x y - 10 x y + 6 x y + 2 x y - x y - x y + 5 x y 16 7 15 8 14 9 13 10 17 5 16 6 15 7 - 4 x y + 2 x y + 6 x y - x y - x y + x y - 8 x y 14 8 15 6 14 7 13 8 12 9 15 5 - 14 x y + 7 x y + 17 x y + 4 x y + 3 x y - 2 x y 14 6 13 7 12 8 14 5 13 6 12 7 - 12 x y - 2 x y - 11 x y + 5 x y - 4 x y + 13 x y 11 8 14 4 13 5 12 6 11 7 10 8 13 4 + 2 x y - x y + 4 x y - 5 x y - x y + 2 x y - x y 11 6 10 7 9 8 11 5 10 6 9 7 11 4 - 3 x y - 5 x y + x y + x y + 6 x y - x y + 2 x y 10 5 9 6 8 7 11 3 10 4 9 5 8 6 - 5 x y + 4 x y - 4 x y - x y + 3 x y - 6 x y + 2 x y 10 3 9 4 8 5 7 6 9 3 7 5 7 4 6 5 - x y + x y + x y - 2 x y + x y + 4 x y - 5 x y + 2 x y 7 3 6 4 5 5 7 2 5 4 5 3 4 4 4 3 + 4 x y - x y + x y - x y - 3 x y + 3 x y + 5 x y - 2 x y 5 4 2 3 3 3 2 2 2 2 / - x y - 2 x y - 4 x y + 3 x y - 2 x y + x y - x y + x - 1) / ( / 12 6 12 5 11 6 12 4 11 5 10 6 11 4 10 5 x y - 2 x y - x y + x y + 2 x y + x y - x y - 2 x y 9 6 9 5 10 3 9 3 8 4 9 2 8 3 7 4 - x y + 2 x y + x y - 2 x y - x y + x y + x y + x y 6 4 5 4 3 2 2 - x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^18*y^10-5*x^18*y^9+x^17*y^10+10*x^18*y^8-5*x^17*y^9+x^16*y^10-10*x^18*y^7+ 10*x^17*y^8-4*x^16*y^9-x^15*y^10+5*x^18*y^6-10*x^17*y^7+6*x^16*y^8+2*x^15*y^9-x ^14*y^10-x^18*y^5+5*x^17*y^6-4*x^16*y^7+2*x^15*y^8+6*x^14*y^9-x^13*y^10-x^17*y^ 5+x^16*y^6-8*x^15*y^7-14*x^14*y^8+7*x^15*y^6+17*x^14*y^7+4*x^13*y^8+3*x^12*y^9-\ 2*x^15*y^5-12*x^14*y^6-2*x^13*y^7-11*x^12*y^8+5*x^14*y^5-4*x^13*y^6+13*x^12*y^7 +2*x^11*y^8-x^14*y^4+4*x^13*y^5-5*x^12*y^6-x^11*y^7+2*x^10*y^8-x^13*y^4-3*x^11* y^6-5*x^10*y^7+x^9*y^8+x^11*y^5+6*x^10*y^6-x^9*y^7+2*x^11*y^4-5*x^10*y^5+4*x^9* y^6-4*x^8*y^7-x^11*y^3+3*x^10*y^4-6*x^9*y^5+2*x^8*y^6-x^10*y^3+x^9*y^4+x^8*y^5-\ 2*x^7*y^6+x^9*y^3+4*x^7*y^5-5*x^7*y^4+2*x^6*y^5+4*x^7*y^3-x^6*y^4+x^5*y^5-x^7*y ^2-3*x^5*y^4+3*x^5*y^3+5*x^4*y^4-2*x^4*y^3-x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-\ 2*x^2*y^2+x^2*y-x*y+x-1)/(x^12*y^6-2*x^12*y^5-x^11*y^6+x^12*y^4+2*x^11*y^5+x^10 *y^6-x^11*y^4-2*x^10*y^5-x^9*y^6+2*x^9*y^5+x^10*y^3-2*x^9*y^3-x^8*y^4+x^9*y^2+x ^8*y^3+x^7*y^4-x^6*y^4+x^5*y^4-x^3*y^2+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5576661576 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 28" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 8 9 7 9 6 9 5 ) | ) C(m, n) x y | = (4 x y - 11 x y + 10 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 6 8 5 7 6 8 4 7 5 6 6 6 5 + 4 x y - 7 x y - 6 x y + 3 x y + 8 x y - x y - x y 7 3 5 5 6 3 5 4 5 3 4 4 5 2 4 3 - 2 x y - x y + x y + 8 x y - 7 x y - 5 x y + x y + 5 x y 4 2 3 3 4 3 2 3 2 2 2 - 2 x y + 4 x y + x y - 2 x y - x y + 2 x y - x y + x y - x + 1 / 6 5 6 4 6 3 5 3 5 2 4 3 4 3 2 ) / (x y - 2 x y + x y + x y - x y - x y + x y - x y + x y / - x y - x + 1) and in Maple notation (4*x^9*y^8-11*x^9*y^7+10*x^9*y^6-3*x^9*y^5+4*x^8*y^6-7*x^8*y^5-6*x^7*y^6+3*x^8* y^4+8*x^7*y^5-x^6*y^6-x^6*y^5-2*x^7*y^3-x^5*y^5+x^6*y^3+8*x^5*y^4-7*x^5*y^3-5*x ^4*y^4+x^5*y^2+5*x^4*y^3-2*x^4*y^2+4*x^3*y^3+x^4*y-2*x^3*y^2-x^3*y+2*x^2*y^2-x^ 2*y+x*y-x+1)/(x^6*y^5-2*x^6*y^4+x^6*y^3+x^5*y^3-x^5*y^2-x^4*y^3+x^4*y-x^3*y+x^2 *y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (3 a - 6 a + 14 a - 17 a + 13 a - 9 a + 2 a + 1) - --------------------------------------------------------- 4 3 2 2 (a - a - 1) (a + 1) (3 a - 2 a + 2) 3 2 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.5557511628 BTW the ratio for words with, 500, letters is, 0.5566493018 ------------------------------------------------ "Theorem Number 29" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 8 9 7 9 6 9 5 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 4 7 5 7 4 6 5 7 3 6 4 5 5 6 3 + x y - x y + 2 x y + 3 x y - x y - 5 x y - 3 x y + 2 x y 5 4 5 3 4 4 4 3 3 3 4 3 2 3 + 6 x y - 3 x y - 3 x y + x y + 4 x y + x y - 2 x y - x y 2 2 2 / 6 5 6 4 6 3 5 3 + 2 x y - x y + x y - x + 1) / (x y - 2 x y + x y + x y / 5 2 4 3 4 3 2 - x y - x y + x y - x y + x y - x y - x + 1) and in Maple notation (x^9*y^8-4*x^9*y^7+6*x^9*y^6-4*x^9*y^5+x^9*y^4-x^7*y^5+2*x^7*y^4+3*x^6*y^5-x^7* y^3-5*x^6*y^4-3*x^5*y^5+2*x^6*y^3+6*x^5*y^4-3*x^5*y^3-3*x^4*y^4+x^4*y^3+4*x^3*y ^3+x^4*y-2*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y-x+1)/(x^6*y^5-2*x^6*y^4+x^6*y^3+x^ 5*y^3-x^5*y^2-x^4*y^3+x^4*y-x^3*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 6 5 4 3 2 2 (a - 3 a + 5 a - 9 a + 9 a - 2 a - 1) --------------------------------------------- 6 5 4 2 3 a - 5 a + a - 5 a + 2 a - 2 3 2 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.5557511628 BTW the ratio for words with, 500, letters is, 0.5565621005 ------------------------------------------------ "Theorem Number 30" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 5 5 4 5 3 4 4 ) | ) C(m, n) x y | = - (x y - 2 x y + 2 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 2 4 3 4 2 3 3 3 2 3 2 2 2 - x y - 4 x y + x y - 4 x y + x y + x y - 2 x y + x y - x y / 3 2 3 2 - 1) / (x y - x y - x y - x y + 1) / and in Maple notation -(x^5*y^5-2*x^5*y^4+2*x^5*y^3+2*x^4*y^4-x^5*y^2-4*x^4*y^3+x^4*y^2-4*x^3*y^3+x^3 *y^2+x^3*y-2*x^2*y^2+x^2*y-x*y-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 5 4 3 2 2 (a + 5 a + 4 a - 3 a - 2 a - 1) - ------------------------------------- 5 4 3 2 2 a + 5 a + 4 a + 3 a + 3 a + 1 2 where a is the root of the polynomial, x + x - 1, and in decimals this is, 0.5527864070 BTW the ratio for words with, 500, letters is, 0.5541162500 ------------------------------------------------ "Theorem Number 31" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 6 7 5 6 6 7 4 ) | ) C(m, n) x y | = - (x y - 3 x y - x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 5 7 3 6 4 5 4 5 3 4 4 5 2 + 2 x y - x y - x y + 3 x y - 5 x y - 2 x y + 2 x y 3 3 4 3 2 2 2 2 / + 4 x y + x y - 3 x y + 2 x y - 2 x y + x y - x + 1) / ( / 3 2 3 2 (x y - x y - x y - x y + 1) (x - 1)) and in Maple notation -(x^7*y^6-3*x^7*y^5-x^6*y^6+3*x^7*y^4+2*x^6*y^5-x^7*y^3-x^6*y^4+3*x^5*y^4-5*x^5 *y^3-2*x^4*y^4+2*x^5*y^2+4*x^3*y^3+x^4*y-3*x^3*y^2+2*x^2*y^2-2*x^2*y+x*y-x+1)/( x^3*y^2-x^3*y-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5539305839 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 32" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 7 9 6 9 5 9 4 8 5 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 4 7 5 6 6 8 3 7 4 6 5 7 3 + 2 x y + 4 x y - x y - x y - 7 x y - 3 x y + 3 x y 6 4 6 3 5 4 5 3 4 4 5 2 4 2 3 3 + 2 x y + x y - 2 x y + 3 x y + 3 x y - x y - x y - 4 x y 4 3 2 2 2 2 / - x y + 3 x y - 2 x y + 2 x y - x y + x - 1) / ( / 4 2 4 3 2 x y - x y - x y + x y + x - 1) and in Maple notation (x^9*y^7-3*x^9*y^6+3*x^9*y^5-x^9*y^4-x^8*y^5+2*x^8*y^4+4*x^7*y^5-x^6*y^6-x^8*y^ 3-7*x^7*y^4-3*x^6*y^5+3*x^7*y^3+2*x^6*y^4+x^6*y^3-2*x^5*y^4+3*x^5*y^3+3*x^4*y^4 -x^5*y^2-x^4*y^2-4*x^3*y^3-x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^4*y^2- x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5538509421 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 33" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 5 6 6 7 4 6 5 ) | ) C(m, n) x y | = (x y + 2 x y - 2 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 3 6 4 5 5 6 3 5 4 5 3 4 4 5 2 + x y + 4 x y + x y - x y - 2 x y + 2 x y + 2 x y - x y 3 3 4 3 2 2 2 2 / - 4 x y - x y + 3 x y - 2 x y + 2 x y - x y + x - 1) / ((x - 1) / 3 2 3 2 (x y - x y - x y - x y + 1)) and in Maple notation (x^7*y^5+2*x^6*y^6-2*x^7*y^4-5*x^6*y^5+x^7*y^3+4*x^6*y^4+x^5*y^5-x^6*y^3-2*x^5* y^4+2*x^5*y^3+2*x^4*y^4-x^5*y^2-4*x^3*y^3-x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y +x-1)/(x-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 5 4 3 2 2 2 (a + 4 a + 4 a - 3 a - 2 a - 1) (3 a - 2) - ------------------------------------------------ 2 4 3 (2 a + 1) (a - a - 1) 2 where a is the root of the polynomial, (x + x - 1) (x - 1), and in decimals this is, 0.5527864076 BTW the ratio for words with, 500, letters is, 0.5536003970 ------------------------------------------------ "Theorem Number 34" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 6 9 5 9 4 8 5 7 6 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 3 8 4 7 5 8 3 7 4 6 5 7 3 - x y + 2 x y + 2 x y - x y - 6 x y - 6 x y + 3 x y 6 4 5 5 6 3 5 4 6 2 5 3 4 4 + 6 x y + 2 x y - 2 x y - 4 x y + x y + 3 x y + 3 x y 5 2 4 2 3 3 4 3 2 2 2 2 - x y - x y - 4 x y - x y + 3 x y - 2 x y + 2 x y - x y + x / 3 2 3 2 - 1) / ((x y - x y - x y - x y + 1) (x - 1)) / and in Maple notation (x^9*y^6-3*x^9*y^5+3*x^9*y^4-x^8*y^5+x^7*y^6-x^9*y^3+2*x^8*y^4+2*x^7*y^5-x^8*y^ 3-6*x^7*y^4-6*x^6*y^5+3*x^7*y^3+6*x^6*y^4+2*x^5*y^5-2*x^6*y^3-4*x^5*y^4+x^6*y^2 +3*x^5*y^3+3*x^4*y^4-x^5*y^2-x^4*y^2-4*x^3*y^3-x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2* y-x*y+x-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5535368130 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 35" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 8 10 7 9 8 10 6 ) | ) C(m, n) x y | = - (x y - 2 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 7 10 5 9 6 10 4 8 6 10 3 8 5 7 6 + 2 x y + 5 x y - x y - 4 x y + x y + x y - 3 x y + x y 8 4 7 5 8 3 6 5 7 3 6 4 5 5 + 3 x y - 3 x y - x y - 3 x y + 4 x y + 4 x y + 4 x y 7 2 6 3 5 4 5 3 4 4 4 3 5 4 2 - 2 x y - x y - 5 x y + 3 x y + 2 x y - x y - x y + x y 3 3 4 3 2 3 2 2 2 / - 4 x y - x y + 2 x y + x y - 2 x y + x y - x y + x - 1) / (( / 9 6 9 5 9 4 8 5 9 3 8 4 8 3 6 3 5 4 x y - 3 x y + 3 x y - x y - x y + 2 x y - x y - x y - x y 6 2 5 2 4 3 3 + x y + x y + x y + x y + x y - 1) (x - 1)) and in Maple notation -(x^10*y^8-2*x^10*y^7-x^9*y^8-x^10*y^6+2*x^9*y^7+5*x^10*y^5-x^9*y^6-4*x^10*y^4+ x^8*y^6+x^10*y^3-3*x^8*y^5+x^7*y^6+3*x^8*y^4-3*x^7*y^5-x^8*y^3-3*x^6*y^5+4*x^7* y^3+4*x^6*y^4+4*x^5*y^5-2*x^7*y^2-x^6*y^3-5*x^5*y^4+3*x^5*y^3+2*x^4*y^4-x^4*y^3 -x^5*y+x^4*y^2-4*x^3*y^3-x^4*y+2*x^3*y^2+x^3*y-2*x^2*y^2+x^2*y-x*y+x-1)/(x^9*y^ 6-3*x^9*y^5+3*x^9*y^4-x^8*y^5-x^9*y^3+2*x^8*y^4-x^8*y^3-x^6*y^3-x^5*y^4+x^6*y^2 +x^5*y^2+x^4*y^3+x^3*y+x*y-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5533963303 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 36" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 6 10 5 9 6 10 4 ) | ) C(m, n) x y | = - (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 5 8 6 10 3 9 4 8 5 7 6 8 4 7 5 - 2 x y - 3 x y - x y + x y + 4 x y + x y - 2 x y - 7 x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 6 3 + 2 x y + 12 x y + 8 x y - x y - 6 x y - 6 x y - x y - x y 5 4 5 3 4 4 5 2 5 4 2 3 3 4 - x y - x y - 3 x y + x y + x y + x y + 4 x y + x y 3 2 2 2 2 / - 3 x y + 2 x y - 2 x y + x y - x + 1) / ( / 4 2 4 3 2 x y - x y - x y + x y + x - 1) and in Maple notation -(x^10*y^6-3*x^10*y^5+x^9*y^6+3*x^10*y^4-2*x^9*y^5-3*x^8*y^6-x^10*y^3+x^9*y^4+4 *x^8*y^5+x^7*y^6-2*x^8*y^4-7*x^7*y^5+2*x^8*y^3+12*x^7*y^4+8*x^6*y^5-x^8*y^2-6*x ^7*y^3-6*x^6*y^4-x^5*y^5-x^6*y^3-x^5*y^4-x^5*y^3-3*x^4*y^4+x^5*y^2+x^5*y+x^4*y^ 2+4*x^3*y^3+x^4*y-3*x^3*y^2+2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^4*y^2-x^4*y-x^3*y^2+x *y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5532575981 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 37" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 10 15 9 15 8 14 9 ) | ) C(m, n) x y | = (x y - 5 x y + 10 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 10 15 7 14 8 13 9 15 6 14 7 - x y - 10 x y + 4 x y + 6 x y + 5 x y - 6 x y 13 8 12 9 15 5 14 6 13 7 12 8 14 5 - 14 x y - x y - x y + 4 x y + 16 x y + 3 x y - x y 13 6 12 7 11 8 13 5 12 6 11 7 10 8 - 9 x y - 3 x y - x y + 2 x y + x y + x y + x y 11 6 9 8 11 5 10 6 9 7 11 4 10 5 + 3 x y + x y - 5 x y - 3 x y - 4 x y + 2 x y + x y 9 6 8 7 10 4 9 5 10 3 9 4 8 5 + 3 x y + 2 x y + 2 x y + 5 x y - x y - 8 x y - 3 x y 7 6 9 3 8 4 7 5 8 3 6 5 7 3 - 2 x y + 3 x y - x y + 4 x y + 2 x y - 2 x y - 3 x y 6 4 5 5 7 2 6 3 5 4 6 2 5 3 - 3 x y - 3 x y + x y + 5 x y + 8 x y - x y - 7 x y 4 4 5 2 4 3 3 3 4 3 2 3 2 2 - 3 x y + 2 x y + x y + 4 x y + x y - 2 x y - x y + 2 x y 2 / 10 6 10 5 9 6 10 4 9 5 - x y + x y - x + 1) / (x y - 3 x y - x y + 3 x y + 2 x y / 10 3 9 4 8 5 8 4 8 3 7 3 6 4 7 2 6 3 - x y - x y + x y - 2 x y + x y - x y - x y + x y + x y 5 4 5 3 5 2 4 3 4 3 2 + x y + x y - x y - x y + x y - x y + x y - x y - x + 1) and in Maple notation (x^15*y^10-5*x^15*y^9+10*x^15*y^8-x^14*y^9-x^13*y^10-10*x^15*y^7+4*x^14*y^8+6*x ^13*y^9+5*x^15*y^6-6*x^14*y^7-14*x^13*y^8-x^12*y^9-x^15*y^5+4*x^14*y^6+16*x^13* y^7+3*x^12*y^8-x^14*y^5-9*x^13*y^6-3*x^12*y^7-x^11*y^8+2*x^13*y^5+x^12*y^6+x^11 *y^7+x^10*y^8+3*x^11*y^6+x^9*y^8-5*x^11*y^5-3*x^10*y^6-4*x^9*y^7+2*x^11*y^4+x^ 10*y^5+3*x^9*y^6+2*x^8*y^7+2*x^10*y^4+5*x^9*y^5-x^10*y^3-8*x^9*y^4-3*x^8*y^5-2* x^7*y^6+3*x^9*y^3-x^8*y^4+4*x^7*y^5+2*x^8*y^3-2*x^6*y^5-3*x^7*y^3-3*x^6*y^4-3*x ^5*y^5+x^7*y^2+5*x^6*y^3+8*x^5*y^4-x^6*y^2-7*x^5*y^3-3*x^4*y^4+2*x^5*y^2+x^4*y^ 3+4*x^3*y^3+x^4*y-2*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y-x+1)/(x^10*y^6-3*x^10*y^5 -x^9*y^6+3*x^10*y^4+2*x^9*y^5-x^10*y^3-x^9*y^4+x^8*y^5-2*x^8*y^4+x^8*y^3-x^7*y^ 3-x^6*y^4+x^7*y^2+x^6*y^3+x^5*y^4+x^5*y^3-x^5*y^2-x^4*y^3+x^4*y-x^3*y+x^2*y-x*y -x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5530165429 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 38" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 6 10 5 10 4 9 5 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 3 9 4 8 5 7 6 9 3 8 4 7 5 8 3 - x y + 2 x y + x y - x y - x y - 2 x y - x y + x y 7 4 6 5 7 3 6 4 5 5 7 2 6 3 5 4 + x y - x y + 2 x y + 4 x y + 3 x y - x y - 2 x y - 4 x y 5 3 4 4 5 2 4 2 3 3 4 3 2 2 2 + 4 x y + 3 x y - 2 x y - x y - 4 x y - x y + 3 x y - 2 x y 2 / 3 2 3 2 + 2 x y - x y + x - 1) / ((x - 1) (x y - x y - x y - x y + 1)) / and in Maple notation (x^10*y^6-3*x^10*y^5+3*x^10*y^4-x^9*y^5-x^10*y^3+2*x^9*y^4+x^8*y^5-x^7*y^6-x^9* y^3-2*x^8*y^4-x^7*y^5+x^8*y^3+x^7*y^4-x^6*y^5+2*x^7*y^3+4*x^6*y^4+3*x^5*y^5-x^7 *y^2-2*x^6*y^3-4*x^5*y^4+4*x^5*y^3+3*x^4*y^4-2*x^5*y^2-x^4*y^2-4*x^3*y^3-x^4*y+ 3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5528446472 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 39" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 6 9 5 8 6 9 4 ) | ) C(m, n) x y | = (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 5 7 6 9 3 8 4 7 5 7 4 6 5 - 2 x y - 2 x y - x y + x y + 6 x y - 8 x y - 5 x y 7 3 6 4 5 5 6 3 5 4 6 2 5 3 + 4 x y + 8 x y + 2 x y - 4 x y - x y + x y + 3 x y 4 4 5 2 4 2 3 3 4 3 2 2 2 2 + 3 x y - 3 x y - x y - 4 x y - x y + 3 x y - 2 x y + 2 x y / 4 2 4 3 2 - x y + x - 1) / (x y - x y - x y + x y + x - 1) / and in Maple notation (x^9*y^6-3*x^9*y^5+x^8*y^6+3*x^9*y^4-2*x^8*y^5-2*x^7*y^6-x^9*y^3+x^8*y^4+6*x^7* y^5-8*x^7*y^4-5*x^6*y^5+4*x^7*y^3+8*x^6*y^4+2*x^5*y^5-4*x^6*y^3-x^5*y^4+x^6*y^2 +3*x^5*y^3+3*x^4*y^4-3*x^5*y^2-x^4*y^2-4*x^3*y^3-x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^ 2*y-x*y+x-1)/(x^4*y^2-x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5528139030 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 40" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 6 10 5 10 4 9 5 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 3 9 4 8 5 7 6 9 3 8 4 7 5 8 3 - x y + 2 x y + x y - x y - x y - 2 x y - x y + x y 7 4 6 5 6 4 5 5 6 3 5 4 5 3 + 2 x y - 4 x y + 9 x y + 2 x y - 4 x y + 3 x y - x y 4 4 5 2 4 3 4 2 3 3 4 3 2 2 2 + 5 x y - 2 x y - 4 x y + x y - 4 x y - x y + 3 x y - 2 x y 2 / 3 2 3 2 + 2 x y - x y + x - 1) / ((x y - x y - x y - x y + 1) (x - 1)) / and in Maple notation (x^10*y^6-3*x^10*y^5+3*x^10*y^4-x^9*y^5-x^10*y^3+2*x^9*y^4+x^8*y^5-x^7*y^6-x^9* y^3-2*x^8*y^4-x^7*y^5+x^8*y^3+2*x^7*y^4-4*x^6*y^5+9*x^6*y^4+2*x^5*y^5-4*x^6*y^3 +3*x^5*y^4-x^5*y^3+5*x^4*y^4-2*x^5*y^2-4*x^4*y^3+x^4*y^2-4*x^3*y^3-x^4*y+3*x^3* y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5521044868 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 41" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 18 10 ) | ) C(m, n) x y | = - (x y - 6 x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 11 18 9 17 10 16 11 18 8 17 9 + 2 x y - 20 x y - 8 x y - x y + 15 x y + 12 x y 16 10 15 11 18 7 17 8 16 9 15 10 + 4 x y + x y - 6 x y - 8 x y - 7 x y - 5 x y 18 6 17 7 16 8 15 9 14 10 16 7 15 8 + x y + 2 x y + 8 x y + 10 x y + x y - 7 x y - 9 x y 14 9 13 10 16 6 15 7 14 8 13 9 16 5 + x y - 2 x y + 4 x y + 2 x y - 8 x y + 2 x y - x y 15 6 14 7 13 8 12 9 15 5 14 6 13 7 + 2 x y + 7 x y + 4 x y - 4 x y - x y + 2 x y - 6 x y 12 8 14 5 13 6 12 7 10 9 14 4 13 5 + 13 x y - 4 x y + 4 x y - 13 x y - x y + x y - 4 x y 12 6 11 7 10 8 13 4 12 5 11 6 + 4 x y + 7 x y + 2 x y + 2 x y - x y - 19 x y 10 7 9 8 12 4 11 5 10 6 9 7 11 4 - 4 x y - 3 x y + x y + 17 x y + 3 x y + x y - 5 x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 - x y + 5 x y + 5 x y + 5 x y + x y - 4 x y - 4 x y 9 4 8 5 7 6 9 3 8 4 7 5 9 2 8 3 - 7 x y + x y + x y + 4 x y - 5 x y - x y - x y + x y 7 4 6 5 8 2 7 3 6 4 5 5 6 3 - 6 x y - 4 x y + 2 x y + 4 x y + 5 x y + 2 x y + 2 x y 5 4 7 6 2 5 3 4 4 4 3 5 4 2 + x y + x y - 3 x y + x y + 5 x y - 4 x y - 2 x y + x y 3 3 3 2 4 2 2 3 2 / - 4 x y + 3 x y - x - 2 x y + x + 2 x y - x y + x - 1) / ( / 14 8 14 7 14 6 14 5 11 7 11 6 11 5 (x - 1) (x y - 3 x y + 3 x y - x y + x y - x y - x y 10 6 11 4 10 5 9 6 10 4 9 5 10 3 9 4 - x y + x y + x y - x y + x y + 2 x y - x y - x y 8 4 8 3 6 5 8 2 7 3 7 2 5 4 6 2 6 + x y - 2 x y - x y + x y - x y + x y + x y + x y - x y 5 4 3 2 3 3 2 - x y - x y - x y + x y + x + x y + x y - 1)) and in Maple notation -(x^18*y^12-6*x^18*y^11+15*x^18*y^10+2*x^17*y^11-20*x^18*y^9-8*x^17*y^10-x^16*y ^11+15*x^18*y^8+12*x^17*y^9+4*x^16*y^10+x^15*y^11-6*x^18*y^7-8*x^17*y^8-7*x^16* y^9-5*x^15*y^10+x^18*y^6+2*x^17*y^7+8*x^16*y^8+10*x^15*y^9+x^14*y^10-7*x^16*y^7 -9*x^15*y^8+x^14*y^9-2*x^13*y^10+4*x^16*y^6+2*x^15*y^7-8*x^14*y^8+2*x^13*y^9-x^ 16*y^5+2*x^15*y^6+7*x^14*y^7+4*x^13*y^8-4*x^12*y^9-x^15*y^5+2*x^14*y^6-6*x^13*y ^7+13*x^12*y^8-4*x^14*y^5+4*x^13*y^6-13*x^12*y^7-x^10*y^9+x^14*y^4-4*x^13*y^5+4 *x^12*y^6+7*x^11*y^7+2*x^10*y^8+2*x^13*y^4-x^12*y^5-19*x^11*y^6-4*x^10*y^7-3*x^ 9*y^8+x^12*y^4+17*x^11*y^5+3*x^10*y^6+x^9*y^7-5*x^11*y^4-x^10*y^5+5*x^9*y^6+5*x ^8*y^7+5*x^10*y^4+x^9*y^5-4*x^8*y^6-4*x^10*y^3-7*x^9*y^4+x^8*y^5+x^7*y^6+4*x^9* y^3-5*x^8*y^4-x^7*y^5-x^9*y^2+x^8*y^3-6*x^7*y^4-4*x^6*y^5+2*x^8*y^2+4*x^7*y^3+5 *x^6*y^4+2*x^5*y^5+2*x^6*y^3+x^5*y^4+x^7*y-3*x^6*y^2+x^5*y^3+5*x^4*y^4-4*x^4*y^ 3-2*x^5*y+x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^4-2*x^2*y^2+x^3+2*x^2*y-x*y+x-1)/(x-1)/ (x^14*y^8-3*x^14*y^7+3*x^14*y^6-x^14*y^5+x^11*y^7-x^11*y^6-x^11*y^5-x^10*y^6+x^ 11*y^4+x^10*y^5-x^9*y^6+x^10*y^4+2*x^9*y^5-x^10*y^3-x^9*y^4+x^8*y^4-2*x^8*y^3-x ^6*y^5+x^8*y^2-x^7*y^3+x^7*y^2+x^5*y^4+x^6*y^2-x^6*y-x^5*y-x^4*y-x^3*y^2+x^3*y+ x^3+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5413452099 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 42" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 18 10 ) | ) C(m, n) x y | = - (x y - 6 x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 11 18 9 17 10 16 11 18 8 17 9 + 2 x y - 20 x y - 8 x y - x y + 15 x y + 12 x y 16 10 15 11 18 7 17 8 16 9 15 10 + 4 x y + x y - 6 x y - 8 x y - 7 x y - 5 x y 18 6 17 7 16 8 15 9 14 10 16 7 15 8 + x y + 2 x y + 8 x y + 10 x y + x y - 7 x y - 9 x y 14 9 13 10 16 6 15 7 14 8 13 9 16 5 + x y - 2 x y + 4 x y + 2 x y - 8 x y + 2 x y - x y 15 6 14 7 13 8 12 9 15 5 14 6 13 7 + 2 x y + 7 x y + 4 x y - 4 x y - x y + 2 x y - 6 x y 12 8 14 5 13 6 12 7 10 9 14 4 13 5 + 13 x y - 4 x y + 4 x y - 13 x y - x y + x y - 4 x y 12 6 11 7 10 8 13 4 12 5 11 6 + 4 x y + 7 x y + 2 x y + 2 x y - x y - 19 x y 10 7 9 8 12 4 11 5 10 6 9 7 11 4 - 4 x y - 3 x y + x y + 17 x y + 3 x y + x y - 5 x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 - x y + 5 x y + 5 x y + 5 x y + x y - 4 x y - 4 x y 9 4 8 5 7 6 9 3 8 4 7 5 9 2 8 3 - 7 x y + x y + x y + 4 x y - 5 x y - x y - x y + x y 7 4 6 5 8 2 7 3 6 4 5 5 6 3 - 6 x y - 4 x y + 2 x y + 4 x y + 5 x y + 2 x y + 2 x y 5 4 7 6 2 5 3 4 4 4 3 5 4 2 + x y + x y - 3 x y + x y + 5 x y - 4 x y - 2 x y + x y 3 3 3 2 4 2 2 3 2 / - 4 x y + 3 x y - x - 2 x y + x + 2 x y - x y + x - 1) / ( / 14 8 14 7 14 6 14 5 11 7 11 6 11 5 (x - 1) (x y - 3 x y + 3 x y - x y + x y - x y - x y 10 6 11 4 10 5 9 6 10 4 9 5 10 3 9 4 - x y + x y + x y - x y + x y + 2 x y - x y - x y 8 4 8 3 6 5 8 2 7 3 7 2 5 4 6 2 6 + x y - 2 x y - x y + x y - x y + x y + x y + x y - x y 5 4 3 2 3 3 2 - x y - x y - x y + x y + x + x y + x y - 1)) and in Maple notation -(x^18*y^12-6*x^18*y^11+15*x^18*y^10+2*x^17*y^11-20*x^18*y^9-8*x^17*y^10-x^16*y ^11+15*x^18*y^8+12*x^17*y^9+4*x^16*y^10+x^15*y^11-6*x^18*y^7-8*x^17*y^8-7*x^16* y^9-5*x^15*y^10+x^18*y^6+2*x^17*y^7+8*x^16*y^8+10*x^15*y^9+x^14*y^10-7*x^16*y^7 -9*x^15*y^8+x^14*y^9-2*x^13*y^10+4*x^16*y^6+2*x^15*y^7-8*x^14*y^8+2*x^13*y^9-x^ 16*y^5+2*x^15*y^6+7*x^14*y^7+4*x^13*y^8-4*x^12*y^9-x^15*y^5+2*x^14*y^6-6*x^13*y ^7+13*x^12*y^8-4*x^14*y^5+4*x^13*y^6-13*x^12*y^7-x^10*y^9+x^14*y^4-4*x^13*y^5+4 *x^12*y^6+7*x^11*y^7+2*x^10*y^8+2*x^13*y^4-x^12*y^5-19*x^11*y^6-4*x^10*y^7-3*x^ 9*y^8+x^12*y^4+17*x^11*y^5+3*x^10*y^6+x^9*y^7-5*x^11*y^4-x^10*y^5+5*x^9*y^6+5*x ^8*y^7+5*x^10*y^4+x^9*y^5-4*x^8*y^6-4*x^10*y^3-7*x^9*y^4+x^8*y^5+x^7*y^6+4*x^9* y^3-5*x^8*y^4-x^7*y^5-x^9*y^2+x^8*y^3-6*x^7*y^4-4*x^6*y^5+2*x^8*y^2+4*x^7*y^3+5 *x^6*y^4+2*x^5*y^5+2*x^6*y^3+x^5*y^4+x^7*y-3*x^6*y^2+x^5*y^3+5*x^4*y^4-4*x^4*y^ 3-2*x^5*y+x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^4-2*x^2*y^2+x^3+2*x^2*y-x*y+x-1)/(x-1)/ (x^14*y^8-3*x^14*y^7+3*x^14*y^6-x^14*y^5+x^11*y^7-x^11*y^6-x^11*y^5-x^10*y^6+x^ 11*y^4+x^10*y^5-x^9*y^6+x^10*y^4+2*x^9*y^5-x^10*y^3-x^9*y^4+x^8*y^4-2*x^8*y^3-x ^6*y^5+x^8*y^2-x^7*y^3+x^7*y^2+x^5*y^4+x^6*y^2-x^6*y-x^5*y-x^4*y-x^3*y^2+x^3*y+ x^3+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5413452099 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 43" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 11 14 10 13 11 14 9 ) | ) C(m, n) x y | = - (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 10 14 8 13 9 12 10 13 8 12 9 13 7 - x y - x y - 5 x y + x y + 11 x y - 2 x y - 8 x y 12 8 11 9 13 6 12 7 11 8 10 9 12 6 - 2 x y + 2 x y + 2 x y + 7 x y - 3 x y + x y - 5 x y 10 8 12 5 11 6 10 7 9 8 11 5 9 7 - 7 x y + x y - 2 x y + 11 x y + x y + 6 x y - 6 x y 11 4 10 5 9 6 8 7 10 4 9 5 10 3 - 3 x y - 11 x y + 9 x y + 2 x y + 7 x y - 6 x y - x y 9 4 8 5 7 6 9 3 8 4 7 5 6 6 + 4 x y - x y - 8 x y - 2 x y - 6 x y + 11 x y - x y 8 3 7 4 8 2 7 3 6 4 5 5 7 2 6 3 + 6 x y + x y - x y - 5 x y - x y - x y + 2 x y - 2 x y 5 4 7 6 2 5 3 4 4 6 5 2 4 3 + 6 x y - x y + 2 x y - 5 x y - 5 x y + x y - x y + 5 x y 5 4 2 3 3 3 2 4 3 2 2 3 2 + x y - 2 x y + 4 x y - 2 x y + x - x y + 2 x y - x - x y / 10 6 10 5 10 4 8 6 10 3 9 4 + x y - x + 1) / (x y - x y - x y - x y + x y - x y / 9 3 8 4 7 5 8 2 7 3 6 4 6 3 5 4 7 + x y + 2 x y + x y - x y - x y + x y - x y - x y + x y 5 3 6 5 2 4 3 5 4 4 3 3 2 - x y - x y + x y + x y + x y - 2 x y - x + x y + x - x y + x y + x - 1) and in Maple notation -(x^14*y^11-3*x^14*y^10+x^13*y^11+3*x^14*y^9-x^13*y^10-x^14*y^8-5*x^13*y^9+x^12 *y^10+11*x^13*y^8-2*x^12*y^9-8*x^13*y^7-2*x^12*y^8+2*x^11*y^9+2*x^13*y^6+7*x^12 *y^7-3*x^11*y^8+x^10*y^9-5*x^12*y^6-7*x^10*y^8+x^12*y^5-2*x^11*y^6+11*x^10*y^7+ x^9*y^8+6*x^11*y^5-6*x^9*y^7-3*x^11*y^4-11*x^10*y^5+9*x^9*y^6+2*x^8*y^7+7*x^10* y^4-6*x^9*y^5-x^10*y^3+4*x^9*y^4-x^8*y^5-8*x^7*y^6-2*x^9*y^3-6*x^8*y^4+11*x^7*y ^5-x^6*y^6+6*x^8*y^3+x^7*y^4-x^8*y^2-5*x^7*y^3-x^6*y^4-x^5*y^5+2*x^7*y^2-2*x^6* y^3+6*x^5*y^4-x^7*y+2*x^6*y^2-5*x^5*y^3-5*x^4*y^4+x^6*y-x^5*y^2+5*x^4*y^3+x^5*y -2*x^4*y^2+4*x^3*y^3-2*x^3*y^2+x^4-x^3*y+2*x^2*y^2-x^3-x^2*y+x*y-x+1)/(x^10*y^6 -x^10*y^5-x^10*y^4-x^8*y^6+x^10*y^3-x^9*y^4+x^9*y^3+2*x^8*y^4+x^7*y^5-x^8*y^2-x ^7*y^3+x^6*y^4-x^6*y^3-x^5*y^4+x^7*y-x^5*y^3-x^6*y+x^5*y^2+x^4*y^3+x^5*y-2*x^4* y-x^4+x^3*y+x^3-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5383651244 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 44" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 11 14 10 13 11 14 9 ) | ) C(m, n) x y | = - (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 10 14 8 13 9 12 10 13 8 12 9 13 7 - x y - x y - 5 x y + x y + 11 x y - 2 x y - 8 x y 12 8 11 9 13 6 12 7 11 8 10 9 12 6 - 2 x y + 2 x y + 2 x y + 7 x y - 3 x y + x y - 5 x y 10 8 12 5 11 6 10 7 9 8 11 5 9 7 - 7 x y + x y - 2 x y + 11 x y + x y + 6 x y - 6 x y 11 4 10 5 9 6 8 7 10 4 9 5 10 3 - 3 x y - 11 x y + 9 x y + 2 x y + 7 x y - 6 x y - x y 9 4 8 5 7 6 9 3 8 4 7 5 6 6 + 4 x y - x y - 8 x y - 2 x y - 6 x y + 11 x y - x y 8 3 7 4 8 2 7 3 6 4 5 5 7 2 6 3 + 6 x y + x y - x y - 5 x y - x y - x y + 2 x y - 2 x y 5 4 7 6 2 5 3 4 4 6 5 2 4 3 + 6 x y - x y + 2 x y - 5 x y - 5 x y + x y - x y + 5 x y 5 4 2 3 3 3 2 4 3 2 2 3 2 + x y - 2 x y + 4 x y - 2 x y + x - x y + 2 x y - x - x y / 9 6 9 5 8 6 9 4 8 5 9 3 + x y - x + 1) / ((x - 1) (x y - x y + x y - x y - x y + x y / 8 4 7 5 8 3 7 3 7 2 6 3 5 4 6 2 - 2 x y - x y + 2 x y + 2 x y - x y + x y + x y - x y 6 5 2 4 3 4 3 3 + x y - x y - x y + x y - x y - x - x y + 1)) and in Maple notation -(x^14*y^11-3*x^14*y^10+x^13*y^11+3*x^14*y^9-x^13*y^10-x^14*y^8-5*x^13*y^9+x^12 *y^10+11*x^13*y^8-2*x^12*y^9-8*x^13*y^7-2*x^12*y^8+2*x^11*y^9+2*x^13*y^6+7*x^12 *y^7-3*x^11*y^8+x^10*y^9-5*x^12*y^6-7*x^10*y^8+x^12*y^5-2*x^11*y^6+11*x^10*y^7+ x^9*y^8+6*x^11*y^5-6*x^9*y^7-3*x^11*y^4-11*x^10*y^5+9*x^9*y^6+2*x^8*y^7+7*x^10* y^4-6*x^9*y^5-x^10*y^3+4*x^9*y^4-x^8*y^5-8*x^7*y^6-2*x^9*y^3-6*x^8*y^4+11*x^7*y ^5-x^6*y^6+6*x^8*y^3+x^7*y^4-x^8*y^2-5*x^7*y^3-x^6*y^4-x^5*y^5+2*x^7*y^2-2*x^6* y^3+6*x^5*y^4-x^7*y+2*x^6*y^2-5*x^5*y^3-5*x^4*y^4+x^6*y-x^5*y^2+5*x^4*y^3+x^5*y -2*x^4*y^2+4*x^3*y^3-2*x^3*y^2+x^4-x^3*y+2*x^2*y^2-x^3-x^2*y+x*y-x+1)/(x-1)/(x^ 9*y^6-x^9*y^5+x^8*y^6-x^9*y^4-x^8*y^5+x^9*y^3-2*x^8*y^4-x^7*y^5+2*x^8*y^3+2*x^7 *y^3-x^7*y^2+x^6*y^3+x^5*y^4-x^6*y^2+x^6*y-x^5*y^2-x^4*y^3+x^4*y-x^3*y-x^3-x*y+ 1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5383651244 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 45" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 22 16 22 15 22 14 ) | ) C(m, n) x y | = - (x y - 7 x y + 21 x y / | / | ----- | ----- | m = 0 \ n = 0 / 21 15 20 16 22 13 21 14 20 15 22 12 - x y - x y - 35 x y + 7 x y + 8 x y + 35 x y 21 13 20 14 19 15 22 11 21 12 - 21 x y - 33 x y - 3 x y - 21 x y + 35 x y 20 13 19 14 18 15 22 10 21 11 20 12 + 89 x y + 23 x y + x y + 7 x y - 35 x y - 164 x y 19 13 18 14 22 9 21 10 20 11 19 12 - 76 x y - 8 x y - x y + 21 x y + 207 x y + 141 x y 18 13 17 14 21 9 20 10 19 11 + 31 x y + 3 x y - 7 x y - 178 x y - 161 x y 18 12 17 13 21 8 20 9 19 10 - 73 x y - 16 x y + x y + 103 x y + 118 x y 18 11 17 12 20 8 19 9 18 10 + 111 x y + 33 x y - 39 x y - 58 x y - 111 x y 17 11 16 12 20 7 19 8 18 9 17 10 - 29 x y + x y + 9 x y + 21 x y + 73 x y + 2 x y 16 11 15 12 20 6 19 7 18 8 17 9 - 7 x y - 3 x y - x y - 6 x y - 31 x y + 14 x y 16 10 15 11 14 12 19 6 18 7 17 8 + 20 x y + 18 x y + x y + x y + 8 x y - 7 x y 16 9 15 10 14 11 18 6 17 7 16 8 - 31 x y - 44 x y - 4 x y - x y - x y + 29 x y 15 9 14 10 13 11 17 6 16 7 15 8 + 57 x y + 9 x y + 2 x y + x y - 17 x y - 44 x y 14 9 13 10 12 11 16 6 15 7 14 8 - 13 x y - 8 x y - x y + 6 x y + 23 x y + 11 x y 13 9 12 10 16 5 15 6 14 7 13 8 + 13 x y + 4 x y - x y - 9 x y - 6 x y - 14 x y 12 9 11 10 15 5 14 6 13 7 12 8 - 10 x y - 2 x y + 2 x y + 3 x y + 15 x y + 14 x y 11 9 14 5 13 6 12 7 11 8 10 9 13 5 + 2 x y - x y - 13 x y - 11 x y + x y - x y + 6 x y 12 6 11 7 10 8 13 4 12 5 11 6 + 7 x y + 7 x y + 2 x y - x y - 4 x y - 19 x y 10 7 9 8 12 4 11 5 10 6 9 7 11 4 - 3 x y + 3 x y + x y + 16 x y + 7 x y - 5 x y - 6 x y 10 5 9 6 8 7 11 3 10 4 9 5 10 3 - 10 x y - x y - x y + x y + 7 x y + 5 x y - 2 x y 9 4 7 6 9 3 8 4 7 5 6 6 7 4 - 3 x y - 3 x y + x y + x y + 6 x y + 2 x y - 4 x y 6 5 6 4 5 5 7 2 6 3 5 4 6 2 4 4 - 2 x y + 3 x y + 3 x y + x y - x y - 2 x y - x y + 2 x y 4 2 3 3 3 2 3 2 2 2 / - x y - 4 x y + 3 x y - x y - 2 x y + 2 x y - x y + x - 1) / ( / 23 16 23 15 22 16 23 14 22 15 23 13 x y - 7 x y - x y + 21 x y + 6 x y - 35 x y 22 14 21 15 23 12 22 13 21 14 20 15 - 14 x y - x y + 35 x y + 14 x y + 5 x y + x y 23 11 21 13 20 14 23 10 22 11 - 21 x y - 10 x y - 4 x y + 7 x y - 14 x y 21 12 20 13 23 9 22 10 21 11 20 12 + 11 x y + 4 x y - x y + 14 x y - 10 x y + 4 x y 19 13 22 9 21 10 20 11 19 12 18 13 + x y - 6 x y + 11 x y - 11 x y - 4 x y - x y 22 8 21 9 20 10 19 11 18 12 21 8 + x y - 10 x y + 9 x y + 6 x y + 5 x y + 5 x y 20 9 19 10 18 11 17 12 21 7 20 8 - 6 x y - 5 x y - 10 x y - x y - x y + 6 x y 19 9 18 10 17 11 16 12 20 7 19 8 + 5 x y + 12 x y + 4 x y + x y - 4 x y - 6 x y 18 9 17 10 16 11 20 6 19 7 18 8 - 13 x y - 8 x y - 4 x y + x y + 4 x y + 13 x y 17 9 16 10 15 11 19 6 18 7 17 8 + 12 x y + 8 x y + 2 x y - x y - 8 x y - 13 x y 16 9 15 10 14 11 18 6 17 7 16 8 - 12 x y - 6 x y - x y + 2 x y + 8 x y + 12 x y 15 9 14 10 17 6 16 7 15 8 14 9 13 10 + 7 x y + x y - 2 x y - 5 x y - 3 x y + 2 x y - x y 16 6 15 7 14 8 13 9 16 5 15 6 14 7 - x y - 3 x y - 3 x y + x y + x y + 5 x y + 2 x y 12 9 15 5 14 6 12 8 14 5 13 6 12 7 + 2 x y - 2 x y - 2 x y - 5 x y + x y + 2 x y + 5 x y 13 5 12 6 11 7 13 4 12 5 11 6 11 5 - 3 x y - 3 x y + x y + x y + x y - 3 x y + 2 x y 9 7 11 4 10 5 11 3 10 4 8 6 10 3 9 4 - x y + x y + x y - x y - 3 x y + x y + 2 x y + 2 x y 7 6 9 3 8 4 7 5 8 3 7 4 6 5 7 3 6 4 + x y - x y - 2 x y + x y + x y + x y - x y - x y - x y 7 2 6 3 6 2 4 2 3 2 3 - x y + x y + x y - 2 x y + x y + x y - x y - x + 1) and in Maple notation -(x^22*y^16-7*x^22*y^15+21*x^22*y^14-x^21*y^15-x^20*y^16-35*x^22*y^13+7*x^21*y^ 14+8*x^20*y^15+35*x^22*y^12-21*x^21*y^13-33*x^20*y^14-3*x^19*y^15-21*x^22*y^11+ 35*x^21*y^12+89*x^20*y^13+23*x^19*y^14+x^18*y^15+7*x^22*y^10-35*x^21*y^11-164*x ^20*y^12-76*x^19*y^13-8*x^18*y^14-x^22*y^9+21*x^21*y^10+207*x^20*y^11+141*x^19* y^12+31*x^18*y^13+3*x^17*y^14-7*x^21*y^9-178*x^20*y^10-161*x^19*y^11-73*x^18*y^ 12-16*x^17*y^13+x^21*y^8+103*x^20*y^9+118*x^19*y^10+111*x^18*y^11+33*x^17*y^12-\ 39*x^20*y^8-58*x^19*y^9-111*x^18*y^10-29*x^17*y^11+x^16*y^12+9*x^20*y^7+21*x^19 *y^8+73*x^18*y^9+2*x^17*y^10-7*x^16*y^11-3*x^15*y^12-x^20*y^6-6*x^19*y^7-31*x^ 18*y^8+14*x^17*y^9+20*x^16*y^10+18*x^15*y^11+x^14*y^12+x^19*y^6+8*x^18*y^7-7*x^ 17*y^8-31*x^16*y^9-44*x^15*y^10-4*x^14*y^11-x^18*y^6-x^17*y^7+29*x^16*y^8+57*x^ 15*y^9+9*x^14*y^10+2*x^13*y^11+x^17*y^6-17*x^16*y^7-44*x^15*y^8-13*x^14*y^9-8*x ^13*y^10-x^12*y^11+6*x^16*y^6+23*x^15*y^7+11*x^14*y^8+13*x^13*y^9+4*x^12*y^10-x ^16*y^5-9*x^15*y^6-6*x^14*y^7-14*x^13*y^8-10*x^12*y^9-2*x^11*y^10+2*x^15*y^5+3* x^14*y^6+15*x^13*y^7+14*x^12*y^8+2*x^11*y^9-x^14*y^5-13*x^13*y^6-11*x^12*y^7+x^ 11*y^8-x^10*y^9+6*x^13*y^5+7*x^12*y^6+7*x^11*y^7+2*x^10*y^8-x^13*y^4-4*x^12*y^5 -19*x^11*y^6-3*x^10*y^7+3*x^9*y^8+x^12*y^4+16*x^11*y^5+7*x^10*y^6-5*x^9*y^7-6*x ^11*y^4-10*x^10*y^5-x^9*y^6-x^8*y^7+x^11*y^3+7*x^10*y^4+5*x^9*y^5-2*x^10*y^3-3* x^9*y^4-3*x^7*y^6+x^9*y^3+x^8*y^4+6*x^7*y^5+2*x^6*y^6-4*x^7*y^4-2*x^6*y^5+3*x^6 *y^4+3*x^5*y^5+x^7*y^2-x^6*y^3-2*x^5*y^4-x^6*y^2+2*x^4*y^4-x^4*y^2-4*x^3*y^3+3* x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^23*y^16-7*x^23*y^15-x^22*y^16+21*x^ 23*y^14+6*x^22*y^15-35*x^23*y^13-14*x^22*y^14-x^21*y^15+35*x^23*y^12+14*x^22*y^ 13+5*x^21*y^14+x^20*y^15-21*x^23*y^11-10*x^21*y^13-4*x^20*y^14+7*x^23*y^10-14*x ^22*y^11+11*x^21*y^12+4*x^20*y^13-x^23*y^9+14*x^22*y^10-10*x^21*y^11+4*x^20*y^ 12+x^19*y^13-6*x^22*y^9+11*x^21*y^10-11*x^20*y^11-4*x^19*y^12-x^18*y^13+x^22*y^ 8-10*x^21*y^9+9*x^20*y^10+6*x^19*y^11+5*x^18*y^12+5*x^21*y^8-6*x^20*y^9-5*x^19* y^10-10*x^18*y^11-x^17*y^12-x^21*y^7+6*x^20*y^8+5*x^19*y^9+12*x^18*y^10+4*x^17* y^11+x^16*y^12-4*x^20*y^7-6*x^19*y^8-13*x^18*y^9-8*x^17*y^10-4*x^16*y^11+x^20*y ^6+4*x^19*y^7+13*x^18*y^8+12*x^17*y^9+8*x^16*y^10+2*x^15*y^11-x^19*y^6-8*x^18*y ^7-13*x^17*y^8-12*x^16*y^9-6*x^15*y^10-x^14*y^11+2*x^18*y^6+8*x^17*y^7+12*x^16* y^8+7*x^15*y^9+x^14*y^10-2*x^17*y^6-5*x^16*y^7-3*x^15*y^8+2*x^14*y^9-x^13*y^10- x^16*y^6-3*x^15*y^7-3*x^14*y^8+x^13*y^9+x^16*y^5+5*x^15*y^6+2*x^14*y^7+2*x^12*y ^9-2*x^15*y^5-2*x^14*y^6-5*x^12*y^8+x^14*y^5+2*x^13*y^6+5*x^12*y^7-3*x^13*y^5-3 *x^12*y^6+x^11*y^7+x^13*y^4+x^12*y^5-3*x^11*y^6+2*x^11*y^5-x^9*y^7+x^11*y^4+x^ 10*y^5-x^11*y^3-3*x^10*y^4+x^8*y^6+2*x^10*y^3+2*x^9*y^4+x^7*y^6-x^9*y^3-2*x^8*y ^4+x^7*y^5+x^8*y^3+x^7*y^4-x^6*y^5-x^7*y^3-x^6*y^4-x^7*y^2+x^6*y^3+x^6*y^2-2*x^ 4*y^2+x^3*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 4 3 2 6 3 2 2 (a - 4 a - 2 a + 3 a + 2 a + 1) (7 a - 8 a + 6 a - 2) ---------------------------------------------------------------- 5 4 3 2 2 6 5 4 3 (6 a + 5 a + 4 a - 3 a + 2 a + 1) (a + a + a - 2 a - 1) 7 4 3 where a is the root of the polynomial, x - 2 x + 2 x - 2 x + 1, and in decimals this is, 0.5188222220 BTW the ratio for words with, 500, letters is, 0.5198670889 ------------------------------------------------ "Theorem Number 46" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 10 13 9 12 10 13 8 ) | ) C(m, n) x y | = (2 x y - 8 x y - x y + 12 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 9 13 7 12 8 11 9 13 6 12 7 11 8 + 2 x y - 8 x y + x y - 3 x y + 2 x y - 5 x y + 6 x y 12 6 11 7 10 8 12 5 11 6 10 7 9 8 + 4 x y + 2 x y + x y - x y - 12 x y - 3 x y + x y 11 5 10 6 9 7 11 4 10 5 9 6 8 7 + 9 x y + 5 x y - x y - 2 x y - 6 x y - 2 x y + 3 x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 + 4 x y + 3 x y - 4 x y - x y - x y - 3 x y + 10 x y 8 4 7 5 8 3 7 4 6 5 7 3 6 4 + 7 x y - 13 x y - 3 x y - x y + 7 x y + 5 x y - 12 x y 5 5 7 2 6 3 5 4 6 2 5 3 4 4 5 2 - x y - x y + 2 x y + x y + 2 x y - 4 x y - 3 x y + x y 4 3 5 4 2 3 3 3 2 2 2 2 - x y + x y + x y - 8 x y + 4 x y - 4 x y + x y - 2 x y - 1) / 10 6 10 5 10 4 10 3 8 5 8 4 7 5 / (x y - 3 x y + 3 x y - x y - x y + 2 x y - x y / 8 3 7 3 6 4 7 2 6 3 5 3 5 2 4 3 - x y + 2 x y - 2 x y - x y + 2 x y - x y + x y + x y 5 4 2 3 2 2 + x y + x y + 2 x y + x y - 1) and in Maple notation (2*x^13*y^10-8*x^13*y^9-x^12*y^10+12*x^13*y^8+2*x^12*y^9-8*x^13*y^7+x^12*y^8-3* x^11*y^9+2*x^13*y^6-5*x^12*y^7+6*x^11*y^8+4*x^12*y^6+2*x^11*y^7+x^10*y^8-x^12*y ^5-12*x^11*y^6-3*x^10*y^7+x^9*y^8+9*x^11*y^5+5*x^10*y^6-x^9*y^7-2*x^11*y^4-6*x^ 10*y^5-2*x^9*y^6+3*x^8*y^7+4*x^10*y^4+3*x^9*y^5-4*x^8*y^6-x^10*y^3-x^9*y^4-3*x^ 8*y^5+10*x^7*y^6+7*x^8*y^4-13*x^7*y^5-3*x^8*y^3-x^7*y^4+7*x^6*y^5+5*x^7*y^3-12* x^6*y^4-x^5*y^5-x^7*y^2+2*x^6*y^3+x^5*y^4+2*x^6*y^2-4*x^5*y^3-3*x^4*y^4+x^5*y^2 -x^4*y^3+x^5*y+x^4*y^2-8*x^3*y^3+4*x^3*y^2-4*x^2*y^2+x^2*y-2*x*y-1)/(x^10*y^6-3 *x^10*y^5+3*x^10*y^4-x^10*y^3-x^8*y^5+2*x^8*y^4-x^7*y^5-x^8*y^3+2*x^7*y^3-2*x^6 *y^4-x^7*y^2+2*x^6*y^3-x^5*y^3+x^5*y^2+x^4*y^3+x^5*y+x^4*y^2+2*x^3*y^2+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 11 9 8 7 6 5 4 3 2 (a - 6 a - 16 a - 25 a - 25 a - 13 a + a + 9 a + 10 a + 4 a + 1) / 2 3 2 2 2 / ((a + 1) (5 a + 8 a + 6 a + 2) (a + 1) a) / 5 4 3 2 where a is the root of the polynomial, x + 2 x + 2 x + x - 1, and in decimals this is, 0.5160132026 BTW the ratio for words with, 500, letters is, 0.5174553869 ------------------------------------------------ "Theorem Number 47" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 9 13 10 14 8 13 9 ) | ) C(m, n) x y | = - (x y + x y - 4 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 10 14 7 13 8 12 9 14 6 13 7 - x y + 6 x y + 9 x y + 3 x y - 4 x y - 7 x y 12 8 11 9 14 5 13 6 12 7 11 8 10 9 - 3 x y - x y + x y + 2 x y + x y + 5 x y + 2 x y 11 7 10 8 11 6 10 7 9 8 11 5 10 6 - 8 x y - 5 x y + 5 x y + 3 x y - 2 x y - x y + x y 9 7 10 5 8 7 10 4 8 6 10 3 9 4 + 3 x y - 2 x y + 2 x y + 2 x y - 5 x y - x y - 2 x y 8 5 7 6 9 3 8 4 7 5 6 6 8 3 + 5 x y + 4 x y + x y - 3 x y - 6 x y - 2 x y + 2 x y 7 4 6 5 8 2 7 3 6 4 5 5 6 3 5 4 + 3 x y + 2 x y - x y - x y + x y - 2 x y - x y + 3 x y 5 3 4 4 5 2 4 2 3 3 3 2 3 2 2 - 3 x y - 3 x y + x y + 2 x y + 4 x y - 3 x y + x y + 2 x y 2 / 15 9 15 8 14 9 15 7 - 2 x y + x y - x + 1) / (x y - 4 x y - x y + 6 x y / 14 8 15 6 14 7 15 5 14 6 12 8 12 7 + 3 x y - 4 x y - 3 x y + x y + x y + x y - 2 x y 12 6 11 7 11 6 10 7 11 5 10 6 9 7 + x y - x y + 2 x y - x y - 2 x y + 2 x y + x y 11 4 9 6 11 3 10 4 9 5 10 3 9 4 8 5 + 2 x y - x y - x y - 2 x y - x y + x y + x y + x y 7 6 9 3 9 2 8 3 6 5 7 3 6 4 6 3 4 2 - x y + x y - x y - x y + x y + x y - x y - x y + 2 x y 3 2 3 - x y - x y + x y + x - 1) and in Maple notation -(x^14*y^9+x^13*y^10-4*x^14*y^8-5*x^13*y^9-x^12*y^10+6*x^14*y^7+9*x^13*y^8+3*x^ 12*y^9-4*x^14*y^6-7*x^13*y^7-3*x^12*y^8-x^11*y^9+x^14*y^5+2*x^13*y^6+x^12*y^7+5 *x^11*y^8+2*x^10*y^9-8*x^11*y^7-5*x^10*y^8+5*x^11*y^6+3*x^10*y^7-2*x^9*y^8-x^11 *y^5+x^10*y^6+3*x^9*y^7-2*x^10*y^5+2*x^8*y^7+2*x^10*y^4-5*x^8*y^6-x^10*y^3-2*x^ 9*y^4+5*x^8*y^5+4*x^7*y^6+x^9*y^3-3*x^8*y^4-6*x^7*y^5-2*x^6*y^6+2*x^8*y^3+3*x^7 *y^4+2*x^6*y^5-x^8*y^2-x^7*y^3+x^6*y^4-2*x^5*y^5-x^6*y^3+3*x^5*y^4-3*x^5*y^3-3* x^4*y^4+x^5*y^2+2*x^4*y^2+4*x^3*y^3-3*x^3*y^2+x^3*y+2*x^2*y^2-2*x^2*y+x*y-x+1)/ (x^15*y^9-4*x^15*y^8-x^14*y^9+6*x^15*y^7+3*x^14*y^8-4*x^15*y^6-3*x^14*y^7+x^15* y^5+x^14*y^6+x^12*y^8-2*x^12*y^7+x^12*y^6-x^11*y^7+2*x^11*y^6-x^10*y^7-2*x^11*y ^5+2*x^10*y^6+x^9*y^7+2*x^11*y^4-x^9*y^6-x^11*y^3-2*x^10*y^4-x^9*y^5+x^10*y^3+x ^9*y^4+x^8*y^5-x^7*y^6+x^9*y^3-x^9*y^2-x^8*y^3+x^6*y^5+x^7*y^3-x^6*y^4-x^6*y^3+ 2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 4 7 6 5 4 3 2 5 3 2 (2 a + a - 2 a + 3 a + 3 a - 3 a - 2 a - 1) (3 a - 4 a + 3 a - 1) / 4 3 2 2 5 4 3 / ((5 a + 4 a - 3 a + 2 a + 1) (a + a - 2 a - 1)) / 6 4 3 where a is the root of the polynomial, x - 2 x + 2 x - 2 x + 1, and in decimals this is, 0.5124611828 BTW the ratio for words with, 500, letters is, 0.5132929531 ------------------------------------------------ "Theorem Number 48" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 9 13 10 14 8 13 9 ) | ) C(m, n) x y | = - (x y + x y - 4 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 10 14 7 13 8 12 9 14 6 13 7 - x y + 6 x y + 9 x y + 3 x y - 4 x y - 7 x y 12 8 11 9 14 5 13 6 12 7 11 8 10 9 - 3 x y - x y + x y + 2 x y + x y + 5 x y + 2 x y 11 7 10 8 11 6 10 7 9 8 11 5 10 6 - 8 x y - 5 x y + 5 x y + 3 x y - 2 x y - x y + x y 9 7 10 5 8 7 10 4 8 6 10 3 9 4 + 3 x y - 2 x y + 2 x y + 2 x y - 5 x y - x y - 2 x y 8 5 7 6 9 3 8 4 7 5 6 6 8 3 + 5 x y + 4 x y + x y - 3 x y - 6 x y - 2 x y + 2 x y 7 4 6 5 8 2 7 3 6 4 5 5 6 3 5 4 + 3 x y + 2 x y - x y - x y + x y - 2 x y - x y + 3 x y 5 3 4 4 5 2 4 2 3 3 3 2 3 2 2 - 3 x y - 3 x y + x y + 2 x y + 4 x y - 3 x y + x y + 2 x y 2 / 15 9 15 8 14 9 15 7 - 2 x y + x y - x + 1) / (x y - 4 x y - x y + 6 x y / 14 8 15 6 14 7 15 5 14 6 12 8 12 7 + 3 x y - 4 x y - 3 x y + x y + x y + x y - 2 x y 12 6 11 7 11 6 10 7 11 5 10 6 9 7 + x y - x y + 2 x y - x y - 2 x y + 2 x y + x y 11 4 9 6 11 3 10 4 9 5 10 3 9 4 8 5 + 2 x y - x y - x y - 2 x y - x y + x y + x y + x y 7 6 9 3 9 2 8 3 6 5 7 3 6 4 6 3 4 2 - x y + x y - x y - x y + x y + x y - x y - x y + 2 x y 3 2 3 - x y - x y + x y + x - 1) and in Maple notation -(x^14*y^9+x^13*y^10-4*x^14*y^8-5*x^13*y^9-x^12*y^10+6*x^14*y^7+9*x^13*y^8+3*x^ 12*y^9-4*x^14*y^6-7*x^13*y^7-3*x^12*y^8-x^11*y^9+x^14*y^5+2*x^13*y^6+x^12*y^7+5 *x^11*y^8+2*x^10*y^9-8*x^11*y^7-5*x^10*y^8+5*x^11*y^6+3*x^10*y^7-2*x^9*y^8-x^11 *y^5+x^10*y^6+3*x^9*y^7-2*x^10*y^5+2*x^8*y^7+2*x^10*y^4-5*x^8*y^6-x^10*y^3-2*x^ 9*y^4+5*x^8*y^5+4*x^7*y^6+x^9*y^3-3*x^8*y^4-6*x^7*y^5-2*x^6*y^6+2*x^8*y^3+3*x^7 *y^4+2*x^6*y^5-x^8*y^2-x^7*y^3+x^6*y^4-2*x^5*y^5-x^6*y^3+3*x^5*y^4-3*x^5*y^3-3* x^4*y^4+x^5*y^2+2*x^4*y^2+4*x^3*y^3-3*x^3*y^2+x^3*y+2*x^2*y^2-2*x^2*y+x*y-x+1)/ (x^15*y^9-4*x^15*y^8-x^14*y^9+6*x^15*y^7+3*x^14*y^8-4*x^15*y^6-3*x^14*y^7+x^15* y^5+x^14*y^6+x^12*y^8-2*x^12*y^7+x^12*y^6-x^11*y^7+2*x^11*y^6-x^10*y^7-2*x^11*y ^5+2*x^10*y^6+x^9*y^7+2*x^11*y^4-x^9*y^6-x^11*y^3-2*x^10*y^4-x^9*y^5+x^10*y^3+x ^9*y^4+x^8*y^5-x^7*y^6+x^9*y^3-x^9*y^2-x^8*y^3+x^6*y^5+x^7*y^3-x^6*y^4-x^6*y^3+ 2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 4 7 6 5 4 3 2 5 3 2 (2 a + a - 2 a + 3 a + 3 a - 3 a - 2 a - 1) (3 a - 4 a + 3 a - 1) / 4 3 2 2 5 4 3 / ((5 a + 4 a - 3 a + 2 a + 1) (a + a - 2 a - 1)) / 6 4 3 where a is the root of the polynomial, x - 2 x + 2 x - 2 x + 1, and in decimals this is, 0.5124611828 BTW the ratio for words with, 500, letters is, 0.5132929531 ------------------------------------------------ "Theorem Number 49" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 20 12 19 13 20 11 19 12 ) | ) C(m, n) x y | = (x y + x y - 7 x y - 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 13 20 10 19 11 18 12 20 9 19 10 - x y + 21 x y + 21 x y + 7 x y - 35 x y - 35 x y 18 11 17 12 20 8 19 9 18 10 17 11 - 22 x y - 2 x y + 35 x y + 35 x y + 41 x y + 12 x y 16 12 20 7 19 8 18 9 17 10 16 11 + x y - 21 x y - 21 x y - 50 x y - 30 x y - 6 x y 20 6 19 7 18 8 17 9 16 10 15 11 + 7 x y + 7 x y + 41 x y + 39 x y + 14 x y + x y 20 5 19 6 18 7 17 8 16 9 15 10 - x y - x y - 22 x y - 25 x y - 15 x y - 5 x y 14 11 18 6 17 7 16 8 15 9 14 10 - x y + 7 x y + 2 x y + 5 x y + 10 x y + 4 x y 13 11 18 5 17 6 16 7 15 8 14 9 13 10 + x y - x y + 8 x y + 4 x y - 8 x y - 4 x y - 5 x y 17 5 16 6 15 7 14 8 13 9 12 10 - 5 x y - 4 x y - 4 x y - 5 x y + 10 x y + 3 x y 17 4 16 5 15 6 14 7 13 8 12 9 11 10 + x y + x y + 15 x y + 14 x y - 7 x y - 9 x y - x y 15 5 14 6 13 7 12 8 11 9 15 4 - 14 x y - 10 x y - 5 x y + 9 x y + 2 x y + 6 x y 13 6 12 7 11 8 10 9 15 3 14 4 + 12 x y - 3 x y - 3 x y - 2 x y - x y + 3 x y 13 5 12 6 11 7 10 8 14 3 13 4 12 5 - 9 x y + x y + 4 x y + 3 x y - x y + 4 x y - 3 x y 11 6 10 7 9 8 13 3 12 4 11 5 10 6 - x y + 2 x y + 2 x y - x y + 3 x y - 2 x y - 5 x y 12 3 11 4 10 5 9 6 8 7 10 4 9 5 - x y + x y + 5 x y - 6 x y - x y - 6 x y + 5 x y 8 6 10 3 9 4 8 5 7 6 9 3 7 5 + x y + 3 x y - 3 x y - x y - 4 x y + 3 x y + 6 x y 6 6 9 2 8 3 7 4 6 5 8 2 7 3 + 2 x y - x y + 2 x y - 5 x y - 6 x y - x y + 3 x y 6 4 5 5 5 4 5 3 4 4 5 2 5 4 2 + 4 x y + 2 x y + 2 x y - 2 x y + 3 x y + x y - x y - 2 x y 3 3 3 2 3 2 2 2 / 15 9 - 4 x y + 3 x y - x y - 2 x y + 2 x y - x y + x - 1) / (x y / 15 8 14 9 15 7 14 8 15 6 14 7 15 5 - 4 x y - x y + 6 x y + 3 x y - 4 x y - 3 x y + x y 14 6 12 8 12 7 12 6 11 7 11 6 10 7 + x y + x y - 2 x y + x y - x y + 2 x y - x y 11 5 10 6 9 7 11 4 9 6 11 3 10 4 - 2 x y + 2 x y + x y + 2 x y - x y - x y - 2 x y 9 5 10 3 9 4 8 5 7 6 9 3 9 2 8 3 6 5 - x y + x y + x y + x y - x y + x y - x y - x y + x y 7 3 6 4 6 3 4 2 3 2 3 + x y - x y - x y + 2 x y - x y - x y + x y + x - 1) and in Maple notation (x^20*y^12+x^19*y^13-7*x^20*y^11-7*x^19*y^12-x^18*y^13+21*x^20*y^10+21*x^19*y^ 11+7*x^18*y^12-35*x^20*y^9-35*x^19*y^10-22*x^18*y^11-2*x^17*y^12+35*x^20*y^8+35 *x^19*y^9+41*x^18*y^10+12*x^17*y^11+x^16*y^12-21*x^20*y^7-21*x^19*y^8-50*x^18*y ^9-30*x^17*y^10-6*x^16*y^11+7*x^20*y^6+7*x^19*y^7+41*x^18*y^8+39*x^17*y^9+14*x^ 16*y^10+x^15*y^11-x^20*y^5-x^19*y^6-22*x^18*y^7-25*x^17*y^8-15*x^16*y^9-5*x^15* y^10-x^14*y^11+7*x^18*y^6+2*x^17*y^7+5*x^16*y^8+10*x^15*y^9+4*x^14*y^10+x^13*y^ 11-x^18*y^5+8*x^17*y^6+4*x^16*y^7-8*x^15*y^8-4*x^14*y^9-5*x^13*y^10-5*x^17*y^5-\ 4*x^16*y^6-4*x^15*y^7-5*x^14*y^8+10*x^13*y^9+3*x^12*y^10+x^17*y^4+x^16*y^5+15*x ^15*y^6+14*x^14*y^7-7*x^13*y^8-9*x^12*y^9-x^11*y^10-14*x^15*y^5-10*x^14*y^6-5*x ^13*y^7+9*x^12*y^8+2*x^11*y^9+6*x^15*y^4+12*x^13*y^6-3*x^12*y^7-3*x^11*y^8-2*x^ 10*y^9-x^15*y^3+3*x^14*y^4-9*x^13*y^5+x^12*y^6+4*x^11*y^7+3*x^10*y^8-x^14*y^3+4 *x^13*y^4-3*x^12*y^5-x^11*y^6+2*x^10*y^7+2*x^9*y^8-x^13*y^3+3*x^12*y^4-2*x^11*y ^5-5*x^10*y^6-x^12*y^3+x^11*y^4+5*x^10*y^5-6*x^9*y^6-x^8*y^7-6*x^10*y^4+5*x^9*y ^5+x^8*y^6+3*x^10*y^3-3*x^9*y^4-x^8*y^5-4*x^7*y^6+3*x^9*y^3+6*x^7*y^5+2*x^6*y^6 -x^9*y^2+2*x^8*y^3-5*x^7*y^4-6*x^6*y^5-x^8*y^2+3*x^7*y^3+4*x^6*y^4+2*x^5*y^5+2* x^5*y^4-2*x^5*y^3+3*x^4*y^4+x^5*y^2-x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^3*y-2 *x^2*y^2+2*x^2*y-x*y+x-1)/(x^15*y^9-4*x^15*y^8-x^14*y^9+6*x^15*y^7+3*x^14*y^8-4 *x^15*y^6-3*x^14*y^7+x^15*y^5+x^14*y^6+x^12*y^8-2*x^12*y^7+x^12*y^6-x^11*y^7+2* x^11*y^6-x^10*y^7-2*x^11*y^5+2*x^10*y^6+x^9*y^7+2*x^11*y^4-x^9*y^6-x^11*y^3-2*x ^10*y^4-x^9*y^5+x^10*y^3+x^9*y^4+x^8*y^5-x^7*y^6+x^9*y^3-x^9*y^2-x^8*y^3+x^6*y^ 5+x^7*y^3-x^6*y^4-x^6*y^3+2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5130510226 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 50" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 20 12 19 13 20 11 19 12 ) | ) C(m, n) x y | = (x y + x y - 7 x y - 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 13 20 10 19 11 18 12 20 9 19 10 - x y + 21 x y + 21 x y + 7 x y - 35 x y - 35 x y 18 11 17 12 20 8 19 9 18 10 17 11 - 22 x y - 2 x y + 35 x y + 35 x y + 41 x y + 12 x y 16 12 20 7 19 8 18 9 17 10 16 11 + x y - 21 x y - 21 x y - 50 x y - 30 x y - 6 x y 20 6 19 7 18 8 17 9 16 10 15 11 + 7 x y + 7 x y + 41 x y + 39 x y + 14 x y + x y 20 5 19 6 18 7 17 8 16 9 15 10 - x y - x y - 22 x y - 25 x y - 15 x y - 5 x y 14 11 18 6 17 7 16 8 15 9 14 10 - x y + 7 x y + 2 x y + 5 x y + 10 x y + 4 x y 13 11 18 5 17 6 16 7 15 8 14 9 13 10 + x y - x y + 8 x y + 4 x y - 8 x y - 4 x y - 5 x y 17 5 16 6 15 7 14 8 13 9 12 10 - 5 x y - 4 x y - 4 x y - 5 x y + 10 x y + 3 x y 17 4 16 5 15 6 14 7 13 8 12 9 11 10 + x y + x y + 15 x y + 14 x y - 7 x y - 9 x y - x y 15 5 14 6 13 7 12 8 11 9 15 4 - 14 x y - 10 x y - 5 x y + 9 x y + 2 x y + 6 x y 13 6 12 7 11 8 10 9 15 3 14 4 + 12 x y - 3 x y - 3 x y - 2 x y - x y + 3 x y 13 5 12 6 11 7 10 8 14 3 13 4 12 5 - 9 x y + x y + 4 x y + 3 x y - x y + 4 x y - 3 x y 11 6 10 7 9 8 13 3 12 4 11 5 10 6 - x y + 2 x y + 2 x y - x y + 3 x y - 2 x y - 5 x y 12 3 11 4 10 5 9 6 8 7 10 4 9 5 - x y + x y + 5 x y - 6 x y - x y - 6 x y + 5 x y 8 6 10 3 9 4 8 5 7 6 9 3 7 5 + x y + 3 x y - 3 x y - x y - 4 x y + 3 x y + 6 x y 6 6 9 2 8 3 7 4 6 5 8 2 7 3 + 2 x y - x y + 2 x y - 5 x y - 6 x y - x y + 3 x y 6 4 5 5 5 4 5 3 4 4 5 2 5 4 2 + 4 x y + 2 x y + 2 x y - 2 x y + 3 x y + x y - x y - 2 x y 3 3 3 2 3 2 2 2 / 14 9 - 4 x y + 3 x y - x y - 2 x y + 2 x y - x y + x - 1) / ((x y / 14 8 14 7 13 8 14 6 13 7 12 8 14 5 - 4 x y + 6 x y - x y - 4 x y + 3 x y - x y + x y 13 6 12 7 13 5 12 6 11 7 12 5 11 6 - 3 x y + 3 x y + x y - 3 x y + x y + x y - 2 x y 11 5 9 7 10 5 9 6 10 4 9 5 8 6 10 3 + x y - x y - x y + 2 x y + 2 x y - x y + x y - x y 8 5 7 6 8 4 7 5 8 3 7 4 6 5 8 2 6 4 - 2 x y + x y + x y - x y + x y + x y - x y - x y + x y 7 2 6 3 6 2 5 2 4 2 3 2 2 - x y + x y - x y - x y - x y + x y - x y - x y + 1) (x - 1)) and in Maple notation (x^20*y^12+x^19*y^13-7*x^20*y^11-7*x^19*y^12-x^18*y^13+21*x^20*y^10+21*x^19*y^ 11+7*x^18*y^12-35*x^20*y^9-35*x^19*y^10-22*x^18*y^11-2*x^17*y^12+35*x^20*y^8+35 *x^19*y^9+41*x^18*y^10+12*x^17*y^11+x^16*y^12-21*x^20*y^7-21*x^19*y^8-50*x^18*y ^9-30*x^17*y^10-6*x^16*y^11+7*x^20*y^6+7*x^19*y^7+41*x^18*y^8+39*x^17*y^9+14*x^ 16*y^10+x^15*y^11-x^20*y^5-x^19*y^6-22*x^18*y^7-25*x^17*y^8-15*x^16*y^9-5*x^15* y^10-x^14*y^11+7*x^18*y^6+2*x^17*y^7+5*x^16*y^8+10*x^15*y^9+4*x^14*y^10+x^13*y^ 11-x^18*y^5+8*x^17*y^6+4*x^16*y^7-8*x^15*y^8-4*x^14*y^9-5*x^13*y^10-5*x^17*y^5-\ 4*x^16*y^6-4*x^15*y^7-5*x^14*y^8+10*x^13*y^9+3*x^12*y^10+x^17*y^4+x^16*y^5+15*x ^15*y^6+14*x^14*y^7-7*x^13*y^8-9*x^12*y^9-x^11*y^10-14*x^15*y^5-10*x^14*y^6-5*x ^13*y^7+9*x^12*y^8+2*x^11*y^9+6*x^15*y^4+12*x^13*y^6-3*x^12*y^7-3*x^11*y^8-2*x^ 10*y^9-x^15*y^3+3*x^14*y^4-9*x^13*y^5+x^12*y^6+4*x^11*y^7+3*x^10*y^8-x^14*y^3+4 *x^13*y^4-3*x^12*y^5-x^11*y^6+2*x^10*y^7+2*x^9*y^8-x^13*y^3+3*x^12*y^4-2*x^11*y ^5-5*x^10*y^6-x^12*y^3+x^11*y^4+5*x^10*y^5-6*x^9*y^6-x^8*y^7-6*x^10*y^4+5*x^9*y ^5+x^8*y^6+3*x^10*y^3-3*x^9*y^4-x^8*y^5-4*x^7*y^6+3*x^9*y^3+6*x^7*y^5+2*x^6*y^6 -x^9*y^2+2*x^8*y^3-5*x^7*y^4-6*x^6*y^5-x^8*y^2+3*x^7*y^3+4*x^6*y^4+2*x^5*y^5+2* x^5*y^4-2*x^5*y^3+3*x^4*y^4+x^5*y^2-x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^3*y-2 *x^2*y^2+2*x^2*y-x*y+x-1)/(x^14*y^9-4*x^14*y^8+6*x^14*y^7-x^13*y^8-4*x^14*y^6+3 *x^13*y^7-x^12*y^8+x^14*y^5-3*x^13*y^6+3*x^12*y^7+x^13*y^5-3*x^12*y^6+x^11*y^7+ x^12*y^5-2*x^11*y^6+x^11*y^5-x^9*y^7-x^10*y^5+2*x^9*y^6+2*x^10*y^4-x^9*y^5+x^8* y^6-x^10*y^3-2*x^8*y^5+x^7*y^6+x^8*y^4-x^7*y^5+x^8*y^3+x^7*y^4-x^6*y^5-x^8*y^2+ x^6*y^4-x^7*y^2+x^6*y^3-x^6*y^2-x^5*y^2-x^4*y^2+x^3*y^2-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5130510226 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 51" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 8 12 7 11 8 12 6 ) | ) C(m, n) x y | = (x y - 4 x y + x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 7 12 5 11 6 11 5 10 6 9 7 11 4 - 2 x y - 2 x y - 2 x y + 6 x y + x y - x y - 3 x y 10 5 9 6 10 4 9 5 8 6 10 3 9 4 - 4 x y + 2 x y + 5 x y + x y - 4 x y - 2 x y - 5 x y 8 5 7 6 9 3 8 4 7 5 6 6 9 2 8 3 + 8 x y + 2 x y + 4 x y - 5 x y - x y + x y - x y + x y 7 4 6 5 7 3 6 4 7 2 6 3 5 4 + 4 x y + 6 x y - 6 x y - 10 x y + 2 x y + 4 x y - x y 5 3 4 4 5 2 4 3 3 3 3 2 3 2 2 - x y - 3 x y + 2 x y + 4 x y + 4 x y - x y - x y + 2 x y 2 / - x y + x y + 1) / ( / 6 2 5 3 6 5 2 5 3 2 3 2 x y - x y - x y + x y - x y + x y - x y - x y - x y + 1) and in Maple notation (x^12*y^8-4*x^12*y^7+x^11*y^8+5*x^12*y^6-2*x^11*y^7-2*x^12*y^5-2*x^11*y^6+6*x^ 11*y^5+x^10*y^6-x^9*y^7-3*x^11*y^4-4*x^10*y^5+2*x^9*y^6+5*x^10*y^4+x^9*y^5-4*x^ 8*y^6-2*x^10*y^3-5*x^9*y^4+8*x^8*y^5+2*x^7*y^6+4*x^9*y^3-5*x^8*y^4-x^7*y^5+x^6* y^6-x^9*y^2+x^8*y^3+4*x^7*y^4+6*x^6*y^5-6*x^7*y^3-10*x^6*y^4+2*x^7*y^2+4*x^6*y^ 3-x^5*y^4-x^5*y^3-3*x^4*y^4+2*x^5*y^2+4*x^4*y^3+4*x^3*y^3-x^3*y^2-x^3*y+2*x^2*y ^2-x^2*y+x*y+1)/(x^6*y^2-x^5*y^3-x^6*y+x^5*y^2-x^5*y+x^3*y^2-x^3*y-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ 11 10 9 bors of a random word of length n tends to n times, - 2 (4 a + 3 a - a 8 7 6 5 4 3 2 / + 4 a + 11 a - a - 5 a + 4 a + 5 a - 3 a - 2 a - 1) / ((a + 1) / 4 6 3 2 (5 a + 2 a + 1) (a + a + a + 1)) 5 2 where a is the root of the polynomial, x + x + x - 1, and in decimals this is, 0.5098970480 BTW the ratio for words with, 500, letters is, 0.5114362394 ------------------------------------------------ "Theorem Number 52" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 8 12 9 13 7 12 8 ) | ) C(m, n) x y | = (x y + x y - 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 6 12 7 13 5 12 6 11 7 12 5 11 6 + 3 x y - x y - x y - x y - 2 x y + 4 x y + 4 x y 10 7 12 4 11 5 10 6 9 7 8 8 10 5 + 2 x y - 2 x y - 2 x y - 5 x y - x y - x y + 5 x y 9 6 8 7 10 4 9 5 8 6 10 3 9 4 - 2 x y + x y - 3 x y + 2 x y - 2 x y + x y + 6 x y 7 6 9 3 8 4 7 5 6 6 9 2 8 3 7 4 + 2 x y - 6 x y + 5 x y - x y - x y + x y - 4 x y + x y 6 5 8 2 7 3 6 4 5 5 7 2 6 3 5 4 + x y + x y + 2 x y + x y + 4 x y - 4 x y + x y - 3 x y 7 6 2 5 3 4 4 5 2 4 3 5 4 2 + x y - 2 x y - 2 x y + 4 x y + 3 x y - 3 x y - 2 x y - x y 3 3 3 2 2 / 9 4 9 3 9 2 7 4 - 3 x y + x y - x y - x y - 1) / (x y - 2 x y + x y + x y / 7 2 6 3 7 5 3 6 3 3 4 3 2 2 - 2 x y - x y + x y - x y + x y - x y + x y + x y + x y + x y - 1) and in Maple notation (x^13*y^8+x^12*y^9-3*x^13*y^7-x^12*y^8+3*x^13*y^6-x^12*y^7-x^13*y^5-x^12*y^6-2* x^11*y^7+4*x^12*y^5+4*x^11*y^6+2*x^10*y^7-2*x^12*y^4-2*x^11*y^5-5*x^10*y^6-x^9* y^7-x^8*y^8+5*x^10*y^5-2*x^9*y^6+x^8*y^7-3*x^10*y^4+2*x^9*y^5-2*x^8*y^6+x^10*y^ 3+6*x^9*y^4+2*x^7*y^6-6*x^9*y^3+5*x^8*y^4-x^7*y^5-x^6*y^6+x^9*y^2-4*x^8*y^3+x^7 *y^4+x^6*y^5+x^8*y^2+2*x^7*y^3+x^6*y^4+4*x^5*y^5-4*x^7*y^2+x^6*y^3-3*x^5*y^4+x^ 7*y-2*x^6*y^2-2*x^5*y^3+4*x^4*y^4+3*x^5*y^2-3*x^4*y^3-2*x^5*y-x^4*y^2-3*x^3*y^3 +x^3*y-x^2*y^2-x*y-1)/(x^9*y^4-2*x^9*y^3+x^9*y^2+x^7*y^4-2*x^7*y^2-x^6*y^3+x^7* y-x^5*y^3+x^6*y-x^3*y^3+x^4*y+x^3*y+x^2*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 11 10 8 6 5 4 2 2 (a - 1) (2 a - 3 a - 2 a + 10 a + 5 a + 5 a - 6 a - 3 a - 1) ----------------------------------------------------------------------- 4 3 7 3 2 (5 a - 4 a - 2 a - 1) (a - 2 a - a - a - 1) 5 4 2 where a is the root of the polynomial, x - x - x - x + 1, and in decimals this is, 0.5035471282 BTW the ratio for words with, 500, letters is, 0.5047135551 ------------------------------------------------ "Theorem Number 53" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 21 13 21 12 20 13 ) | ) C(m, n) x y | = - (x y - 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 21 11 20 12 19 13 21 10 20 11 + 15 x y + 12 x y + 2 x y - 20 x y - 29 x y 19 12 21 9 20 10 19 11 18 12 21 8 - 8 x y + 15 x y + 34 x y + 7 x y + x y - 6 x y 20 9 19 10 18 11 17 12 21 7 20 8 - 15 x y + 13 x y - 2 x y - 2 x y + x y - 8 x y 19 9 18 10 17 11 16 12 20 7 19 8 - 30 x y - 7 x y + 4 x y + 2 x y + 13 x y + 18 x y 18 9 17 10 16 11 15 12 20 6 19 7 + 30 x y + 5 x y - 3 x y - x y - 6 x y + 3 x y 18 8 17 9 16 10 20 5 19 6 18 7 - 45 x y - 18 x y - 11 x y + x y - 7 x y + 34 x y 17 8 16 9 15 10 14 11 19 5 18 6 + 13 x y + 27 x y + 7 x y - 3 x y + 2 x y - 13 x y 16 8 15 9 14 10 18 5 17 6 16 7 - 8 x y - 7 x y + 7 x y + 2 x y - x y - 26 x y 15 8 14 9 12 11 17 5 16 6 15 7 - 9 x y - 7 x y - x y - 2 x y + 29 x y + 23 x y 14 8 13 9 12 10 17 4 16 5 15 6 + 11 x y + 6 x y + x y + x y - 13 x y - 22 x y 14 7 13 8 12 9 11 10 16 4 15 5 - 8 x y - 18 x y + 3 x y - 2 x y + 4 x y + 12 x y 14 6 13 7 12 8 11 9 16 3 15 4 - 13 x y + 17 x y - 2 x y + 4 x y - x y - 3 x y 14 5 13 6 12 7 11 8 10 9 14 4 + 19 x y - x y - 3 x y + 3 x y + 4 x y - 5 x y 13 5 12 6 11 7 10 8 14 3 13 4 12 5 - 10 x y + x y - 13 x y - 2 x y - x y + 11 x y + x y 11 6 10 7 9 8 13 3 12 4 11 5 + 19 x y - 2 x y + 3 x y - 7 x y + x y - 16 x y 10 6 9 7 13 2 12 3 11 4 10 5 9 6 - 7 x y - 4 x y + 2 x y - 2 x y + 3 x y + 7 x y - 4 x y 8 7 12 2 11 3 9 5 8 6 7 7 11 2 - 2 x y + x y + x y + 10 x y - x y + 2 x y + x y 9 4 8 5 7 6 10 2 9 3 8 4 7 5 - 8 x y + 7 x y - 2 x y + x y + 4 x y - 2 x y + x y 6 6 10 7 4 6 5 9 8 2 7 3 6 4 + 3 x y - x y + 3 x y - 5 x y - x y - x y - 8 x y - 2 x y 5 5 8 7 2 6 3 5 4 7 5 3 4 4 - 6 x y - x y + 2 x y - x y + x y + x y + x y - 5 x y 6 4 3 5 4 2 3 3 4 3 2 2 3 + 2 x y + 4 x y + x y + 2 x y + 3 x y - 2 x y - x y + x y - x / 20 11 20 10 20 9 20 8 20 7 + x y + 1) / (x y - 6 x y + 15 x y - 20 x y + 15 x y / 18 9 17 10 20 6 18 8 17 9 16 10 20 5 - 2 x y + x y - 6 x y + 10 x y - 5 x y - x y + x y 18 7 17 8 16 9 18 6 17 7 16 8 - 20 x y + 9 x y + 3 x y + 20 x y - 6 x y - 2 x y 15 9 18 5 17 6 16 7 15 8 18 4 17 5 - x y - 10 x y - x y - x y + x y + 2 x y + 3 x y 16 6 15 7 13 9 17 4 16 5 15 6 14 7 - x y + 5 x y - x y - x y + 5 x y - 9 x y + x y 13 8 16 4 15 5 14 6 13 7 16 3 15 4 + 2 x y - 4 x y + 2 x y - 4 x y + 2 x y + x y + 4 x y 14 5 13 6 12 7 11 8 15 3 14 4 12 6 + 6 x y - 5 x y - x y + x y - 2 x y - 4 x y + 3 x y 11 7 14 3 13 4 12 5 11 6 13 3 12 4 - x y + x y + 2 x y - x y + x y + x y - 2 x y 11 5 13 2 11 4 10 5 9 6 8 7 12 2 - 3 x y - x y + x y + 2 x y - x y + x y + x y 11 3 10 4 9 5 8 6 11 2 10 3 9 4 7 6 + 2 x y - 2 x y + x y - x y - x y - 2 x y + x y - x y 10 2 9 3 7 5 9 2 8 3 6 5 9 8 2 7 3 + 2 x y - x y + x y + x y - x y - x y - x y + x y + x y 7 2 5 4 5 3 5 2 3 3 3 2 2 3 - 2 x y + x y - x y - x y - x y + x y + x y + x + x y - 1) and in Maple notation -(x^21*y^13-6*x^21*y^12-2*x^20*y^13+15*x^21*y^11+12*x^20*y^12+2*x^19*y^13-20*x^ 21*y^10-29*x^20*y^11-8*x^19*y^12+15*x^21*y^9+34*x^20*y^10+7*x^19*y^11+x^18*y^12 -6*x^21*y^8-15*x^20*y^9+13*x^19*y^10-2*x^18*y^11-2*x^17*y^12+x^21*y^7-8*x^20*y^ 8-30*x^19*y^9-7*x^18*y^10+4*x^17*y^11+2*x^16*y^12+13*x^20*y^7+18*x^19*y^8+30*x^ 18*y^9+5*x^17*y^10-3*x^16*y^11-x^15*y^12-6*x^20*y^6+3*x^19*y^7-45*x^18*y^8-18*x ^17*y^9-11*x^16*y^10+x^20*y^5-7*x^19*y^6+34*x^18*y^7+13*x^17*y^8+27*x^16*y^9+7* x^15*y^10-3*x^14*y^11+2*x^19*y^5-13*x^18*y^6-8*x^16*y^8-7*x^15*y^9+7*x^14*y^10+ 2*x^18*y^5-x^17*y^6-26*x^16*y^7-9*x^15*y^8-7*x^14*y^9-x^12*y^11-2*x^17*y^5+29*x ^16*y^6+23*x^15*y^7+11*x^14*y^8+6*x^13*y^9+x^12*y^10+x^17*y^4-13*x^16*y^5-22*x^ 15*y^6-8*x^14*y^7-18*x^13*y^8+3*x^12*y^9-2*x^11*y^10+4*x^16*y^4+12*x^15*y^5-13* x^14*y^6+17*x^13*y^7-2*x^12*y^8+4*x^11*y^9-x^16*y^3-3*x^15*y^4+19*x^14*y^5-x^13 *y^6-3*x^12*y^7+3*x^11*y^8+4*x^10*y^9-5*x^14*y^4-10*x^13*y^5+x^12*y^6-13*x^11*y ^7-2*x^10*y^8-x^14*y^3+11*x^13*y^4+x^12*y^5+19*x^11*y^6-2*x^10*y^7+3*x^9*y^8-7* x^13*y^3+x^12*y^4-16*x^11*y^5-7*x^10*y^6-4*x^9*y^7+2*x^13*y^2-2*x^12*y^3+3*x^11 *y^4+7*x^10*y^5-4*x^9*y^6-2*x^8*y^7+x^12*y^2+x^11*y^3+10*x^9*y^5-x^8*y^6+2*x^7* y^7+x^11*y^2-8*x^9*y^4+7*x^8*y^5-2*x^7*y^6+x^10*y^2+4*x^9*y^3-2*x^8*y^4+x^7*y^5 +3*x^6*y^6-x^10*y+3*x^7*y^4-5*x^6*y^5-x^9*y-x^8*y^2-8*x^7*y^3-2*x^6*y^4-6*x^5*y ^5-x^8*y+2*x^7*y^2-x^6*y^3+x^5*y^4+x^7*y+x^5*y^3-5*x^4*y^4+2*x^6*y+4*x^4*y^3+x^ 5*y+2*x^4*y^2+3*x^3*y^3-2*x^4*y-x^3*y+x^2*y^2-x^3+x*y+1)/(x^20*y^11-6*x^20*y^10 +15*x^20*y^9-20*x^20*y^8+15*x^20*y^7-2*x^18*y^9+x^17*y^10-6*x^20*y^6+10*x^18*y^ 8-5*x^17*y^9-x^16*y^10+x^20*y^5-20*x^18*y^7+9*x^17*y^8+3*x^16*y^9+20*x^18*y^6-6 *x^17*y^7-2*x^16*y^8-x^15*y^9-10*x^18*y^5-x^17*y^6-x^16*y^7+x^15*y^8+2*x^18*y^4 +3*x^17*y^5-x^16*y^6+5*x^15*y^7-x^13*y^9-x^17*y^4+5*x^16*y^5-9*x^15*y^6+x^14*y^ 7+2*x^13*y^8-4*x^16*y^4+2*x^15*y^5-4*x^14*y^6+2*x^13*y^7+x^16*y^3+4*x^15*y^4+6* x^14*y^5-5*x^13*y^6-x^12*y^7+x^11*y^8-2*x^15*y^3-4*x^14*y^4+3*x^12*y^6-x^11*y^7 +x^14*y^3+2*x^13*y^4-x^12*y^5+x^11*y^6+x^13*y^3-2*x^12*y^4-3*x^11*y^5-x^13*y^2+ x^11*y^4+2*x^10*y^5-x^9*y^6+x^8*y^7+x^12*y^2+2*x^11*y^3-2*x^10*y^4+x^9*y^5-x^8* y^6-x^11*y^2-2*x^10*y^3+x^9*y^4-x^7*y^6+2*x^10*y^2-x^9*y^3+x^7*y^5+x^9*y^2-x^8* y^3-x^6*y^5-x^9*y+x^8*y^2+x^7*y^3-2*x^7*y^2+x^5*y^4-x^5*y^3-x^5*y^2-x^3*y^3+x^3 *y+x^2*y^2+x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, 2 (a + 2 a - a 11 10 9 8 7 6 5 4 3 - 8 a - 14 a - 8 a + 8 a + 22 a + 16 a - 4 a - 14 a - 8 a 2 / 6 5 4 2 + 3 a + 2 a + 1) / ((a + 1) (a + 2 a + a - a - 1) / 6 5 4 2 (7 a + 6 a + 5 a - 3 a - 2 a - 1)) 7 6 5 3 2 where a is the root of the polynomial, x + x + x - x - x - x + 1, and in decimals this is, 0.4969815398 BTW the ratio for words with, 500, letters is, 0.4977828792 ------------------------------------------------ "Theorem Number 54" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 13 19 12 19 11 18 12 ) | ) C(m, n) x y | = (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 19 10 18 11 17 12 19 9 18 10 17 11 - 10 x y - 4 x y - x y + 5 x y + 6 x y + 7 x y 19 8 18 9 17 10 16 11 18 8 17 9 - x y - 4 x y - 19 x y - 3 x y + x y + 26 x y 16 10 17 8 16 9 15 10 17 7 16 8 + 14 x y - 19 x y - 25 x y - 8 x y + 7 x y + 21 x y 15 9 17 6 16 7 15 8 14 9 16 6 + 29 x y - x y - 8 x y - 39 x y - 7 x y + x y 15 7 14 8 13 9 15 6 14 7 13 8 + 23 x y + 20 x y + 8 x y - 5 x y - 19 x y - 32 x y 12 9 14 6 13 7 12 8 13 6 12 7 + x y + 6 x y + 49 x y + x y - 35 x y - 10 x y 11 8 13 5 12 6 11 7 13 4 12 5 - 4 x y + 11 x y + 13 x y + 18 x y - x y - 5 x y 11 6 10 7 11 5 10 6 9 7 11 4 - 31 x y - 6 x y + 24 x y + 14 x y + 4 x y - 7 x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 - 13 x y - 10 x y + x y + 6 x y + 14 x y + 8 x y - x y 9 4 8 5 7 6 9 3 8 4 7 5 8 3 - 13 x y - 13 x y - 2 x y + 5 x y + 8 x y + 8 x y - 4 x y 7 4 6 5 8 2 7 3 6 4 5 5 7 2 - 12 x y - 8 x y + x y + 8 x y + 11 x y - x y - 2 x y 6 3 5 4 6 2 5 3 4 4 6 5 2 4 3 - 5 x y + x y + 3 x y + x y + 5 x y - x y - 2 x y - 2 x y 5 4 2 3 3 4 3 2 2 2 2 + x y - x y - 4 x y - x y + 3 x y - 2 x y + 2 x y - x y + x - 1 / 14 8 14 7 14 6 13 7 14 5 13 6 12 7 ) / (x y - 3 x y + 3 x y + x y - x y - 2 x y - x y / 13 5 12 6 12 5 11 6 12 4 11 5 10 6 + x y + 3 x y - 3 x y - x y + x y + 2 x y + x y 11 4 10 5 10 4 10 3 8 5 8 4 8 3 - x y - 3 x y + 3 x y - x y - x y + 2 x y - 2 x y 7 4 8 2 6 2 5 3 6 5 2 5 4 2 4 - x y + x y + x y + x y - x y - x y + x y + x y - x y 3 2 - x y + x y + x - 1) and in Maple notation (x^19*y^13-5*x^19*y^12+10*x^19*y^11+x^18*y^12-10*x^19*y^10-4*x^18*y^11-x^17*y^ 12+5*x^19*y^9+6*x^18*y^10+7*x^17*y^11-x^19*y^8-4*x^18*y^9-19*x^17*y^10-3*x^16*y ^11+x^18*y^8+26*x^17*y^9+14*x^16*y^10-19*x^17*y^8-25*x^16*y^9-8*x^15*y^10+7*x^ 17*y^7+21*x^16*y^8+29*x^15*y^9-x^17*y^6-8*x^16*y^7-39*x^15*y^8-7*x^14*y^9+x^16* y^6+23*x^15*y^7+20*x^14*y^8+8*x^13*y^9-5*x^15*y^6-19*x^14*y^7-32*x^13*y^8+x^12* y^9+6*x^14*y^6+49*x^13*y^7+x^12*y^8-35*x^13*y^6-10*x^12*y^7-4*x^11*y^8+11*x^13* y^5+13*x^12*y^6+18*x^11*y^7-x^13*y^4-5*x^12*y^5-31*x^11*y^6-6*x^10*y^7+24*x^11* y^5+14*x^10*y^6+4*x^9*y^7-7*x^11*y^4-13*x^10*y^5-10*x^9*y^6+x^8*y^7+6*x^10*y^4+ 14*x^9*y^5+8*x^8*y^6-x^10*y^3-13*x^9*y^4-13*x^8*y^5-2*x^7*y^6+5*x^9*y^3+8*x^8*y ^4+8*x^7*y^5-4*x^8*y^3-12*x^7*y^4-8*x^6*y^5+x^8*y^2+8*x^7*y^3+11*x^6*y^4-x^5*y^ 5-2*x^7*y^2-5*x^6*y^3+x^5*y^4+3*x^6*y^2+x^5*y^3+5*x^4*y^4-x^6*y-2*x^5*y^2-2*x^4 *y^3+x^5*y-x^4*y^2-4*x^3*y^3-x^4*y+3*x^3*y^2-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^14*y ^8-3*x^14*y^7+3*x^14*y^6+x^13*y^7-x^14*y^5-2*x^13*y^6-x^12*y^7+x^13*y^5+3*x^12* y^6-3*x^12*y^5-x^11*y^6+x^12*y^4+2*x^11*y^5+x^10*y^6-x^11*y^4-3*x^10*y^5+3*x^10 *y^4-x^10*y^3-x^8*y^5+2*x^8*y^4-2*x^8*y^3-x^7*y^4+x^8*y^2+x^6*y^2+x^5*y^3-x^6*y -x^5*y^2+x^5*y+x^4*y^2-x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, 2 (a + 2 a - 2 a 11 10 9 8 7 6 5 4 3 2 - 6 a - a + 4 a - 5 a - 12 a + a + 6 a - 2 a - 6 a + 3 a + 2 a 6 4 2 / 5 4 2 + 1) (7 a - 5 a + 3 a - 2) / ((6 a + 5 a + 2 a + 1) / 8 4 3 (a + a - a - 1)) 7 5 3 where a is the root of the polynomial, x - x + x - 2 x + 1, and in decimals this is, 0.4953488882 BTW the ratio for words with, 500, letters is, 0.4966536393 ------------------------------------------------ "Theorem Number 55" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 9 12 8 11 9 12 7 ) | ) C(m, n) x y | = (x y - 4 x y - x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 8 10 9 12 6 11 7 10 8 12 5 11 6 + 3 x y - 2 x y - 4 x y - 3 x y + 5 x y + x y + x y 10 7 9 8 10 6 9 7 10 5 9 6 9 5 8 6 - 3 x y + 2 x y - x y - 4 x y + x y + x y + 2 x y + x y 9 4 8 5 7 6 8 4 7 5 6 6 8 3 7 4 - x y - 3 x y - x y + 3 x y + 5 x y + 2 x y - x y - 7 x y 6 5 7 3 6 4 5 5 7 2 6 3 6 2 5 3 - 3 x y + 4 x y + x y + 2 x y - x y + x y - x y - 2 x y 4 4 4 3 4 2 3 3 4 3 2 2 2 3 2 + 3 x y - 4 x y - x y - 4 x y + x y + x y - 2 x y + x + x y / 12 9 12 8 12 7 11 8 12 6 11 7 - x y - 1) / (x y - 3 x y + 3 x y - x y - x y + 3 x y / 11 6 11 5 10 6 10 5 10 4 7 6 8 4 7 5 - 3 x y + x y - x y + 2 x y - x y - x y - x y + x y 8 3 6 5 7 3 6 4 7 2 6 2 5 3 4 2 4 + x y + x y + x y - 2 x y - x y + x y + x y + x y - x y 3 2 3 2 - x y + x + x y + x y - 1) and in Maple notation (x^12*y^9-4*x^12*y^8-x^11*y^9+6*x^12*y^7+3*x^11*y^8-2*x^10*y^9-4*x^12*y^6-3*x^ 11*y^7+5*x^10*y^8+x^12*y^5+x^11*y^6-3*x^10*y^7+2*x^9*y^8-x^10*y^6-4*x^9*y^7+x^ 10*y^5+x^9*y^6+2*x^9*y^5+x^8*y^6-x^9*y^4-3*x^8*y^5-x^7*y^6+3*x^8*y^4+5*x^7*y^5+ 2*x^6*y^6-x^8*y^3-7*x^7*y^4-3*x^6*y^5+4*x^7*y^3+x^6*y^4+2*x^5*y^5-x^7*y^2+x^6*y ^3-x^6*y^2-2*x^5*y^3+3*x^4*y^4-4*x^4*y^3-x^4*y^2-4*x^3*y^3+x^4*y+x^3*y^2-2*x^2* y^2+x^3+x^2*y-x*y-1)/(x^12*y^9-3*x^12*y^8+3*x^12*y^7-x^11*y^8-x^12*y^6+3*x^11*y ^7-3*x^11*y^6+x^11*y^5-x^10*y^6+2*x^10*y^5-x^10*y^4-x^7*y^6-x^8*y^4+x^7*y^5+x^8 *y^3+x^6*y^5+x^7*y^3-2*x^6*y^4-x^7*y^2+x^6*y^2+x^5*y^3+x^4*y^2-x^4*y-x^3*y^2+x^ 3+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 9 7 6 4 3 2 2 (a - a + 2 a - 5 a - 5 a + 3 a + 2 a + 1) ------------------------------------------------- 4 4 3 2 (5 a + 2 a + 1) (a + 2 a + a + a + 1) 5 2 where a is the root of the polynomial, x + x + x - 1, and in decimals this is, 0.4926205404 BTW the ratio for words with, 500, letters is, 0.4939332226 ------------------------------------------------ "Theorem Number 56" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 17 12 18 10 ) | ) C(m, n) x y | = (x y - 4 x y + x y + 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 11 16 12 18 9 17 10 16 11 15 12 - 4 x y + 2 x y - 8 x y + 6 x y - 9 x y - 4 x y 18 8 17 9 16 10 15 11 18 7 17 8 + 7 x y - 5 x y + 15 x y + 13 x y - 4 x y + 5 x y 16 9 15 10 14 11 18 6 17 7 16 8 - 11 x y - 13 x y + 4 x y + x y - 6 x y + 3 x y 14 10 17 6 16 7 15 8 14 9 17 5 - 15 x y + 4 x y - x y + 7 x y + 19 x y - x y 16 6 15 7 14 8 13 9 16 5 15 6 + 3 x y - 2 x y - 7 x y + 3 x y - 3 x y - 2 x y 14 7 13 8 12 9 16 4 15 5 14 6 13 7 - 4 x y - 9 x y - 2 x y + x y + x y + 4 x y + 7 x y 12 8 11 9 14 5 13 6 12 7 11 8 10 9 + 5 x y - x y - 2 x y + x y - 7 x y + x y + 4 x y 14 4 13 5 12 6 11 7 10 8 14 3 13 4 + 2 x y - 2 x y + 8 x y + 7 x y - 5 x y - x y + x y 12 5 11 6 10 7 9 8 13 3 11 5 - 5 x y - 12 x y - 3 x y - 3 x y - 2 x y + 5 x y 10 6 9 7 13 2 12 3 11 4 10 5 9 6 + 2 x y + 8 x y + x y + 2 x y + 2 x y + 8 x y - x y 8 7 12 2 11 3 10 4 9 5 8 6 10 3 - x y - x y - 2 x y - 7 x y - 7 x y - 2 x y + 2 x y 9 4 8 5 7 6 10 2 9 3 8 4 7 5 6 6 + 3 x y + 5 x y + x y - 2 x y - x y - 7 x y - 9 x y - x y 10 9 2 8 3 7 4 6 5 9 7 3 6 4 + x y + x y + 6 x y + 16 x y + 6 x y + x y - 7 x y - 2 x y 5 5 9 8 6 3 7 6 2 5 3 4 4 6 - x y - x - 2 x y - 4 x y - x y + x y - 2 x y - 5 x y - x y 5 2 4 3 6 5 4 2 3 3 4 3 2 4 - x y + 2 x y + x + 2 x y + 2 x y + 4 x y - x y - 3 x y + x 3 2 2 3 2 / 12 9 12 8 + x y + 2 x y - x - 2 x y + x y - x + 1) / ((x y - 3 x y / 12 7 11 8 12 6 11 7 11 6 11 5 10 6 + 3 x y - x y - x y + 3 x y - 3 x y + x y - x y 10 5 10 4 7 6 8 4 7 5 8 3 6 5 7 3 + 2 x y - x y - x y - x y + x y + x y + x y + x y 6 4 7 2 6 2 5 3 4 2 4 3 2 3 2 - 2 x y - x y + x y + x y + x y - x y - x y + x + x y + x y - 1) (x - 1)) and in Maple notation (x^18*y^12-4*x^18*y^11+x^17*y^12+7*x^18*y^10-4*x^17*y^11+2*x^16*y^12-8*x^18*y^9 +6*x^17*y^10-9*x^16*y^11-4*x^15*y^12+7*x^18*y^8-5*x^17*y^9+15*x^16*y^10+13*x^15 *y^11-4*x^18*y^7+5*x^17*y^8-11*x^16*y^9-13*x^15*y^10+4*x^14*y^11+x^18*y^6-6*x^ 17*y^7+3*x^16*y^8-15*x^14*y^10+4*x^17*y^6-x^16*y^7+7*x^15*y^8+19*x^14*y^9-x^17* y^5+3*x^16*y^6-2*x^15*y^7-7*x^14*y^8+3*x^13*y^9-3*x^16*y^5-2*x^15*y^6-4*x^14*y^ 7-9*x^13*y^8-2*x^12*y^9+x^16*y^4+x^15*y^5+4*x^14*y^6+7*x^13*y^7+5*x^12*y^8-x^11 *y^9-2*x^14*y^5+x^13*y^6-7*x^12*y^7+x^11*y^8+4*x^10*y^9+2*x^14*y^4-2*x^13*y^5+8 *x^12*y^6+7*x^11*y^7-5*x^10*y^8-x^14*y^3+x^13*y^4-5*x^12*y^5-12*x^11*y^6-3*x^10 *y^7-3*x^9*y^8-2*x^13*y^3+5*x^11*y^5+2*x^10*y^6+8*x^9*y^7+x^13*y^2+2*x^12*y^3+2 *x^11*y^4+8*x^10*y^5-x^9*y^6-x^8*y^7-x^12*y^2-2*x^11*y^3-7*x^10*y^4-7*x^9*y^5-2 *x^8*y^6+2*x^10*y^3+3*x^9*y^4+5*x^8*y^5+x^7*y^6-2*x^10*y^2-x^9*y^3-7*x^8*y^4-9* x^7*y^5-x^6*y^6+x^10*y+x^9*y^2+6*x^8*y^3+16*x^7*y^4+6*x^6*y^5+x^9*y-7*x^7*y^3-2 *x^6*y^4-x^5*y^5-x^9-2*x^8*y-4*x^6*y^3-x^7*y+x^6*y^2-2*x^5*y^3-5*x^4*y^4-x^6*y- x^5*y^2+2*x^4*y^3+x^6+2*x^5*y+2*x^4*y^2+4*x^3*y^3-x^4*y-3*x^3*y^2+x^4+x^3*y+2*x ^2*y^2-x^3-2*x^2*y+x*y-x+1)/(x^12*y^9-3*x^12*y^8+3*x^12*y^7-x^11*y^8-x^12*y^6+3 *x^11*y^7-3*x^11*y^6+x^11*y^5-x^10*y^6+2*x^10*y^5-x^10*y^4-x^7*y^6-x^8*y^4+x^7* y^5+x^8*y^3+x^6*y^5+x^7*y^3-2*x^6*y^4-x^7*y^2+x^6*y^2+x^5*y^3+x^4*y^2-x^4*y-x^3 *y^2+x^3+x^2*y+x*y-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 5 4 2 14 13 12 11 10 9 8 (6 a - 5 a + 3 a - 2) (a + 2 a + 3 a + 2 a + 6 a + 8 a + 3 a 7 6 5 4 3 2 / - 2 a + 6 a + 2 a - 8 a - 6 a + 3 a + 2 a + 1) / ( / 4 2 8 5 4 3 (5 a + 2 a + 1) (a + 2 a + a - a - 1)) 5 2 where a is the root of the polynomial, (x + x + x - 1) (x - 1), and in decimals this is, 0.4926205406 BTW the ratio for words with, 500, letters is, 0.4935278608 ------------------------------------------------ "Theorem Number 57" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 1, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = - (x y - 4 x y + x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 16 6 15 7 14 8 13 9 16 5 15 6 - 3 x y - 4 x y + 2 x y + 2 x y - x y + x y + 2 x y 14 7 13 8 12 9 15 5 14 6 13 7 12 8 - 9 x y + 3 x y + x y - 3 x y + 15 x y - x y - 2 x y 15 4 14 5 13 6 12 7 11 8 14 4 13 5 + x y - 11 x y - 5 x y + 3 x y - x y + 3 x y + 5 x y 12 6 11 7 10 8 12 5 10 7 13 3 12 4 - 2 x y + 3 x y + x y - 8 x y - 5 x y - x y + 14 x y 11 5 10 6 9 7 12 3 11 4 10 5 9 6 - 3 x y + 9 x y + x y - 6 x y - 2 x y - 3 x y - 4 x y 11 3 10 4 9 5 8 6 11 2 9 4 8 5 + 4 x y - 4 x y + x y + 4 x y - x y + 6 x y - 4 x y 7 6 10 2 9 3 8 4 7 5 9 2 8 3 7 4 - x y + 2 x y - 4 x y + x y - x y - x y + 2 x y + 2 x y 6 5 9 8 2 7 3 6 4 5 5 6 3 5 4 + 5 x y + x y - 2 x y + x y - 8 x y - x y + 5 x y + x y 7 6 2 5 3 4 4 5 2 4 3 4 2 3 3 - x y - x y - 2 x y - 5 x y + 2 x y + 2 x y + x y + 4 x y 4 3 2 2 2 2 / 11 4 11 3 + x y - 3 x y + 2 x y - 2 x y + x y - x + 1) / (x y - 2 x y / 11 2 9 3 8 4 8 3 7 4 9 7 3 7 4 2 + x y + x y - x y + x y + x y - x y - x y + x y + x y 4 3 2 - x y - x y + x y + x - 1) and in Maple notation -(x^16*y^9-4*x^16*y^8+x^15*y^9+6*x^16*y^7-3*x^15*y^8-4*x^16*y^6+2*x^15*y^7+2*x^ 14*y^8-x^13*y^9+x^16*y^5+2*x^15*y^6-9*x^14*y^7+3*x^13*y^8+x^12*y^9-3*x^15*y^5+ 15*x^14*y^6-x^13*y^7-2*x^12*y^8+x^15*y^4-11*x^14*y^5-5*x^13*y^6+3*x^12*y^7-x^11 *y^8+3*x^14*y^4+5*x^13*y^5-2*x^12*y^6+3*x^11*y^7+x^10*y^8-8*x^12*y^5-5*x^10*y^7 -x^13*y^3+14*x^12*y^4-3*x^11*y^5+9*x^10*y^6+x^9*y^7-6*x^12*y^3-2*x^11*y^4-3*x^ 10*y^5-4*x^9*y^6+4*x^11*y^3-4*x^10*y^4+x^9*y^5+4*x^8*y^6-x^11*y^2+6*x^9*y^4-4*x ^8*y^5-x^7*y^6+2*x^10*y^2-4*x^9*y^3+x^8*y^4-x^7*y^5-x^9*y^2+2*x^8*y^3+2*x^7*y^4 +5*x^6*y^5+x^9*y-2*x^8*y^2+x^7*y^3-8*x^6*y^4-x^5*y^5+5*x^6*y^3+x^5*y^4-x^7*y-x^ 6*y^2-2*x^5*y^3-5*x^4*y^4+2*x^5*y^2+2*x^4*y^3+x^4*y^2+4*x^3*y^3+x^4*y-3*x^3*y^2 +2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^11*y^4-2*x^11*y^3+x^11*y^2+x^9*y^3-x^8*y^4+x^8*y ^3+x^7*y^4-x^9*y-x^7*y^3+x^7*y+x^4*y^2-x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 16 14 12 bors of a random word of length n tends to n times, 2 (a - 3 a - 3 a 11 10 9 8 7 6 5 4 3 / + 3 a + 8 a - a - 8 a + 3 a - 9 a + 8 a + 10 a - 10 a + 1) / / 6 2 8 6 4 3 ((7 a - 3 a + 2) (a + a - a + a + 1)) 7 3 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.4780162458 BTW the ratio for words with, 500, letters is, 0.4790746569 ------------------------------------------------ "Theorem Number 58" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = - (x y - 4 x y + x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 16 6 15 7 14 8 13 9 16 5 15 6 - 3 x y - 4 x y + 2 x y + 2 x y - x y + x y + 2 x y 14 7 13 8 12 9 15 5 14 6 13 7 12 8 - 9 x y + 3 x y + x y - 3 x y + 15 x y - x y - 2 x y 15 4 14 5 13 6 12 7 11 8 14 4 13 5 + x y - 11 x y - 5 x y + 3 x y - x y + 3 x y + 5 x y 12 6 11 7 10 8 12 5 10 7 13 3 12 4 - 2 x y + 3 x y + x y - 8 x y - 5 x y - x y + 14 x y 11 5 10 6 9 7 12 3 11 4 10 5 9 6 - 3 x y + 9 x y + x y - 6 x y - 2 x y - 3 x y - 4 x y 11 3 10 4 9 5 8 6 11 2 9 4 8 5 + 4 x y - 4 x y + x y + 4 x y - x y + 6 x y - 4 x y 7 6 10 2 9 3 8 4 7 5 9 2 8 3 7 4 - x y + 2 x y - 4 x y + x y - x y - x y + 2 x y + 2 x y 6 5 9 8 2 7 3 6 4 5 5 6 3 5 4 + 5 x y + x y - 2 x y + x y - 8 x y - x y + 5 x y + x y 7 6 2 5 3 4 4 5 2 4 3 4 2 3 3 - x y - x y - 2 x y - 5 x y + 2 x y + 2 x y + x y + 4 x y 4 3 2 2 2 2 / 11 4 11 3 + x y - 3 x y + 2 x y - 2 x y + x y - x + 1) / (x y - 2 x y / 11 2 9 3 8 4 8 3 7 4 9 7 3 7 4 2 + x y + x y - x y + x y + x y - x y - x y + x y + x y 4 3 2 - x y - x y + x y + x - 1) and in Maple notation -(x^16*y^9-4*x^16*y^8+x^15*y^9+6*x^16*y^7-3*x^15*y^8-4*x^16*y^6+2*x^15*y^7+2*x^ 14*y^8-x^13*y^9+x^16*y^5+2*x^15*y^6-9*x^14*y^7+3*x^13*y^8+x^12*y^9-3*x^15*y^5+ 15*x^14*y^6-x^13*y^7-2*x^12*y^8+x^15*y^4-11*x^14*y^5-5*x^13*y^6+3*x^12*y^7-x^11 *y^8+3*x^14*y^4+5*x^13*y^5-2*x^12*y^6+3*x^11*y^7+x^10*y^8-8*x^12*y^5-5*x^10*y^7 -x^13*y^3+14*x^12*y^4-3*x^11*y^5+9*x^10*y^6+x^9*y^7-6*x^12*y^3-2*x^11*y^4-3*x^ 10*y^5-4*x^9*y^6+4*x^11*y^3-4*x^10*y^4+x^9*y^5+4*x^8*y^6-x^11*y^2+6*x^9*y^4-4*x ^8*y^5-x^7*y^6+2*x^10*y^2-4*x^9*y^3+x^8*y^4-x^7*y^5-x^9*y^2+2*x^8*y^3+2*x^7*y^4 +5*x^6*y^5+x^9*y-2*x^8*y^2+x^7*y^3-8*x^6*y^4-x^5*y^5+5*x^6*y^3+x^5*y^4-x^7*y-x^ 6*y^2-2*x^5*y^3-5*x^4*y^4+2*x^5*y^2+2*x^4*y^3+x^4*y^2+4*x^3*y^3+x^4*y-3*x^3*y^2 +2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^11*y^4-2*x^11*y^3+x^11*y^2+x^9*y^3-x^8*y^4+x^8*y ^3+x^7*y^4-x^9*y-x^7*y^3+x^7*y+x^4*y^2-x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 16 14 12 bors of a random word of length n tends to n times, 2 (a - 3 a - 3 a 11 10 9 8 7 6 5 4 3 / + 3 a + 8 a - a - 8 a + 3 a - 9 a + 8 a + 10 a - 10 a + 1) / / 6 2 8 6 4 3 ((7 a - 3 a + 2) (a + a - a + a + 1)) 7 3 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.4780162458 BTW the ratio for words with, 500, letters is, 0.4790746569 ------------------------------------------------ "Theorem Number 59" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [2, 1, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 7 9 6 8 7 9 5 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 6 9 4 8 5 7 6 8 4 7 5 6 6 - 5 x y - 2 x y + 5 x y + x y - 3 x y - 2 x y - 2 x y 8 3 7 4 6 5 7 3 5 5 7 2 6 3 5 4 + x y + 3 x y + 2 x y - 3 x y - 3 x y + x y + x y + 4 x y 6 2 5 3 4 4 6 5 2 5 4 2 3 3 - 2 x y - x y - 2 x y + x y + x y - x y + x y + 4 x y 3 2 3 2 2 2 / 8 4 8 3 - 3 x y + x y + 2 x y - 2 x y + x y - x + 1) / (x y - x y / 7 4 7 2 6 3 6 2 5 3 6 5 2 5 4 2 - x y + x y + 2 x y - x y - x y - x y - x y + x y + 2 x y 3 2 3 - x y - x y + x y + x - 1) and in Maple notation -(2*x^9*y^7-6*x^9*y^6+2*x^8*y^7+6*x^9*y^5-5*x^8*y^6-2*x^9*y^4+5*x^8*y^5+x^7*y^6 -3*x^8*y^4-2*x^7*y^5-2*x^6*y^6+x^8*y^3+3*x^7*y^4+2*x^6*y^5-3*x^7*y^3-3*x^5*y^5+ x^7*y^2+x^6*y^3+4*x^5*y^4-2*x^6*y^2-x^5*y^3-2*x^4*y^4+x^6*y+x^5*y^2-x^5*y+x^4*y ^2+4*x^3*y^3-3*x^3*y^2+x^3*y+2*x^2*y^2-2*x^2*y+x*y-x+1)/(x^8*y^4-x^8*y^3-x^7*y^ 4+x^7*y^2+2*x^6*y^3-x^6*y^2-x^5*y^3-x^6*y-x^5*y^2+x^5*y+2*x^4*y^2-x^3*y^2-x^3*y +x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.4747919204 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 60" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 10 13 9 13 8 12 9 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 11 9 13 6 11 8 12 6 11 7 - 4 x y - 2 x y + x y + x y - 5 x y + 2 x y + 8 x y 10 8 12 5 11 6 10 7 9 8 11 5 10 6 - 3 x y - x y - 4 x y + 11 x y - x y - x y - 15 x y 9 7 11 4 10 5 9 6 10 4 9 5 8 6 + x y + x y + 9 x y - 6 x y - 2 x y + 12 x y + 4 x y 9 4 8 5 7 6 8 4 7 5 8 3 7 4 - 6 x y - 6 x y + 2 x y + x y - 6 x y + x y + 7 x y 6 5 7 3 6 4 5 5 7 2 6 3 5 4 6 2 + 3 x y - 4 x y + x y - 2 x y + x y - 3 x y - x y - x y 5 3 4 4 6 5 2 4 3 5 3 3 3 2 + 2 x y - 5 x y + x y + x y + 4 x y - x y + 4 x y - 3 x y 3 2 2 2 / 8 4 8 3 7 4 + x y + 2 x y - 2 x y + x y - x + 1) / (x y - x y - x y / 7 2 6 3 6 2 5 3 6 5 2 5 4 2 3 2 + x y + 2 x y - x y - x y - x y - x y + x y + 2 x y - x y 3 - x y + x y + x - 1) and in Maple notation -(x^13*y^10-4*x^13*y^9+6*x^13*y^8+x^12*y^9-4*x^13*y^7-2*x^12*y^8+x^11*y^9+x^13* y^6-5*x^11*y^8+2*x^12*y^6+8*x^11*y^7-3*x^10*y^8-x^12*y^5-4*x^11*y^6+11*x^10*y^7 -x^9*y^8-x^11*y^5-15*x^10*y^6+x^9*y^7+x^11*y^4+9*x^10*y^5-6*x^9*y^6-2*x^10*y^4+ 12*x^9*y^5+4*x^8*y^6-6*x^9*y^4-6*x^8*y^5+2*x^7*y^6+x^8*y^4-6*x^7*y^5+x^8*y^3+7* x^7*y^4+3*x^6*y^5-4*x^7*y^3+x^6*y^4-2*x^5*y^5+x^7*y^2-3*x^6*y^3-x^5*y^4-x^6*y^2 +2*x^5*y^3-5*x^4*y^4+x^6*y+x^5*y^2+4*x^4*y^3-x^5*y+4*x^3*y^3-3*x^3*y^2+x^3*y+2* x^2*y^2-2*x^2*y+x*y-x+1)/(x^8*y^4-x^8*y^3-x^7*y^4+x^7*y^2+2*x^6*y^3-x^6*y^2-x^5 *y^3-x^6*y-x^5*y^2+x^5*y+2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.4745698218 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 61" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 10 14 9 13 10 ) | ) C(m, n) x y | = - (x y - 3 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 9 14 7 13 8 12 9 14 6 13 7 + 10 x y + 10 x y - 20 x y + x y - 15 x y + 20 x y 12 8 11 9 14 5 13 6 12 7 11 8 - 3 x y - 2 x y + 9 x y - 10 x y + 2 x y + 7 x y 14 4 13 5 12 6 11 7 10 8 12 5 - 2 x y + 2 x y + 2 x y - 10 x y + 2 x y - 3 x y 11 6 10 7 12 4 11 5 10 6 9 7 11 4 + 8 x y - 6 x y + x y - 4 x y + 6 x y + x y + x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 9 4 - x y - 3 x y + x y - 2 x y + 6 x y - x y + x y - 6 x y 8 5 7 6 9 3 8 4 7 5 6 6 7 4 6 5 + x y + x y + 2 x y - x y - 2 x y + 2 x y + x y - 3 x y 5 5 5 4 4 4 5 2 4 3 5 4 2 + 2 x y - 4 x y + 3 x y + 2 x y - 2 x y - x y - 2 x y 3 3 4 3 2 2 2 2 / 10 7 - 4 x y + x y + 2 x y - 2 x y + x y - x y + x - 1) / (x y / 11 5 10 6 11 4 10 5 11 3 10 4 8 6 + x y - 2 x y - 2 x y + 2 x y + x y - x y + x y 9 4 8 5 9 3 8 4 8 3 8 2 7 3 6 3 6 2 + x y - 2 x y - x y + x y - x y + x y - x y + x y - x y 5 3 5 4 2 - x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^14*y^10-3*x^14*y^9-2*x^13*y^10+10*x^13*y^9+10*x^14*y^7-20*x^13*y^8+x^12*y^9 -15*x^14*y^6+20*x^13*y^7-3*x^12*y^8-2*x^11*y^9+9*x^14*y^5-10*x^13*y^6+2*x^12*y^ 7+7*x^11*y^8-2*x^14*y^4+2*x^13*y^5+2*x^12*y^6-10*x^11*y^7+2*x^10*y^8-3*x^12*y^5 +8*x^11*y^6-6*x^10*y^7+x^12*y^4-4*x^11*y^5+6*x^10*y^6+x^9*y^7+x^11*y^4-x^10*y^5 -3*x^9*y^6+x^8*y^7-2*x^10*y^4+6*x^9*y^5-x^8*y^6+x^10*y^3-6*x^9*y^4+x^8*y^5+x^7* y^6+2*x^9*y^3-x^8*y^4-2*x^7*y^5+2*x^6*y^6+x^7*y^4-3*x^6*y^5+2*x^5*y^5-4*x^5*y^4 +3*x^4*y^4+2*x^5*y^2-2*x^4*y^3-x^5*y-2*x^4*y^2-4*x^3*y^3+x^4*y+2*x^3*y^2-2*x^2* y^2+x^2*y-x*y+x-1)/(x^10*y^7+x^11*y^5-2*x^10*y^6-2*x^11*y^4+2*x^10*y^5+x^11*y^3 -x^10*y^4+x^8*y^6+x^9*y^4-2*x^8*y^5-x^9*y^3+x^8*y^4-x^8*y^3+x^8*y^2-x^7*y^3+x^6 *y^3-x^6*y^2-x^5*y^3+x^5*y-x^4*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 10 9 7 6 4 3 2 2 (a - 1) (a - a + a - 2 a + 4 a - 4 a + 3 a + a + 1) - ------------------------------------------------------------- 6 3 6 5 3 2 (7 a + 4 a - 2 a + 2) (a + a + 2 a + a + 1) 7 4 2 where a is the root of the polynomial, x + x - x + 2 x - 1, and in decimals this is, 0.4714071908 BTW the ratio for words with, 500, letters is, 0.4727121881 ------------------------------------------------ "Theorem Number 62" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 10 14 9 13 10 ) | ) C(m, n) x y | = - (x y - 3 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 9 14 7 13 8 12 9 14 6 13 7 + 10 x y + 10 x y - 20 x y + x y - 15 x y + 20 x y 12 8 11 9 14 5 13 6 12 7 11 8 - 3 x y - 2 x y + 9 x y - 10 x y + 2 x y + 7 x y 14 4 13 5 12 6 11 7 10 8 12 5 - 2 x y + 2 x y + 2 x y - 10 x y + 2 x y - 3 x y 11 6 10 7 12 4 11 5 10 6 9 7 11 4 + 8 x y - 6 x y + x y - 4 x y + 6 x y + x y + x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 9 4 - x y - 3 x y + x y - 2 x y + 6 x y - x y + x y - 6 x y 8 5 7 6 9 3 8 4 7 5 6 6 7 4 6 5 + x y + x y + 2 x y - x y - 2 x y + 2 x y + x y - 3 x y 5 5 5 4 4 4 5 2 4 3 5 4 2 + 2 x y - 4 x y + 3 x y + 2 x y - 2 x y - x y - 2 x y 3 3 4 3 2 2 2 2 / 10 7 - 4 x y + x y + 2 x y - 2 x y + x y - x y + x - 1) / (x y / 11 5 10 6 11 4 10 5 11 3 10 4 8 6 + x y - 2 x y - 2 x y + 2 x y + x y - x y + x y 9 4 8 5 9 3 8 4 8 3 8 2 7 3 6 3 6 2 + x y - 2 x y - x y + x y - x y + x y - x y + x y - x y 5 3 5 4 2 - x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^14*y^10-3*x^14*y^9-2*x^13*y^10+10*x^13*y^9+10*x^14*y^7-20*x^13*y^8+x^12*y^9 -15*x^14*y^6+20*x^13*y^7-3*x^12*y^8-2*x^11*y^9+9*x^14*y^5-10*x^13*y^6+2*x^12*y^ 7+7*x^11*y^8-2*x^14*y^4+2*x^13*y^5+2*x^12*y^6-10*x^11*y^7+2*x^10*y^8-3*x^12*y^5 +8*x^11*y^6-6*x^10*y^7+x^12*y^4-4*x^11*y^5+6*x^10*y^6+x^9*y^7+x^11*y^4-x^10*y^5 -3*x^9*y^6+x^8*y^7-2*x^10*y^4+6*x^9*y^5-x^8*y^6+x^10*y^3-6*x^9*y^4+x^8*y^5+x^7* y^6+2*x^9*y^3-x^8*y^4-2*x^7*y^5+2*x^6*y^6+x^7*y^4-3*x^6*y^5+2*x^5*y^5-4*x^5*y^4 +3*x^4*y^4+2*x^5*y^2-2*x^4*y^3-x^5*y-2*x^4*y^2-4*x^3*y^3+x^4*y+2*x^3*y^2-2*x^2* y^2+x^2*y-x*y+x-1)/(x^10*y^7+x^11*y^5-2*x^10*y^6-2*x^11*y^4+2*x^10*y^5+x^11*y^3 -x^10*y^4+x^8*y^6+x^9*y^4-2*x^8*y^5-x^9*y^3+x^8*y^4-x^8*y^3+x^8*y^2-x^7*y^3+x^6 *y^3-x^6*y^2-x^5*y^3+x^5*y-x^4*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 10 9 7 6 4 3 2 2 (a - 1) (a - a + a - 2 a + 4 a - 4 a + 3 a + a + 1) - ------------------------------------------------------------- 6 3 6 5 3 2 (7 a + 4 a - 2 a + 2) (a + a + 2 a + a + 1) 7 4 2 where a is the root of the polynomial, x + x - x + 2 x - 1, and in decimals this is, 0.4714071908 BTW the ratio for words with, 500, letters is, 0.4727121881 ------------------------------------------------ "Theorem Number 63" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 21 14 21 13 21 12 ) | ) C(m, n) x y | = (2 x y - 10 x y + 20 x y / | / | ----- | ----- | m = 0 \ n = 0 / 20 13 21 11 20 12 19 13 21 10 - 2 x y - 20 x y + 10 x y + 5 x y + 10 x y 20 11 19 12 21 9 20 10 19 11 18 12 - 20 x y - 22 x y - 2 x y + 20 x y + 34 x y - 5 x y 17 13 20 9 19 10 18 11 20 8 19 9 + x y - 10 x y - 16 x y + 22 x y + 2 x y - 11 x y 18 10 17 11 16 12 19 8 18 9 17 10 - 36 x y - 9 x y - x y + 14 x y + 24 x y + 9 x y 16 11 15 12 19 7 18 8 17 9 16 10 + 2 x y + x y - 4 x y - x y + 14 x y + x y 15 11 18 7 17 8 16 9 15 10 18 6 - 3 x y - 6 x y - 23 x y - 4 x y - x y + 2 x y 17 7 16 8 15 9 14 10 17 6 15 8 + 2 x y + x y + 2 x y + 2 x y + 10 x y + 13 x y 14 9 17 5 16 6 15 7 14 8 13 9 - 9 x y - 4 x y + 5 x y - 15 x y + 17 x y - 6 x y 12 10 16 5 15 6 14 7 13 8 12 9 11 10 + x y - 6 x y - x y - 14 x y + 8 x y - 8 x y - x y 16 4 15 5 14 6 13 7 12 8 11 9 + 2 x y + 2 x y + x y + 9 x y + 11 x y + 2 x y 15 4 14 5 13 6 12 7 11 8 10 9 + 4 x y + 5 x y - 14 x y + 8 x y - 4 x y + 2 x y 15 3 14 4 13 5 12 6 11 7 10 8 - 2 x y - 2 x y + x y - 20 x y + 7 x y - 9 x y 12 5 11 6 10 7 13 3 11 5 10 6 + 7 x y + 4 x y + 18 x y + 2 x y - 11 x y - 9 x y 9 7 8 8 12 3 11 4 10 5 9 6 8 7 - 2 x y - x y + x y - 3 x y - 7 x y + 9 x y - 2 x y 11 3 10 4 9 5 8 6 7 7 11 2 10 3 + 6 x y + x y + x y + 16 x y - 2 x y + x y + 4 x y 9 4 8 5 7 6 11 10 2 9 3 8 4 - 14 x y - 17 x y + 5 x y - x y + x y + 3 x y - 3 x y 6 6 9 2 8 3 7 4 6 5 10 9 8 2 + 8 x y + 4 x y + 4 x y - 5 x y - 4 x y - x - x y + 2 x y 7 3 6 4 7 2 6 3 5 4 6 2 5 3 - 2 x y - 12 x y + 3 x y + 5 x y - 4 x y + 2 x y + 3 x y 4 4 5 2 4 3 5 4 2 3 3 3 / - 8 x y + x y + 4 x y - x y + 2 x y + 2 x y - x y + 1) / ( / 16 9 16 8 16 7 15 8 16 6 15 7 14 8 x y - 3 x y + 3 x y - x y - x y + 3 x y + 3 x y 15 6 14 7 15 5 14 6 13 7 12 8 14 5 - 3 x y - 7 x y + x y + 3 x y - 3 x y + x y + 3 x y 13 6 12 7 14 4 13 5 12 6 11 7 13 4 + 7 x y + x y - 2 x y - 4 x y - 5 x y - 2 x y - x y 11 6 10 7 13 3 12 4 11 5 10 6 12 3 + x y + 2 x y + x y + 4 x y + 4 x y - x y + x y 11 4 10 5 9 6 12 2 11 3 10 4 9 5 - 2 x y - 5 x y - 4 x y - 2 x y - x y + 2 x y + 5 x y 8 6 11 2 10 3 9 4 8 5 11 10 2 9 3 + x y - x y + 2 x y + 2 x y - x y + x y + x y - 2 x y 8 4 7 5 8 3 7 4 10 9 8 2 7 3 - 3 x y - 2 x y + 2 x y + 5 x y - x - x y + x y - x y 6 4 7 2 6 3 6 2 5 3 4 4 5 2 5 - x y - x y - x y + 2 x y + 3 x y - x y - x y - x y 3 3 3 + 2 x y - x y - 2 x y + 1) and in Maple notation (2*x^21*y^14-10*x^21*y^13+20*x^21*y^12-2*x^20*y^13-20*x^21*y^11+10*x^20*y^12+5* x^19*y^13+10*x^21*y^10-20*x^20*y^11-22*x^19*y^12-2*x^21*y^9+20*x^20*y^10+34*x^ 19*y^11-5*x^18*y^12+x^17*y^13-10*x^20*y^9-16*x^19*y^10+22*x^18*y^11+2*x^20*y^8-\ 11*x^19*y^9-36*x^18*y^10-9*x^17*y^11-x^16*y^12+14*x^19*y^8+24*x^18*y^9+9*x^17*y ^10+2*x^16*y^11+x^15*y^12-4*x^19*y^7-x^18*y^8+14*x^17*y^9+x^16*y^10-3*x^15*y^11 -6*x^18*y^7-23*x^17*y^8-4*x^16*y^9-x^15*y^10+2*x^18*y^6+2*x^17*y^7+x^16*y^8+2*x ^15*y^9+2*x^14*y^10+10*x^17*y^6+13*x^15*y^8-9*x^14*y^9-4*x^17*y^5+5*x^16*y^6-15 *x^15*y^7+17*x^14*y^8-6*x^13*y^9+x^12*y^10-6*x^16*y^5-x^15*y^6-14*x^14*y^7+8*x^ 13*y^8-8*x^12*y^9-x^11*y^10+2*x^16*y^4+2*x^15*y^5+x^14*y^6+9*x^13*y^7+11*x^12*y ^8+2*x^11*y^9+4*x^15*y^4+5*x^14*y^5-14*x^13*y^6+8*x^12*y^7-4*x^11*y^8+2*x^10*y^ 9-2*x^15*y^3-2*x^14*y^4+x^13*y^5-20*x^12*y^6+7*x^11*y^7-9*x^10*y^8+7*x^12*y^5+4 *x^11*y^6+18*x^10*y^7+2*x^13*y^3-11*x^11*y^5-9*x^10*y^6-2*x^9*y^7-x^8*y^8+x^12* y^3-3*x^11*y^4-7*x^10*y^5+9*x^9*y^6-2*x^8*y^7+6*x^11*y^3+x^10*y^4+x^9*y^5+16*x^ 8*y^6-2*x^7*y^7+x^11*y^2+4*x^10*y^3-14*x^9*y^4-17*x^8*y^5+5*x^7*y^6-x^11*y+x^10 *y^2+3*x^9*y^3-3*x^8*y^4+8*x^6*y^6+4*x^9*y^2+4*x^8*y^3-5*x^7*y^4-4*x^6*y^5-x^10 -x^9*y+2*x^8*y^2-2*x^7*y^3-12*x^6*y^4+3*x^7*y^2+5*x^6*y^3-4*x^5*y^4+2*x^6*y^2+3 *x^5*y^3-8*x^4*y^4+x^5*y^2+4*x^4*y^3-x^5*y+2*x^4*y^2+2*x^3*y^3-x^3*y+1)/(x^16*y ^9-3*x^16*y^8+3*x^16*y^7-x^15*y^8-x^16*y^6+3*x^15*y^7+3*x^14*y^8-3*x^15*y^6-7*x ^14*y^7+x^15*y^5+3*x^14*y^6-3*x^13*y^7+x^12*y^8+3*x^14*y^5+7*x^13*y^6+x^12*y^7-\ 2*x^14*y^4-4*x^13*y^5-5*x^12*y^6-2*x^11*y^7-x^13*y^4+x^11*y^6+2*x^10*y^7+x^13*y ^3+4*x^12*y^4+4*x^11*y^5-x^10*y^6+x^12*y^3-2*x^11*y^4-5*x^10*y^5-4*x^9*y^6-2*x^ 12*y^2-x^11*y^3+2*x^10*y^4+5*x^9*y^5+x^8*y^6-x^11*y^2+2*x^10*y^3+2*x^9*y^4-x^8* y^5+x^11*y+x^10*y^2-2*x^9*y^3-3*x^8*y^4-2*x^7*y^5+2*x^8*y^3+5*x^7*y^4-x^10-x^9* y+x^8*y^2-x^7*y^3-x^6*y^4-x^7*y^2-x^6*y^3+2*x^6*y^2+3*x^5*y^3-x^4*y^4-x^5*y^2-x ^5*y+2*x^3*y^3-x^3*y-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 14 bors of a random word of length n tends to n times, 2 (a + 2 a - 2 a 12 11 10 9 8 7 6 5 4 3 - a - 2 a - 2 a - 8 a + 10 a + 4 a - 8 a + 6 a + 8 a - 10 a / 6 4 3 2 + 1) / ((7 a + 5 a - 4 a + 3 a - 2) / 8 7 6 5 4 3 (a + a + a + a + 2 a - a - 1)) 7 5 4 3 where a is the root of the polynomial, x + x - x + x - 2 x + 1, and in decimals this is, 0.4671635910 BTW the ratio for words with, 500, letters is, 0.4683922549 ------------------------------------------------ "Theorem Number 64" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 13 18 12 18 11 17 12 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 17 11 16 12 18 9 16 11 18 8 - 3 x y - 2 x y + x y - 3 x y - 4 x y + 6 x y 17 9 16 10 15 11 18 7 17 8 16 9 - x y + 5 x y + x y - 4 x y + 9 x y - 3 x y 15 10 14 11 18 6 17 7 16 8 15 9 14 10 - 6 x y - x y + x y - 12 x y + 5 x y + 8 x y + x y 17 6 16 7 14 9 13 10 12 11 17 5 16 6 + 6 x y - 9 x y + x y - 3 x y - x y - x y + 8 x y 15 7 14 8 13 9 12 10 16 5 15 6 14 7 - 4 x y - 3 x y + 5 x y + x y - 4 x y + x y + 7 x y 13 8 12 9 11 10 16 4 15 5 14 6 13 7 - 3 x y + x y - x y + x y - x y - 7 x y + 10 x y 12 8 11 9 15 4 14 5 13 6 12 7 + 5 x y + 3 x y + x y + 2 x y - 15 x y - 10 x y 11 8 10 9 14 4 13 5 12 6 11 7 14 3 + 3 x y + 2 x y - x y + 7 x y + x y - 11 x y + x y 13 4 12 5 11 6 10 7 9 8 13 3 12 4 - 3 x y + 8 x y + 9 x y - x y + 2 x y + 2 x y - 9 x y 10 6 9 7 12 3 11 4 10 5 9 6 8 7 - 7 x y - 3 x y + 4 x y - 8 x y + 8 x y + x y - x y 11 3 10 4 8 6 7 7 10 3 9 4 8 5 + 5 x y - x y - 3 x y - 2 x y - 2 x y + 2 x y - 4 x y 7 6 10 2 9 3 8 4 7 5 6 6 9 2 + x y + x y - 3 x y + 15 x y + 2 x y - 3 x y + x y 8 3 7 4 6 5 8 2 6 4 5 5 7 2 - 7 x y + 3 x y + 4 x y - x y + 3 x y + 6 x y - 5 x y 6 3 5 4 7 6 2 5 3 4 4 5 2 - 3 x y - 2 x y + x y - 2 x y - 3 x y + 5 x y - 2 x y 4 3 4 2 3 3 4 2 2 / 11 6 - 4 x y - 3 x y - 3 x y + x y - x y - x y - 1) / (x y / 11 5 11 4 10 5 10 4 9 5 10 3 9 4 8 4 - 2 x y + x y - x y + 2 x y + x y - x y - 2 x y - x y 9 2 8 3 8 2 7 3 7 6 2 5 3 5 2 5 + x y + 2 x y - x y - x y + x y + x y - x y - x y + x y 4 2 3 3 4 2 2 + x y - x y + x y + x y + x y - 1) and in Maple notation (x^18*y^13-4*x^18*y^12+6*x^18*y^11+x^17*y^12-3*x^18*y^10-2*x^17*y^11+x^16*y^12-\ 3*x^18*y^9-4*x^16*y^11+6*x^18*y^8-x^17*y^9+5*x^16*y^10+x^15*y^11-4*x^18*y^7+9*x ^17*y^8-3*x^16*y^9-6*x^15*y^10-x^14*y^11+x^18*y^6-12*x^17*y^7+5*x^16*y^8+8*x^15 *y^9+x^14*y^10+6*x^17*y^6-9*x^16*y^7+x^14*y^9-3*x^13*y^10-x^12*y^11-x^17*y^5+8* x^16*y^6-4*x^15*y^7-3*x^14*y^8+5*x^13*y^9+x^12*y^10-4*x^16*y^5+x^15*y^6+7*x^14* y^7-3*x^13*y^8+x^12*y^9-x^11*y^10+x^16*y^4-x^15*y^5-7*x^14*y^6+10*x^13*y^7+5*x^ 12*y^8+3*x^11*y^9+x^15*y^4+2*x^14*y^5-15*x^13*y^6-10*x^12*y^7+3*x^11*y^8+2*x^10 *y^9-x^14*y^4+7*x^13*y^5+x^12*y^6-11*x^11*y^7+x^14*y^3-3*x^13*y^4+8*x^12*y^5+9* x^11*y^6-x^10*y^7+2*x^9*y^8+2*x^13*y^3-9*x^12*y^4-7*x^10*y^6-3*x^9*y^7+4*x^12*y ^3-8*x^11*y^4+8*x^10*y^5+x^9*y^6-x^8*y^7+5*x^11*y^3-x^10*y^4-3*x^8*y^6-2*x^7*y^ 7-2*x^10*y^3+2*x^9*y^4-4*x^8*y^5+x^7*y^6+x^10*y^2-3*x^9*y^3+15*x^8*y^4+2*x^7*y^ 5-3*x^6*y^6+x^9*y^2-7*x^8*y^3+3*x^7*y^4+4*x^6*y^5-x^8*y^2+3*x^6*y^4+6*x^5*y^5-5 *x^7*y^2-3*x^6*y^3-2*x^5*y^4+x^7*y-2*x^6*y^2-3*x^5*y^3+5*x^4*y^4-2*x^5*y^2-4*x^ 4*y^3-3*x^4*y^2-3*x^3*y^3+x^4*y-x^2*y^2-x*y-1)/(x^11*y^6-2*x^11*y^5+x^11*y^4-x^ 10*y^5+2*x^10*y^4+x^9*y^5-x^10*y^3-2*x^9*y^4-x^8*y^4+x^9*y^2+2*x^8*y^3-x^8*y^2- x^7*y^3+x^7*y+x^6*y^2-x^5*y^3-x^5*y^2+x^5*y+x^4*y^2-x^3*y^3+x^4*y+x^2*y^2+x*y-1 ) As the length of the word goes to infinity, the average number of good neigh\ 13 12 bors of a random word of length n tends to n times, - 2 (4 a - 5 a 11 10 9 7 6 5 4 3 2 + 10 a - 10 a + 10 a - 4 a + 5 a - 6 a + 6 a + 4 a - 3 a - 2 a / 5 4 3 2 - 1) / ((6 a - 5 a + 8 a - 3 a + 2 a + 1) / 8 6 5 4 3 2 (a + a + a + a + 3 a + a + a + 1)) 6 5 4 3 2 where a is the root of the polynomial, x - x + 2 x - x + x + x - 1, and in decimals this is, 0.4666590412 BTW the ratio for words with, 500, letters is, 0.4678874293 ------------------------------------------------ "Theorem Number 65" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 13 19 12 19 11 ) | ) C(m, n) x y | = (2 x y - 10 x y + 20 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 12 19 10 18 11 17 12 19 9 18 10 - 4 x y - 20 x y + 20 x y + 4 x y + 10 x y - 40 x y 17 11 19 8 18 9 17 10 16 11 18 8 - 16 x y - 2 x y + 40 x y + 18 x y - 8 x y - 20 x y 17 9 16 10 15 11 18 7 17 8 16 9 + 8 x y + 34 x y + 2 x y + 4 x y - 32 x y - 52 x y 15 10 17 7 16 8 15 9 14 10 17 6 - 12 x y + 24 x y + 28 x y + 22 x y - 4 x y - 6 x y 16 7 15 8 14 9 13 10 16 6 15 7 + 8 x y - 14 x y + 30 x y - 6 x y - 14 x y + 2 x y 14 8 13 9 16 5 15 6 14 7 13 8 - 70 x y + 14 x y + 4 x y - 4 x y + 70 x y - 14 x y 12 9 15 5 14 6 13 7 12 8 11 9 + 14 x y + 6 x y - 31 x y + 24 x y - 29 x y - 6 x y 15 4 14 5 13 6 12 7 11 8 14 4 - 2 x y + 6 x y - 36 x y + 8 x y + 12 x y - x y 13 5 12 6 11 7 10 8 13 4 12 5 + 22 x y + 10 x y + 6 x y + 14 x y - 4 x y + 2 x y 11 6 10 7 12 4 11 5 10 6 9 7 - 22 x y - 42 x y - 4 x y + 10 x y + 25 x y + 2 x y 8 8 12 3 10 5 9 6 8 7 12 2 10 4 + x y - 2 x y + 14 x y + 6 x y - 2 x y + x y - 9 x y 9 5 8 6 7 7 10 3 9 4 8 5 7 6 - 12 x y - 20 x y + 2 x y - 2 x y + 4 x y + 26 x y - 2 x y 9 3 8 4 7 5 6 6 9 2 8 3 7 4 - 2 x y - 2 x y + 8 x y - 8 x y + 2 x y - 4 x y - 4 x y 6 5 7 3 6 4 6 3 5 4 6 2 5 3 + 6 x y - 2 x y + 6 x y - 4 x y + 2 x y - x y - 2 x y 4 4 5 2 4 3 5 4 2 3 3 / + 8 x y - 2 x y - 4 x y + 2 x y - 2 x y - 2 x y - 1) / ( / 16 10 16 9 16 8 15 9 16 7 15 8 14 9 x y - 4 x y + 6 x y - 2 x y - 4 x y + 8 x y + 2 x y 16 6 15 7 14 8 15 6 13 8 15 5 + x y - 12 x y - 5 x y + 8 x y - 4 x y - 2 x y 14 6 13 7 12 8 14 5 13 6 12 7 + 10 x y + 12 x y + x y - 10 x y - 10 x y - 2 x y 14 4 13 5 12 6 11 7 13 4 12 5 + 3 x y - 2 x y - 3 x y - 2 x y + 6 x y + 8 x y 11 6 10 7 13 3 12 4 11 5 10 6 + 8 x y - 2 x y - 2 x y - 3 x y - 8 x y + 4 x y 12 3 10 5 9 6 12 2 11 3 10 4 9 5 - 2 x y - 6 x y + 4 x y + x y + 2 x y + 8 x y - 6 x y 8 6 10 3 9 4 8 5 9 3 8 4 7 5 - 2 x y - 4 x y + 4 x y + 2 x y - 4 x y + 3 x y + 4 x y 9 2 8 3 7 4 8 2 7 3 7 2 6 3 + 2 x y - 4 x y - 8 x y + x y + 2 x y + 2 x y + 4 x y 6 2 5 3 4 4 5 4 2 3 3 - 4 x y - 4 x y + x y + 2 x y + 2 x y - 2 x y + 2 x y - 1) and in Maple notation (2*x^19*y^13-10*x^19*y^12+20*x^19*y^11-4*x^18*y^12-20*x^19*y^10+20*x^18*y^11+4* x^17*y^12+10*x^19*y^9-40*x^18*y^10-16*x^17*y^11-2*x^19*y^8+40*x^18*y^9+18*x^17* y^10-8*x^16*y^11-20*x^18*y^8+8*x^17*y^9+34*x^16*y^10+2*x^15*y^11+4*x^18*y^7-32* x^17*y^8-52*x^16*y^9-12*x^15*y^10+24*x^17*y^7+28*x^16*y^8+22*x^15*y^9-4*x^14*y^ 10-6*x^17*y^6+8*x^16*y^7-14*x^15*y^8+30*x^14*y^9-6*x^13*y^10-14*x^16*y^6+2*x^15 *y^7-70*x^14*y^8+14*x^13*y^9+4*x^16*y^5-4*x^15*y^6+70*x^14*y^7-14*x^13*y^8+14*x ^12*y^9+6*x^15*y^5-31*x^14*y^6+24*x^13*y^7-29*x^12*y^8-6*x^11*y^9-2*x^15*y^4+6* x^14*y^5-36*x^13*y^6+8*x^12*y^7+12*x^11*y^8-x^14*y^4+22*x^13*y^5+10*x^12*y^6+6* x^11*y^7+14*x^10*y^8-4*x^13*y^4+2*x^12*y^5-22*x^11*y^6-42*x^10*y^7-4*x^12*y^4+ 10*x^11*y^5+25*x^10*y^6+2*x^9*y^7+x^8*y^8-2*x^12*y^3+14*x^10*y^5+6*x^9*y^6-2*x^ 8*y^7+x^12*y^2-9*x^10*y^4-12*x^9*y^5-20*x^8*y^6+2*x^7*y^7-2*x^10*y^3+4*x^9*y^4+ 26*x^8*y^5-2*x^7*y^6-2*x^9*y^3-2*x^8*y^4+8*x^7*y^5-8*x^6*y^6+2*x^9*y^2-4*x^8*y^ 3-4*x^7*y^4+6*x^6*y^5-2*x^7*y^3+6*x^6*y^4-4*x^6*y^3+2*x^5*y^4-x^6*y^2-2*x^5*y^3 +8*x^4*y^4-2*x^5*y^2-4*x^4*y^3+2*x^5*y-2*x^4*y^2-2*x^3*y^3-1)/(x^16*y^10-4*x^16 *y^9+6*x^16*y^8-2*x^15*y^9-4*x^16*y^7+8*x^15*y^8+2*x^14*y^9+x^16*y^6-12*x^15*y^ 7-5*x^14*y^8+8*x^15*y^6-4*x^13*y^8-2*x^15*y^5+10*x^14*y^6+12*x^13*y^7+x^12*y^8-\ 10*x^14*y^5-10*x^13*y^6-2*x^12*y^7+3*x^14*y^4-2*x^13*y^5-3*x^12*y^6-2*x^11*y^7+ 6*x^13*y^4+8*x^12*y^5+8*x^11*y^6-2*x^10*y^7-2*x^13*y^3-3*x^12*y^4-8*x^11*y^5+4* x^10*y^6-2*x^12*y^3-6*x^10*y^5+4*x^9*y^6+x^12*y^2+2*x^11*y^3+8*x^10*y^4-6*x^9*y ^5-2*x^8*y^6-4*x^10*y^3+4*x^9*y^4+2*x^8*y^5-4*x^9*y^3+3*x^8*y^4+4*x^7*y^5+2*x^9 *y^2-4*x^8*y^3-8*x^7*y^4+x^8*y^2+2*x^7*y^3+2*x^7*y^2+4*x^6*y^3-4*x^6*y^2-4*x^5* y^3+x^4*y^4+2*x^5*y+2*x^4*y^2-2*x^3*y^3+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 14 bors of a random word of length n tends to n times, - (a - 4 a + 7 a 13 12 11 10 9 8 7 6 5 - 8 a + 14 a - 28 a + 35 a - 22 a - 13 a + 38 a - 29 a + 6 a 4 3 / 4 3 2 4 3 2 + 9 a - 8 a + 1) / ((a - a - 1) (5 a - 6 a + 3 a - 1)) / 5 4 3 where a is the root of the polynomial, 2 x - 3 x + 2 x - 2 x + 1, and in decimals this is, 0.4643858402 BTW the ratio for words with, 500, letters is, 0.4659677472 ------------------------------------------------ "Theorem Number 66" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 20 14 20 13 19 14 20 12 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 9 x y / | / | ----- | ----- | m = 0 \ n = 0 / 19 13 20 11 19 12 18 13 20 10 19 11 + 6 x y - 5 x y - 15 x y - x y - 5 x y + 21 x y 18 12 17 13 20 9 19 10 18 11 17 12 + 7 x y - x y + 9 x y - 20 x y - 23 x y + 3 x y 20 8 19 9 18 10 17 11 20 7 19 8 - 5 x y + 16 x y + 44 x y + x y + x y - 11 x y 18 9 17 10 15 12 19 7 18 8 17 9 - 50 x y - 13 x y - x y + 5 x y + 31 x y + 17 x y 16 10 15 11 19 6 18 7 17 8 16 9 15 10 + x y + x y - x y - 7 x y - 7 x y - 9 x y + 8 x y 18 6 17 7 16 8 15 9 14 10 18 5 17 6 - 2 x y - x y + 26 x y - 21 x y - 6 x y + x y + x y 16 7 15 8 14 9 16 6 14 8 13 9 - 34 x y + 18 x y + 22 x y + 21 x y - 33 x y + x y 16 5 15 6 14 7 13 8 12 9 15 5 - 5 x y - 11 x y + 25 x y - 3 x y - 2 x y + 8 x y 14 6 13 7 12 8 11 9 15 4 14 5 - 6 x y + 5 x y + 10 x y + x y - 2 x y - 6 x y 13 6 12 7 11 8 14 4 13 5 12 6 - 10 x y - 17 x y + 3 x y + 5 x y + 15 x y + 8 x y 11 7 10 8 14 3 13 4 12 5 11 6 - 10 x y - 3 x y - x y - 11 x y + 4 x y + 9 x y 10 7 9 8 13 3 12 4 11 5 10 6 9 7 + 4 x y + x y + 3 x y - x y - 6 x y + 9 x y + 3 x y 12 3 11 4 10 5 9 6 12 2 11 3 10 4 - 3 x y + 3 x y - 17 x y - 2 x y + x y + x y + 6 x y 9 5 8 6 11 2 10 3 9 4 8 5 9 3 - 5 x y - 4 x y - x y + x y + 4 x y + 5 x y - 2 x y 7 5 9 2 8 3 7 4 6 5 8 2 7 3 6 4 + 4 x y + x y + x y - 11 x y - 4 x y - 2 x y + 8 x y - x y 5 5 6 3 7 6 2 5 3 4 4 5 2 4 3 + 2 x y + 2 x y - x y + x y - 2 x y + 5 x y - x y - 4 x y 5 3 3 3 2 3 2 2 2 / + x y - 4 x y + 3 x y - x y - 2 x y + 2 x y - x y + x - 1) / ( / 17 11 17 10 17 9 16 10 17 8 16 9 17 7 x y - 5 x y + 10 x y + x y - 10 x y - 4 x y + 5 x y 16 8 15 9 17 6 16 7 15 8 16 6 15 7 + 6 x y + x y - x y - 4 x y - 4 x y + x y + 6 x y 15 6 14 7 13 8 15 5 14 6 13 7 13 6 - 4 x y + x y + x y + x y - 2 x y - 4 x y + 5 x y 12 7 14 4 13 5 12 6 11 7 14 3 13 4 + x y + 2 x y - x y - 3 x y - x y - x y - 2 x y 12 5 11 6 13 3 12 4 12 3 10 5 9 6 12 2 + 4 x y + x y + x y - 2 x y - x y - x y - x y + x y 11 3 10 4 9 5 11 2 10 3 9 4 8 5 8 4 + x y + 2 x y + x y - x y - x y - x y + 2 x y - x y 7 5 9 2 8 3 7 4 7 3 6 4 6 3 7 6 2 - x y + x y - x y - x y + 3 x y + x y + x y - x y - x y 5 3 5 2 5 4 2 3 2 3 - x y - x y + x y + 2 x y - x y - x y + x y + x - 1) and in Maple notation (x^20*y^14-5*x^20*y^13-x^19*y^14+9*x^20*y^12+6*x^19*y^13-5*x^20*y^11-15*x^19*y^ 12-x^18*y^13-5*x^20*y^10+21*x^19*y^11+7*x^18*y^12-x^17*y^13+9*x^20*y^9-20*x^19* y^10-23*x^18*y^11+3*x^17*y^12-5*x^20*y^8+16*x^19*y^9+44*x^18*y^10+x^17*y^11+x^ 20*y^7-11*x^19*y^8-50*x^18*y^9-13*x^17*y^10-x^15*y^12+5*x^19*y^7+31*x^18*y^8+17 *x^17*y^9+x^16*y^10+x^15*y^11-x^19*y^6-7*x^18*y^7-7*x^17*y^8-9*x^16*y^9+8*x^15* y^10-2*x^18*y^6-x^17*y^7+26*x^16*y^8-21*x^15*y^9-6*x^14*y^10+x^18*y^5+x^17*y^6-\ 34*x^16*y^7+18*x^15*y^8+22*x^14*y^9+21*x^16*y^6-33*x^14*y^8+x^13*y^9-5*x^16*y^5 -11*x^15*y^6+25*x^14*y^7-3*x^13*y^8-2*x^12*y^9+8*x^15*y^5-6*x^14*y^6+5*x^13*y^7 +10*x^12*y^8+x^11*y^9-2*x^15*y^4-6*x^14*y^5-10*x^13*y^6-17*x^12*y^7+3*x^11*y^8+ 5*x^14*y^4+15*x^13*y^5+8*x^12*y^6-10*x^11*y^7-3*x^10*y^8-x^14*y^3-11*x^13*y^4+4 *x^12*y^5+9*x^11*y^6+4*x^10*y^7+x^9*y^8+3*x^13*y^3-x^12*y^4-6*x^11*y^5+9*x^10*y ^6+3*x^9*y^7-3*x^12*y^3+3*x^11*y^4-17*x^10*y^5-2*x^9*y^6+x^12*y^2+x^11*y^3+6*x^ 10*y^4-5*x^9*y^5-4*x^8*y^6-x^11*y^2+x^10*y^3+4*x^9*y^4+5*x^8*y^5-2*x^9*y^3+4*x^ 7*y^5+x^9*y^2+x^8*y^3-11*x^7*y^4-4*x^6*y^5-2*x^8*y^2+8*x^7*y^3-x^6*y^4+2*x^5*y^ 5+2*x^6*y^3-x^7*y+x^6*y^2-2*x^5*y^3+5*x^4*y^4-x^5*y^2-4*x^4*y^3+x^5*y-4*x^3*y^3 +3*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^17*y^11-5*x^17*y^10+10*x^17*y^9+ x^16*y^10-10*x^17*y^8-4*x^16*y^9+5*x^17*y^7+6*x^16*y^8+x^15*y^9-x^17*y^6-4*x^16 *y^7-4*x^15*y^8+x^16*y^6+6*x^15*y^7-4*x^15*y^6+x^14*y^7+x^13*y^8+x^15*y^5-2*x^ 14*y^6-4*x^13*y^7+5*x^13*y^6+x^12*y^7+2*x^14*y^4-x^13*y^5-3*x^12*y^6-x^11*y^7-x ^14*y^3-2*x^13*y^4+4*x^12*y^5+x^11*y^6+x^13*y^3-2*x^12*y^4-x^12*y^3-x^10*y^5-x^ 9*y^6+x^12*y^2+x^11*y^3+2*x^10*y^4+x^9*y^5-x^11*y^2-x^10*y^3-x^9*y^4+2*x^8*y^5- x^8*y^4-x^7*y^5+x^9*y^2-x^8*y^3-x^7*y^4+3*x^7*y^3+x^6*y^4+x^6*y^3-x^7*y-x^6*y^2 -x^5*y^3-x^5*y^2+x^5*y+2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, - 2 (a - a - 4 a 11 10 9 8 7 6 5 4 3 + 11 a - 8 a + 8 a - 12 a - 2 a + 24 a - 16 a - 5 a + 7 a - 1) / 5 4 3 2 6 4 3 / ((6 a - 5 a + 8 a - 6 a + 2) (2 a - a + 2 a + 1)) / 6 5 4 3 where a is the root of the polynomial, x - x + 2 x - 2 x + 2 x - 1, and in decimals this is, 0.4622959834 BTW the ratio for words with, 500, letters is, 0.4638851145 ------------------------------------------------ "Theorem Number 67" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 20 14 20 13 19 14 20 12 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 9 x y / | / | ----- | ----- | m = 0 \ n = 0 / 19 13 20 11 19 12 18 13 20 10 19 11 + 6 x y - 5 x y - 15 x y - x y - 5 x y + 21 x y 18 12 17 13 20 9 19 10 18 11 17 12 + 7 x y - x y + 9 x y - 20 x y - 23 x y + 3 x y 20 8 19 9 18 10 17 11 20 7 19 8 - 5 x y + 16 x y + 44 x y + x y + x y - 11 x y 18 9 17 10 15 12 19 7 18 8 17 9 - 50 x y - 13 x y - x y + 5 x y + 31 x y + 17 x y 16 10 15 11 19 6 18 7 17 8 16 9 15 10 + x y + x y - x y - 7 x y - 7 x y - 9 x y + 8 x y 18 6 17 7 16 8 15 9 14 10 18 5 17 6 - 2 x y - x y + 26 x y - 21 x y - 6 x y + x y + x y 16 7 15 8 14 9 16 6 14 8 13 9 - 34 x y + 18 x y + 22 x y + 21 x y - 33 x y + x y 16 5 15 6 14 7 13 8 12 9 15 5 - 5 x y - 11 x y + 25 x y - 3 x y - 2 x y + 8 x y 14 6 13 7 12 8 11 9 15 4 14 5 - 6 x y + 5 x y + 10 x y + x y - 2 x y - 6 x y 13 6 12 7 11 8 14 4 13 5 12 6 - 10 x y - 17 x y + 3 x y + 5 x y + 15 x y + 8 x y 11 7 10 8 14 3 13 4 12 5 11 6 - 10 x y - 3 x y - x y - 11 x y + 4 x y + 9 x y 10 7 9 8 13 3 12 4 11 5 10 6 9 7 + 4 x y + x y + 3 x y - x y - 6 x y + 9 x y + 3 x y 12 3 11 4 10 5 9 6 12 2 11 3 10 4 - 3 x y + 3 x y - 17 x y - 2 x y + x y + x y + 6 x y 9 5 8 6 11 2 10 3 9 4 8 5 9 3 - 5 x y - 4 x y - x y + x y + 4 x y + 5 x y - 2 x y 7 5 9 2 8 3 7 4 6 5 8 2 7 3 6 4 + 4 x y + x y + x y - 11 x y - 4 x y - 2 x y + 8 x y - x y 5 5 6 3 7 6 2 5 3 4 4 5 2 4 3 + 2 x y + 2 x y - x y + x y - 2 x y + 5 x y - x y - 4 x y 5 3 3 3 2 3 2 2 2 / + x y - 4 x y + 3 x y - x y - 2 x y + 2 x y - x y + x - 1) / ( / 17 11 17 10 17 9 16 10 17 8 16 9 17 7 x y - 5 x y + 10 x y + x y - 10 x y - 4 x y + 5 x y 16 8 15 9 17 6 16 7 15 8 16 6 15 7 + 6 x y + x y - x y - 4 x y - 4 x y + x y + 6 x y 15 6 14 7 13 8 15 5 14 6 13 7 13 6 - 4 x y + x y + x y + x y - 2 x y - 4 x y + 5 x y 12 7 14 4 13 5 12 6 11 7 14 3 13 4 + x y + 2 x y - x y - 3 x y - x y - x y - 2 x y 12 5 11 6 13 3 12 4 12 3 10 5 9 6 12 2 + 4 x y + x y + x y - 2 x y - x y - x y - x y + x y 11 3 10 4 9 5 11 2 10 3 9 4 8 5 8 4 + x y + 2 x y + x y - x y - x y - x y + 2 x y - x y 7 5 9 2 8 3 7 4 7 3 6 4 6 3 7 6 2 - x y + x y - x y - x y + 3 x y + x y + x y - x y - x y 5 3 5 2 5 4 2 3 2 3 - x y - x y + x y + 2 x y - x y - x y + x y + x - 1) and in Maple notation (x^20*y^14-5*x^20*y^13-x^19*y^14+9*x^20*y^12+6*x^19*y^13-5*x^20*y^11-15*x^19*y^ 12-x^18*y^13-5*x^20*y^10+21*x^19*y^11+7*x^18*y^12-x^17*y^13+9*x^20*y^9-20*x^19* y^10-23*x^18*y^11+3*x^17*y^12-5*x^20*y^8+16*x^19*y^9+44*x^18*y^10+x^17*y^11+x^ 20*y^7-11*x^19*y^8-50*x^18*y^9-13*x^17*y^10-x^15*y^12+5*x^19*y^7+31*x^18*y^8+17 *x^17*y^9+x^16*y^10+x^15*y^11-x^19*y^6-7*x^18*y^7-7*x^17*y^8-9*x^16*y^9+8*x^15* y^10-2*x^18*y^6-x^17*y^7+26*x^16*y^8-21*x^15*y^9-6*x^14*y^10+x^18*y^5+x^17*y^6-\ 34*x^16*y^7+18*x^15*y^8+22*x^14*y^9+21*x^16*y^6-33*x^14*y^8+x^13*y^9-5*x^16*y^5 -11*x^15*y^6+25*x^14*y^7-3*x^13*y^8-2*x^12*y^9+8*x^15*y^5-6*x^14*y^6+5*x^13*y^7 +10*x^12*y^8+x^11*y^9-2*x^15*y^4-6*x^14*y^5-10*x^13*y^6-17*x^12*y^7+3*x^11*y^8+ 5*x^14*y^4+15*x^13*y^5+8*x^12*y^6-10*x^11*y^7-3*x^10*y^8-x^14*y^3-11*x^13*y^4+4 *x^12*y^5+9*x^11*y^6+4*x^10*y^7+x^9*y^8+3*x^13*y^3-x^12*y^4-6*x^11*y^5+9*x^10*y ^6+3*x^9*y^7-3*x^12*y^3+3*x^11*y^4-17*x^10*y^5-2*x^9*y^6+x^12*y^2+x^11*y^3+6*x^ 10*y^4-5*x^9*y^5-4*x^8*y^6-x^11*y^2+x^10*y^3+4*x^9*y^4+5*x^8*y^5-2*x^9*y^3+4*x^ 7*y^5+x^9*y^2+x^8*y^3-11*x^7*y^4-4*x^6*y^5-2*x^8*y^2+8*x^7*y^3-x^6*y^4+2*x^5*y^ 5+2*x^6*y^3-x^7*y+x^6*y^2-2*x^5*y^3+5*x^4*y^4-x^5*y^2-4*x^4*y^3+x^5*y-4*x^3*y^3 +3*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^17*y^11-5*x^17*y^10+10*x^17*y^9+ x^16*y^10-10*x^17*y^8-4*x^16*y^9+5*x^17*y^7+6*x^16*y^8+x^15*y^9-x^17*y^6-4*x^16 *y^7-4*x^15*y^8+x^16*y^6+6*x^15*y^7-4*x^15*y^6+x^14*y^7+x^13*y^8+x^15*y^5-2*x^ 14*y^6-4*x^13*y^7+5*x^13*y^6+x^12*y^7+2*x^14*y^4-x^13*y^5-3*x^12*y^6-x^11*y^7-x ^14*y^3-2*x^13*y^4+4*x^12*y^5+x^11*y^6+x^13*y^3-2*x^12*y^4-x^12*y^3-x^10*y^5-x^ 9*y^6+x^12*y^2+x^11*y^3+2*x^10*y^4+x^9*y^5-x^11*y^2-x^10*y^3-x^9*y^4+2*x^8*y^5- x^8*y^4-x^7*y^5+x^9*y^2-x^8*y^3-x^7*y^4+3*x^7*y^3+x^6*y^4+x^6*y^3-x^7*y-x^6*y^2 -x^5*y^3-x^5*y^2+x^5*y+2*x^4*y^2-x^3*y^2-x^3*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, - 2 (a - a - 4 a 11 10 9 8 7 6 5 4 3 + 11 a - 8 a + 8 a - 12 a - 2 a + 24 a - 16 a - 5 a + 7 a - 1) / 5 4 3 2 6 4 3 / ((6 a - 5 a + 8 a - 6 a + 2) (2 a - a + 2 a + 1)) / 6 5 4 3 where a is the root of the polynomial, x - x + 2 x - 2 x + 2 x - 1, and in decimals this is, 0.4622959834 BTW the ratio for words with, 500, letters is, 0.4638851145 ------------------------------------------------ "Theorem Number 68" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 12 16 11 16 10 ) | ) C(m, n) x y | = (x y - 2 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 11 16 9 15 10 16 8 15 9 14 10 + 6 x y + 10 x y - 20 x y - 14 x y + 22 x y - 2 x y 16 7 15 8 16 6 14 8 13 9 12 10 + 10 x y - 8 x y - 2 x y + 11 x y - 6 x y - 2 x y 16 5 15 6 14 7 13 8 12 9 16 4 - 2 x y + 2 x y - 18 x y + 10 x y + 6 x y + x y 15 5 14 6 13 7 12 8 11 9 15 4 - 4 x y + 16 x y + 6 x y - 8 x y - 8 x y + 2 x y 14 5 13 6 12 7 11 8 14 4 13 5 - 8 x y - 18 x y - 2 x y + 18 x y - x y + 10 x y 12 6 11 7 10 8 14 3 13 4 12 5 + 22 x y - 8 x y + 2 x y + 2 x y - 4 x y - 20 x y 11 6 10 7 13 3 12 4 11 5 10 6 9 7 - 12 x y - 2 x y + 2 x y + x y + 18 x y + x y + 6 x y 8 8 12 3 11 4 10 5 9 6 12 2 11 3 + x y + 2 x y - 10 x y - 2 x y - 8 x y + x y + 2 x y 10 4 9 5 8 6 7 7 10 3 9 4 7 6 + x y + 8 x y + 3 x y + 4 x y + 2 x y - 10 x y - 4 x y 10 2 9 3 7 5 8 3 7 4 6 5 9 - 2 x y + 6 x y + 10 x y - 8 x y - 10 x y - 4 x y - 2 x y 8 2 7 3 6 4 5 5 7 2 6 3 5 4 + 3 x y - 2 x y + 9 x y - 2 x y - 2 x y - 8 x y - 4 x y 7 6 2 4 4 4 3 6 4 2 3 3 3 + 2 x y - 2 x y - 4 x y - 4 x y + x + 2 x y - 8 x y + 2 x y 2 2 / 12 8 12 7 12 6 8 6 9 4 - 4 x y - 2 x y - 1) / (x y - 2 x y + x y - 2 x y - 2 x y / 8 5 9 3 8 4 8 3 9 8 2 4 4 5 2 + 2 x y + 4 x y + 3 x y - 4 x y - 2 x y + x y + x y + 2 x y 6 4 2 3 + x + 2 x y + 2 x y - 1) and in Maple notation (x^16*y^12-2*x^16*y^11-2*x^16*y^10+6*x^15*y^11+10*x^16*y^9-20*x^15*y^10-14*x^16 *y^8+22*x^15*y^9-2*x^14*y^10+10*x^16*y^7-8*x^15*y^8-2*x^16*y^6+11*x^14*y^8-6*x^ 13*y^9-2*x^12*y^10-2*x^16*y^5+2*x^15*y^6-18*x^14*y^7+10*x^13*y^8+6*x^12*y^9+x^ 16*y^4-4*x^15*y^5+16*x^14*y^6+6*x^13*y^7-8*x^12*y^8-8*x^11*y^9+2*x^15*y^4-8*x^ 14*y^5-18*x^13*y^6-2*x^12*y^7+18*x^11*y^8-x^14*y^4+10*x^13*y^5+22*x^12*y^6-8*x^ 11*y^7+2*x^10*y^8+2*x^14*y^3-4*x^13*y^4-20*x^12*y^5-12*x^11*y^6-2*x^10*y^7+2*x^ 13*y^3+x^12*y^4+18*x^11*y^5+x^10*y^6+6*x^9*y^7+x^8*y^8+2*x^12*y^3-10*x^11*y^4-2 *x^10*y^5-8*x^9*y^6+x^12*y^2+2*x^11*y^3+x^10*y^4+8*x^9*y^5+3*x^8*y^6+4*x^7*y^7+ 2*x^10*y^3-10*x^9*y^4-4*x^7*y^6-2*x^10*y^2+6*x^9*y^3+10*x^7*y^5-8*x^8*y^3-10*x^ 7*y^4-4*x^6*y^5-2*x^9*y+3*x^8*y^2-2*x^7*y^3+9*x^6*y^4-2*x^5*y^5-2*x^7*y^2-8*x^6 *y^3-4*x^5*y^4+2*x^7*y-2*x^6*y^2-4*x^4*y^4-4*x^4*y^3+x^6+2*x^4*y^2-8*x^3*y^3+2* x^3*y-4*x^2*y^2-2*x*y-1)/(x^12*y^8-2*x^12*y^7+x^12*y^6-2*x^8*y^6-2*x^9*y^4+2*x^ 8*y^5+4*x^9*y^3+3*x^8*y^4-4*x^8*y^3-2*x^9*y+x^8*y^2+x^4*y^4+2*x^5*y^2+x^6+2*x^4 *y^2+2*x^3*y-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 14 bors of a random word of length n tends to n times, (a + 4 a + 13 a 13 12 11 10 9 8 7 6 5 + 26 a + 38 a + 36 a + 13 a - 20 a - 50 a - 50 a - 31 a - 4 a 4 3 2 / 2 3 2 + 16 a + 16 a + 12 a + 4 a + 1) / ((a + 1) (3 a + 5 a + 6 a + 3) / 2 3 2 (a + 1) a ) 6 5 4 3 where a is the root of the polynomial, x + 2 x + 3 x + 2 x - 1, and in decimals this is, 0.4605645036 BTW the ratio for words with, 500, letters is, 0.4618798372 ------------------------------------------------ "Theorem Number 69" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 7 9 6 9 5 9 4 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 6 9 3 7 5 6 6 7 4 6 5 7 3 6 4 + x y + x y - 3 x y + x y + 3 x y + 2 x y - x y - 5 x y 5 5 6 3 5 4 4 4 5 2 4 3 5 4 2 + 4 x y + 2 x y - 5 x y + 2 x y + 2 x y - 2 x y - x y - x y 3 3 4 3 2 2 2 2 / 8 5 - 4 x y + x y + 2 x y - 2 x y + x y - x y + x - 1) / (x y / 8 4 7 5 8 3 7 4 6 3 6 2 5 3 5 4 - 2 x y + x y + x y - x y + x y - x y - x y + x y - x y 2 + x y - x y - x + 1) and in Maple notation -(x^9*y^7-4*x^9*y^6+6*x^9*y^5-4*x^9*y^4+x^7*y^6+x^9*y^3-3*x^7*y^5+x^6*y^6+3*x^7 *y^4+2*x^6*y^5-x^7*y^3-5*x^6*y^4+4*x^5*y^5+2*x^6*y^3-5*x^5*y^4+2*x^4*y^4+2*x^5* y^2-2*x^4*y^3-x^5*y-x^4*y^2-4*x^3*y^3+x^4*y+2*x^3*y^2-2*x^2*y^2+x^2*y-x*y+x-1)/ (x^8*y^5-2*x^8*y^4+x^7*y^5+x^8*y^3-x^7*y^4+x^6*y^3-x^6*y^2-x^5*y^3+x^5*y-x^4*y+ x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 6 4 3 2 a - 2 a + 3 a - 6 a + 2 a + 1 ---------------------------------- 3 3 2 (2 a - a + 1) (2 a + a + 1) 4 2 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.4472135973 BTW the ratio for words with, 500, letters is, 0.4489177212 ------------------------------------------------ "Theorem Number 70" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 10 16 9 15 10 ) | ) C(m, n) x y | = (2 x y - 10 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 8 15 9 14 10 16 7 15 8 14 9 + 20 x y + 20 x y + 2 x y - 20 x y - 40 x y - 10 x y 16 6 15 7 14 8 13 9 16 5 15 6 + 10 x y + 40 x y + 21 x y - 2 x y - 2 x y - 20 x y 14 7 13 8 12 9 15 5 14 6 13 7 - 24 x y + 10 x y + 2 x y + 4 x y + 16 x y - 20 x y 12 8 14 5 13 6 12 7 14 4 13 5 - 13 x y - 6 x y + 20 x y + 30 x y + x y - 10 x y 12 6 10 8 13 4 12 5 10 7 12 4 - 34 x y + 3 x y + 2 x y + 22 x y - 10 x y - 9 x y 10 6 9 7 8 8 12 3 10 5 9 6 8 7 + 17 x y + 2 x y + x y + 2 x y - 20 x y - 8 x y + 2 x y 10 4 9 5 8 6 7 7 10 3 9 4 8 5 + 14 x y + 8 x y - 12 x y + 2 x y - 4 x y - 2 x y + 14 x y 7 6 9 3 8 4 7 5 6 6 9 2 7 4 + 2 x y + 2 x y - 4 x y - 10 x y + 2 x y - 2 x y + 10 x y 6 5 8 2 7 3 6 4 5 5 7 2 5 4 - 4 x y - x y - 2 x y + 2 x y + 4 x y - 2 x y - 8 x y 6 2 4 4 5 2 4 3 3 3 4 3 2 2 - x y - x y + 2 x y - 4 x y - 6 x y + 2 x y + 2 x y - 3 x y / 10 4 9 5 8 6 10 3 9 4 10 2 - 2 x y - 1) / (x y - 2 x y - x y - 2 x y + 4 x y + x y / 9 3 8 4 8 3 7 3 6 4 7 2 6 3 - 2 x y + 3 x y - 2 x y + 2 x y + x y - 2 x y - 2 x y 6 2 5 3 5 2 4 3 2 2 + x y - 2 x y + 2 x y + 2 x y + 2 x y + x y - 1) and in Maple notation (2*x^16*y^10-10*x^16*y^9-4*x^15*y^10+20*x^16*y^8+20*x^15*y^9+2*x^14*y^10-20*x^ 16*y^7-40*x^15*y^8-10*x^14*y^9+10*x^16*y^6+40*x^15*y^7+21*x^14*y^8-2*x^13*y^9-2 *x^16*y^5-20*x^15*y^6-24*x^14*y^7+10*x^13*y^8+2*x^12*y^9+4*x^15*y^5+16*x^14*y^6 -20*x^13*y^7-13*x^12*y^8-6*x^14*y^5+20*x^13*y^6+30*x^12*y^7+x^14*y^4-10*x^13*y^ 5-34*x^12*y^6+3*x^10*y^8+2*x^13*y^4+22*x^12*y^5-10*x^10*y^7-9*x^12*y^4+17*x^10* y^6+2*x^9*y^7+x^8*y^8+2*x^12*y^3-20*x^10*y^5-8*x^9*y^6+2*x^8*y^7+14*x^10*y^4+8* x^9*y^5-12*x^8*y^6+2*x^7*y^7-4*x^10*y^3-2*x^9*y^4+14*x^8*y^5+2*x^7*y^6+2*x^9*y^ 3-4*x^8*y^4-10*x^7*y^5+2*x^6*y^6-2*x^9*y^2+10*x^7*y^4-4*x^6*y^5-x^8*y^2-2*x^7*y ^3+2*x^6*y^4+4*x^5*y^5-2*x^7*y^2-8*x^5*y^4-x^6*y^2-x^4*y^4+2*x^5*y^2-4*x^4*y^3-\ 6*x^3*y^3+2*x^4*y+2*x^3*y-3*x^2*y^2-2*x*y-1)/(x^10*y^4-2*x^9*y^5-x^8*y^6-2*x^10 *y^3+4*x^9*y^4+x^10*y^2-2*x^9*y^3+3*x^8*y^4-2*x^8*y^3+2*x^7*y^3+x^6*y^4-2*x^7*y ^2-2*x^6*y^3+x^6*y^2-2*x^5*y^3+2*x^5*y^2+2*x^4*y+2*x^3*y+x^2*y^2-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 4 10 9 8 7 6 5 4 3 2 (a + 2 a + 5 a - 4 a - a - 14 a - a - 4 a + 5 a + 2 a + 1) 2 / 2 2 2 2 2 (4 a + 3 a + 1) a / ((a + 1) (6 a + 1) (a + 1) ) / 4 3 2 where a is the root of the polynomial, 2 x + 2 x + x - 1, and in decimals this is, 0.4335497588 BTW the ratio for words with, 500, letters is, 0.4353937480 ------------------------------------------------ "Theorem Number 71" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 11 17 10 16 11 ) | ) C(m, n) x y | = - (x y - 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 12 17 9 16 10 15 11 17 8 16 9 - 2 x y + 15 x y - 4 x y + 9 x y - 20 x y + 5 x y 15 10 14 11 17 7 15 9 14 10 17 6 - 13 x y - x y + 15 x y + x y + 3 x y - 6 x y 16 7 15 8 14 9 13 10 17 5 16 6 15 7 - 5 x y + 15 x y + x y - x y + x y + 4 x y - 13 x y 14 8 13 9 16 5 15 6 14 7 13 8 - 14 x y + 6 x y - x y + x y + 21 x y - 13 x y 12 9 15 5 14 6 13 7 12 8 15 4 - 2 x y + 3 x y - 13 x y + 12 x y + 7 x y - x y 14 5 13 6 12 7 11 8 13 5 12 6 + 3 x y - 3 x y - 10 x y + 2 x y - 2 x y + 8 x y 11 7 10 8 13 4 12 5 11 6 10 7 12 4 - 3 x y + x y + x y - 4 x y - 6 x y - x y + x y 11 5 10 6 9 7 11 4 10 5 9 6 11 3 + 16 x y + x y - 3 x y - 12 x y - 6 x y + 7 x y + 3 x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 + 9 x y - 2 x y - 3 x y - 5 x y - 7 x y + 7 x y - 3 x y 10 2 9 3 8 4 7 5 6 6 9 2 8 3 + x y + 8 x y - 2 x y + 6 x y + x y - 3 x y - 6 x y 7 4 6 5 8 2 7 3 6 4 5 5 8 - 5 x y - 2 x y + 5 x y + 3 x y + 2 x y + 3 x y - x y 7 2 6 3 5 4 7 6 2 5 3 4 4 6 - 2 x y - 3 x y - 2 x y + x y + 2 x y - x y + 3 x y - x y 5 2 5 4 2 3 3 3 2 3 2 2 2 - x y + x y - 2 x y - 4 x y + 3 x y - x y - 2 x y + 2 x y / 13 7 12 8 13 6 12 7 11 8 - x y + x - 1) / (x y - x y - 3 x y + 2 x y + x y / 13 5 11 7 13 4 12 5 11 6 10 7 12 4 + 3 x y - x y - x y - 2 x y - x y - x y + x y 11 5 11 4 10 5 9 6 11 3 10 4 9 5 + 2 x y - 2 x y + 3 x y + 2 x y + x y - 2 x y - 2 x y 10 3 9 4 8 5 10 2 8 4 9 2 8 3 7 4 + x y - x y - 2 x y - x y + 2 x y + x y + x y + x y 8 2 7 3 8 6 3 7 6 2 5 3 6 5 2 - 2 x y - x y + x y - 2 x y - x y + x y + x y + x y + x y 5 4 2 3 2 3 - x y - 2 x y + x y + x y - x y - x + 1) and in Maple notation -(x^17*y^11-6*x^17*y^10+x^16*y^11-2*x^15*y^12+15*x^17*y^9-4*x^16*y^10+9*x^15*y^ 11-20*x^17*y^8+5*x^16*y^9-13*x^15*y^10-x^14*y^11+15*x^17*y^7+x^15*y^9+3*x^14*y^ 10-6*x^17*y^6-5*x^16*y^7+15*x^15*y^8+x^14*y^9-x^13*y^10+x^17*y^5+4*x^16*y^6-13* x^15*y^7-14*x^14*y^8+6*x^13*y^9-x^16*y^5+x^15*y^6+21*x^14*y^7-13*x^13*y^8-2*x^ 12*y^9+3*x^15*y^5-13*x^14*y^6+12*x^13*y^7+7*x^12*y^8-x^15*y^4+3*x^14*y^5-3*x^13 *y^6-10*x^12*y^7+2*x^11*y^8-2*x^13*y^5+8*x^12*y^6-3*x^11*y^7+x^10*y^8+x^13*y^4-\ 4*x^12*y^5-6*x^11*y^6-x^10*y^7+x^12*y^4+16*x^11*y^5+x^10*y^6-3*x^9*y^7-12*x^11* y^4-6*x^10*y^5+7*x^9*y^6+3*x^11*y^3+9*x^10*y^4-2*x^9*y^5-3*x^8*y^6-5*x^10*y^3-7 *x^9*y^4+7*x^8*y^5-3*x^7*y^6+x^10*y^2+8*x^9*y^3-2*x^8*y^4+6*x^7*y^5+x^6*y^6-3*x ^9*y^2-6*x^8*y^3-5*x^7*y^4-2*x^6*y^5+5*x^8*y^2+3*x^7*y^3+2*x^6*y^4+3*x^5*y^5-x^ 8*y-2*x^7*y^2-3*x^6*y^3-2*x^5*y^4+x^7*y+2*x^6*y^2-x^5*y^3+3*x^4*y^4-x^6*y-x^5*y ^2+x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^13*y ^7-x^12*y^8-3*x^13*y^6+2*x^12*y^7+x^11*y^8+3*x^13*y^5-x^11*y^7-x^13*y^4-2*x^12* y^5-x^11*y^6-x^10*y^7+x^12*y^4+2*x^11*y^5-2*x^11*y^4+3*x^10*y^5+2*x^9*y^6+x^11* y^3-2*x^10*y^4-2*x^9*y^5+x^10*y^3-x^9*y^4-2*x^8*y^5-x^10*y^2+2*x^8*y^4+x^9*y^2+ x^8*y^3+x^7*y^4-2*x^8*y^2-x^7*y^3+x^8*y-2*x^6*y^3-x^7*y+x^6*y^2+x^5*y^3+x^6*y+x ^5*y^2-x^5*y-2*x^4*y^2+x^3*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 2 12 11 10 9 8 7 6 5 4 3 (a - 2 a - 3 a + 4 a - 3 a - a + 9 a - 8 a - 5 a + 7 a - 1) / 6 4 3 2 6 4 3 / ((7 a - 5 a + 8 a - 6 a + 2) (a - a + 2 a + 1)) / 7 5 4 3 where a is the root of the polynomial, x - x + 2 x - 2 x + 2 x - 1, and in decimals this is, 0.4302842106 BTW the ratio for words with, 500, letters is, 0.4317146490 ------------------------------------------------ "Theorem Number 72" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 11 17 10 16 11 ) | ) C(m, n) x y | = - (x y - 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 12 17 9 16 10 15 11 17 8 16 9 - 2 x y + 15 x y - 4 x y + 9 x y - 20 x y + 5 x y 15 10 14 11 17 7 15 9 14 10 17 6 - 13 x y - x y + 15 x y + x y + 3 x y - 6 x y 16 7 15 8 14 9 13 10 17 5 16 6 15 7 - 5 x y + 15 x y + x y - x y + x y + 4 x y - 13 x y 14 8 13 9 16 5 15 6 14 7 13 8 - 14 x y + 6 x y - x y + x y + 21 x y - 13 x y 12 9 15 5 14 6 13 7 12 8 15 4 - 2 x y + 3 x y - 13 x y + 12 x y + 7 x y - x y 14 5 13 6 12 7 11 8 13 5 12 6 + 3 x y - 3 x y - 10 x y + 2 x y - 2 x y + 8 x y 11 7 10 8 13 4 12 5 11 6 10 7 12 4 - 3 x y + x y + x y - 4 x y - 6 x y - x y + x y 11 5 10 6 9 7 11 4 10 5 9 6 11 3 + 16 x y + x y - 3 x y - 12 x y - 6 x y + 7 x y + 3 x y 10 4 9 5 8 6 10 3 9 4 8 5 7 6 + 9 x y - 2 x y - 3 x y - 5 x y - 7 x y + 7 x y - 3 x y 10 2 9 3 8 4 7 5 6 6 9 2 8 3 + x y + 8 x y - 2 x y + 6 x y + x y - 3 x y - 6 x y 7 4 6 5 8 2 7 3 6 4 5 5 8 - 5 x y - 2 x y + 5 x y + 3 x y + 2 x y + 3 x y - x y 7 2 6 3 5 4 7 6 2 5 3 4 4 6 - 2 x y - 3 x y - 2 x y + x y + 2 x y - x y + 3 x y - x y 5 2 5 4 2 3 3 3 2 3 2 2 2 - x y + x y - 2 x y - 4 x y + 3 x y - x y - 2 x y + 2 x y / 13 7 12 8 13 6 12 7 11 8 - x y + x - 1) / (x y - x y - 3 x y + 2 x y + x y / 13 5 11 7 13 4 12 5 11 6 10 7 12 4 + 3 x y - x y - x y - 2 x y - x y - x y + x y 11 5 11 4 10 5 9 6 11 3 10 4 9 5 + 2 x y - 2 x y + 3 x y + 2 x y + x y - 2 x y - 2 x y 10 3 9 4 8 5 10 2 8 4 9 2 8 3 7 4 + x y - x y - 2 x y - x y + 2 x y + x y + x y + x y 8 2 7 3 8 6 3 7 6 2 5 3 6 5 2 - 2 x y - x y + x y - 2 x y - x y + x y + x y + x y + x y 5 4 2 3 2 3 - x y - 2 x y + x y + x y - x y - x + 1) and in Maple notation -(x^17*y^11-6*x^17*y^10+x^16*y^11-2*x^15*y^12+15*x^17*y^9-4*x^16*y^10+9*x^15*y^ 11-20*x^17*y^8+5*x^16*y^9-13*x^15*y^10-x^14*y^11+15*x^17*y^7+x^15*y^9+3*x^14*y^ 10-6*x^17*y^6-5*x^16*y^7+15*x^15*y^8+x^14*y^9-x^13*y^10+x^17*y^5+4*x^16*y^6-13* x^15*y^7-14*x^14*y^8+6*x^13*y^9-x^16*y^5+x^15*y^6+21*x^14*y^7-13*x^13*y^8-2*x^ 12*y^9+3*x^15*y^5-13*x^14*y^6+12*x^13*y^7+7*x^12*y^8-x^15*y^4+3*x^14*y^5-3*x^13 *y^6-10*x^12*y^7+2*x^11*y^8-2*x^13*y^5+8*x^12*y^6-3*x^11*y^7+x^10*y^8+x^13*y^4-\ 4*x^12*y^5-6*x^11*y^6-x^10*y^7+x^12*y^4+16*x^11*y^5+x^10*y^6-3*x^9*y^7-12*x^11* y^4-6*x^10*y^5+7*x^9*y^6+3*x^11*y^3+9*x^10*y^4-2*x^9*y^5-3*x^8*y^6-5*x^10*y^3-7 *x^9*y^4+7*x^8*y^5-3*x^7*y^6+x^10*y^2+8*x^9*y^3-2*x^8*y^4+6*x^7*y^5+x^6*y^6-3*x ^9*y^2-6*x^8*y^3-5*x^7*y^4-2*x^6*y^5+5*x^8*y^2+3*x^7*y^3+2*x^6*y^4+3*x^5*y^5-x^ 8*y-2*x^7*y^2-3*x^6*y^3-2*x^5*y^4+x^7*y+2*x^6*y^2-x^5*y^3+3*x^4*y^4-x^6*y-x^5*y ^2+x^5*y-2*x^4*y^2-4*x^3*y^3+3*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^13*y ^7-x^12*y^8-3*x^13*y^6+2*x^12*y^7+x^11*y^8+3*x^13*y^5-x^11*y^7-x^13*y^4-2*x^12* y^5-x^11*y^6-x^10*y^7+x^12*y^4+2*x^11*y^5-2*x^11*y^4+3*x^10*y^5+2*x^9*y^6+x^11* y^3-2*x^10*y^4-2*x^9*y^5+x^10*y^3-x^9*y^4-2*x^8*y^5-x^10*y^2+2*x^8*y^4+x^9*y^2+ x^8*y^3+x^7*y^4-2*x^8*y^2-x^7*y^3+x^8*y-2*x^6*y^3-x^7*y+x^6*y^2+x^5*y^3+x^6*y+x ^5*y^2-x^5*y-2*x^4*y^2+x^3*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, - 2 12 11 10 9 8 7 6 5 4 3 (a - 2 a - 3 a + 4 a - 3 a - a + 9 a - 8 a - 5 a + 7 a - 1) / 6 4 3 2 6 4 3 / ((7 a - 5 a + 8 a - 6 a + 2) (a - a + 2 a + 1)) / 7 5 4 3 where a is the root of the polynomial, x - x + 2 x - 2 x + 2 x - 1, and in decimals this is, 0.4302842106 BTW the ratio for words with, 500, letters is, 0.4317146490 ------------------------------------------------ "Theorem Number 73" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 18 10 17 11 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 10 18 8 17 9 16 10 18 7 17 8 - 5 x y - 5 x y + 10 x y - 2 x y + 4 x y - 11 x y 16 9 15 10 18 6 17 7 16 8 15 9 + 9 x y + 2 x y - x y + 9 x y - 18 x y - 6 x y 17 6 16 7 15 8 14 9 17 5 16 6 - 7 x y + 22 x y + 3 x y + 5 x y + 4 x y - 18 x y 15 7 14 8 13 9 17 4 16 5 15 6 + 8 x y - 16 x y + 3 x y - x y + 9 x y - 12 x y 14 7 13 8 16 4 15 5 14 6 13 7 + 13 x y - 10 x y - 2 x y + 6 x y + 7 x y + 10 x y 12 8 15 4 14 5 12 7 11 8 14 4 - 6 x y - x y - 14 x y + 16 x y - 5 x y + 5 x y 13 5 12 6 11 7 13 4 12 5 11 6 - 5 x y - 11 x y + 10 x y + 2 x y - 2 x y - 4 x y 10 7 12 4 11 5 10 6 9 7 12 3 11 4 - 7 x y + 2 x y + x y + 19 x y - x y + 2 x y - 6 x y 10 5 9 6 12 2 11 3 10 4 9 5 8 6 - 16 x y - x y - x y + 4 x y + 4 x y - 2 x y - 3 x y 11 2 10 3 9 4 8 5 11 10 2 9 3 + x y - x y + 10 x y + 9 x y - x y + x y - 6 x y 8 4 7 5 8 3 7 4 6 5 8 2 7 3 - 6 x y + 6 x y - 2 x y - 2 x y + x y + x y - 3 x y 6 4 8 7 2 5 4 7 6 2 5 3 4 4 + 2 x y + x y - x y + 4 x y + x y - 3 x y - 3 x y - 5 x y 5 2 4 3 6 4 2 3 3 5 3 2 3 2 2 + x y + 4 x y + x + x y + 4 x y - x - 2 x y - x y + 2 x y 2 / 12 7 12 6 11 6 12 4 - x y + x y - x + 1) / (x y - 2 x y + 2 x y + 2 x y / 11 5 10 6 12 3 11 4 10 5 11 3 11 2 10 3 - 4 x y + x y - x y + x y - x y + x y + x y - x y 8 5 11 10 2 9 3 8 4 9 2 8 3 7 4 8 - x y - x y + x y + x y + x y - x y - x y - x y + x y 7 2 7 6 2 6 6 4 2 5 3 2 + x y - x y - 3 x y + 2 x y + x + x y - x - x y + x y - x y - x + 1) and in Maple notation (x^18*y^12-4*x^18*y^11+5*x^18*y^10+x^17*y^11-5*x^17*y^10-5*x^18*y^8+10*x^17*y^9 -2*x^16*y^10+4*x^18*y^7-11*x^17*y^8+9*x^16*y^9+2*x^15*y^10-x^18*y^6+9*x^17*y^7-\ 18*x^16*y^8-6*x^15*y^9-7*x^17*y^6+22*x^16*y^7+3*x^15*y^8+5*x^14*y^9+4*x^17*y^5-\ 18*x^16*y^6+8*x^15*y^7-16*x^14*y^8+3*x^13*y^9-x^17*y^4+9*x^16*y^5-12*x^15*y^6+ 13*x^14*y^7-10*x^13*y^8-2*x^16*y^4+6*x^15*y^5+7*x^14*y^6+10*x^13*y^7-6*x^12*y^8 -x^15*y^4-14*x^14*y^5+16*x^12*y^7-5*x^11*y^8+5*x^14*y^4-5*x^13*y^5-11*x^12*y^6+ 10*x^11*y^7+2*x^13*y^4-2*x^12*y^5-4*x^11*y^6-7*x^10*y^7+2*x^12*y^4+x^11*y^5+19* x^10*y^6-x^9*y^7+2*x^12*y^3-6*x^11*y^4-16*x^10*y^5-x^9*y^6-x^12*y^2+4*x^11*y^3+ 4*x^10*y^4-2*x^9*y^5-3*x^8*y^6+x^11*y^2-x^10*y^3+10*x^9*y^4+9*x^8*y^5-x^11*y+x^ 10*y^2-6*x^9*y^3-6*x^8*y^4+6*x^7*y^5-2*x^8*y^3-2*x^7*y^4+x^6*y^5+x^8*y^2-3*x^7* y^3+2*x^6*y^4+x^8*y-x^7*y^2+4*x^5*y^4+x^7*y-3*x^6*y^2-3*x^5*y^3-5*x^4*y^4+x^5*y ^2+4*x^4*y^3+x^6+x^4*y^2+4*x^3*y^3-x^5-2*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y-x+1) /(x^12*y^7-2*x^12*y^6+2*x^11*y^6+2*x^12*y^4-4*x^11*y^5+x^10*y^6-x^12*y^3+x^11*y ^4-x^10*y^5+x^11*y^3+x^11*y^2-x^10*y^3-x^8*y^5-x^11*y+x^10*y^2+x^9*y^3+x^8*y^4- x^9*y^2-x^8*y^3-x^7*y^4+x^8*y+x^7*y^2-x^7*y-3*x^6*y^2+2*x^6*y+x^6+x^4*y^2-x^5-x ^3*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, 2 (a - 3 a + a 11 10 9 8 7 6 5 4 3 - 7 a + 7 a - 7 a + 10 a - 16 a + 9 a - 9 a + 13 a - 10 a 2 / 6 4 3 2 + 2 a + 1) / ((7 a + 5 a - 4 a + 3 a - 2 a + 2) / 7 6 5 3 2 (a + a + a + a + a + 1)) 7 5 4 3 2 where a is the root of the polynomial, x + x - x + x - x + 2 x - 1, and in decimals this is, 0.4198460104 BTW the ratio for words with, 500, letters is, 0.4218053786 ------------------------------------------------ "Theorem Number 74" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [2, 1, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 18 10 17 11 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 10 18 8 17 9 16 10 18 7 17 8 - 5 x y - 5 x y + 10 x y - 2 x y + 4 x y - 11 x y 16 9 15 10 18 6 17 7 16 8 15 9 + 9 x y + 2 x y - x y + 9 x y - 18 x y - 6 x y 17 6 16 7 15 8 14 9 17 5 16 6 - 7 x y + 22 x y + 3 x y + 5 x y + 4 x y - 18 x y 15 7 14 8 13 9 17 4 16 5 15 6 + 8 x y - 16 x y + 3 x y - x y + 9 x y - 12 x y 14 7 13 8 16 4 15 5 14 6 13 7 + 13 x y - 10 x y - 2 x y + 6 x y + 7 x y + 10 x y 12 8 15 4 14 5 12 7 11 8 14 4 - 6 x y - x y - 14 x y + 16 x y - 5 x y + 5 x y 13 5 12 6 11 7 13 4 12 5 11 6 - 5 x y - 11 x y + 10 x y + 2 x y - 2 x y - 4 x y 10 7 12 4 11 5 10 6 9 7 12 3 11 4 - 7 x y + 2 x y + x y + 19 x y - x y + 2 x y - 6 x y 10 5 9 6 12 2 11 3 10 4 9 5 8 6 - 16 x y - x y - x y + 4 x y + 4 x y - 2 x y - 3 x y 11 2 10 3 9 4 8 5 11 10 2 9 3 + x y - x y + 10 x y + 9 x y - x y + x y - 6 x y 8 4 7 5 8 3 7 4 6 5 8 2 7 3 - 6 x y + 6 x y - 2 x y - 2 x y + x y + x y - 3 x y 6 4 8 7 2 5 4 7 6 2 5 3 4 4 + 2 x y + x y - x y + 4 x y + x y - 3 x y - 3 x y - 5 x y 5 2 4 3 6 4 2 3 3 5 3 2 3 2 2 + x y + 4 x y + x + x y + 4 x y - x - 2 x y - x y + 2 x y 2 / 12 7 12 6 11 6 12 4 - x y + x y - x + 1) / (x y - 2 x y + 2 x y + 2 x y / 11 5 10 6 12 3 11 4 10 5 11 3 11 2 10 3 - 4 x y + x y - x y + x y - x y + x y + x y - x y 8 5 11 10 2 9 3 8 4 9 2 8 3 7 4 8 - x y - x y + x y + x y + x y - x y - x y - x y + x y 7 2 7 6 2 6 6 4 2 5 3 2 + x y - x y - 3 x y + 2 x y + x + x y - x - x y + x y - x y - x + 1) and in Maple notation (x^18*y^12-4*x^18*y^11+5*x^18*y^10+x^17*y^11-5*x^17*y^10-5*x^18*y^8+10*x^17*y^9 -2*x^16*y^10+4*x^18*y^7-11*x^17*y^8+9*x^16*y^9+2*x^15*y^10-x^18*y^6+9*x^17*y^7-\ 18*x^16*y^8-6*x^15*y^9-7*x^17*y^6+22*x^16*y^7+3*x^15*y^8+5*x^14*y^9+4*x^17*y^5-\ 18*x^16*y^6+8*x^15*y^7-16*x^14*y^8+3*x^13*y^9-x^17*y^4+9*x^16*y^5-12*x^15*y^6+ 13*x^14*y^7-10*x^13*y^8-2*x^16*y^4+6*x^15*y^5+7*x^14*y^6+10*x^13*y^7-6*x^12*y^8 -x^15*y^4-14*x^14*y^5+16*x^12*y^7-5*x^11*y^8+5*x^14*y^4-5*x^13*y^5-11*x^12*y^6+ 10*x^11*y^7+2*x^13*y^4-2*x^12*y^5-4*x^11*y^6-7*x^10*y^7+2*x^12*y^4+x^11*y^5+19* x^10*y^6-x^9*y^7+2*x^12*y^3-6*x^11*y^4-16*x^10*y^5-x^9*y^6-x^12*y^2+4*x^11*y^3+ 4*x^10*y^4-2*x^9*y^5-3*x^8*y^6+x^11*y^2-x^10*y^3+10*x^9*y^4+9*x^8*y^5-x^11*y+x^ 10*y^2-6*x^9*y^3-6*x^8*y^4+6*x^7*y^5-2*x^8*y^3-2*x^7*y^4+x^6*y^5+x^8*y^2-3*x^7* y^3+2*x^6*y^4+x^8*y-x^7*y^2+4*x^5*y^4+x^7*y-3*x^6*y^2-3*x^5*y^3-5*x^4*y^4+x^5*y ^2+4*x^4*y^3+x^6+x^4*y^2+4*x^3*y^3-x^5-2*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y-x+1) /(x^12*y^7-2*x^12*y^6+2*x^11*y^6+2*x^12*y^4-4*x^11*y^5+x^10*y^6-x^12*y^3+x^11*y ^4-x^10*y^5+x^11*y^3+x^11*y^2-x^10*y^3-x^8*y^5-x^11*y+x^10*y^2+x^9*y^3+x^8*y^4- x^9*y^2-x^8*y^3-x^7*y^4+x^8*y+x^7*y^2-x^7*y-3*x^6*y^2+2*x^6*y+x^6+x^4*y^2-x^5-x ^3*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 12 bors of a random word of length n tends to n times, 2 (a - 3 a + a 11 10 9 8 7 6 5 4 3 - 7 a + 7 a - 7 a + 10 a - 16 a + 9 a - 9 a + 13 a - 10 a 2 / 6 4 3 2 + 2 a + 1) / ((7 a + 5 a - 4 a + 3 a - 2 a + 2) / 7 6 5 3 2 (a + a + a + a + a + 1)) 7 5 4 3 2 where a is the root of the polynomial, x + x - x + x - x + 2 x - 1, and in decimals this is, 0.4198460104 BTW the ratio for words with, 500, letters is, 0.4218053786 ------------------------------------------------ "Theorem Number 75" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 17 12 ) | ) C(m, n) x y | = (2 x y - 10 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 17 11 16 12 18 9 17 10 16 11 + 20 x y - 15 x y + x y - 20 x y + 30 x y - 8 x y 18 8 17 9 16 10 15 11 18 7 17 8 + 10 x y - 30 x y + 25 x y - 3 x y - 2 x y + 15 x y 16 9 15 10 14 11 17 7 16 8 15 9 - 40 x y + 14 x y - 2 x y - 3 x y + 35 x y - 29 x y 14 10 13 11 16 7 15 8 14 9 13 10 + 9 x y - x y - 16 x y + 36 x y - 18 x y + 6 x y 16 6 15 7 14 8 13 9 12 10 15 6 + 3 x y - 29 x y + 20 x y - 13 x y + 2 x y + 14 x y 14 7 13 8 12 9 15 5 14 6 13 7 - 12 x y + 12 x y - 13 x y - 3 x y + 3 x y - 5 x y 12 8 11 9 13 6 12 7 11 8 13 5 + 24 x y - 5 x y + 4 x y - 10 x y + 14 x y - 5 x y 12 6 11 7 10 8 13 4 12 5 11 6 - 14 x y - 14 x y + 4 x y + 2 x y + 14 x y + 5 x y 10 7 9 8 12 4 11 5 9 7 8 8 12 3 - 6 x y + x y - 2 x y + 4 x y + 4 x y + x y - x y 11 4 10 5 9 6 11 3 10 4 9 5 8 6 - 7 x y + 4 x y - 6 x y + 3 x y - 5 x y + x y - 8 x y 10 3 9 4 8 5 7 6 10 2 9 3 8 4 + 5 x y - 2 x y + 16 x y - 2 x y - 2 x y + 2 x y - 10 x y 7 5 8 3 7 4 8 2 7 3 6 4 5 5 8 + 4 x y - x y - 8 x y + 3 x y + 5 x y - 4 x y + 2 x y - x y 6 3 5 4 5 3 4 4 4 3 4 2 3 3 + 3 x y - 3 x y + x y - 7 x y + 4 x y + 4 x y + 2 x y 4 3 2 2 / 13 7 13 6 12 7 - 2 x y - x y + x y + 1) / (2 x y - 6 x y + 3 x y / 13 5 12 6 11 7 13 4 12 5 11 6 12 4 + 6 x y - 8 x y + x y - 2 x y + 7 x y - 4 x y - 2 x y 11 5 10 6 11 4 10 5 9 6 11 3 10 4 + 6 x y - 2 x y - 4 x y + 2 x y - 2 x y + x y - x y 9 5 8 6 10 3 9 4 8 5 10 2 9 3 + 3 x y - x y + 3 x y - x y + 3 x y - 2 x y - x y 8 4 7 5 9 2 7 4 8 2 7 3 6 4 8 6 3 - 2 x y + x y + x y - x y + x y + x y + x y - x y - x y 5 3 5 2 5 4 2 4 3 2 2 - 2 x y + 3 x y - x y + 2 x y - 2 x y - x y + x y - 2 x y + 1) and in Maple notation (2*x^18*y^12-10*x^18*y^11+3*x^17*y^12+20*x^18*y^10-15*x^17*y^11+x^16*y^12-20*x^ 18*y^9+30*x^17*y^10-8*x^16*y^11+10*x^18*y^8-30*x^17*y^9+25*x^16*y^10-3*x^15*y^ 11-2*x^18*y^7+15*x^17*y^8-40*x^16*y^9+14*x^15*y^10-2*x^14*y^11-3*x^17*y^7+35*x^ 16*y^8-29*x^15*y^9+9*x^14*y^10-x^13*y^11-16*x^16*y^7+36*x^15*y^8-18*x^14*y^9+6* x^13*y^10+3*x^16*y^6-29*x^15*y^7+20*x^14*y^8-13*x^13*y^9+2*x^12*y^10+14*x^15*y^ 6-12*x^14*y^7+12*x^13*y^8-13*x^12*y^9-3*x^15*y^5+3*x^14*y^6-5*x^13*y^7+24*x^12* y^8-5*x^11*y^9+4*x^13*y^6-10*x^12*y^7+14*x^11*y^8-5*x^13*y^5-14*x^12*y^6-14*x^ 11*y^7+4*x^10*y^8+2*x^13*y^4+14*x^12*y^5+5*x^11*y^6-6*x^10*y^7+x^9*y^8-2*x^12*y ^4+4*x^11*y^5+4*x^9*y^7+x^8*y^8-x^12*y^3-7*x^11*y^4+4*x^10*y^5-6*x^9*y^6+3*x^11 *y^3-5*x^10*y^4+x^9*y^5-8*x^8*y^6+5*x^10*y^3-2*x^9*y^4+16*x^8*y^5-2*x^7*y^6-2*x ^10*y^2+2*x^9*y^3-10*x^8*y^4+4*x^7*y^5-x^8*y^3-8*x^7*y^4+3*x^8*y^2+5*x^7*y^3-4* x^6*y^4+2*x^5*y^5-x^8*y+3*x^6*y^3-3*x^5*y^4+x^5*y^3-7*x^4*y^4+4*x^4*y^3+4*x^4*y ^2+2*x^3*y^3-2*x^4*y-x^3*y+x^2*y^2+1)/(2*x^13*y^7-6*x^13*y^6+3*x^12*y^7+6*x^13* y^5-8*x^12*y^6+x^11*y^7-2*x^13*y^4+7*x^12*y^5-4*x^11*y^6-2*x^12*y^4+6*x^11*y^5-\ 2*x^10*y^6-4*x^11*y^4+2*x^10*y^5-2*x^9*y^6+x^11*y^3-x^10*y^4+3*x^9*y^5-x^8*y^6+ 3*x^10*y^3-x^9*y^4+3*x^8*y^5-2*x^10*y^2-x^9*y^3-2*x^8*y^4+x^7*y^5+x^9*y^2-x^7*y ^4+x^8*y^2+x^7*y^3+x^6*y^4-x^8*y-x^6*y^3-2*x^5*y^3+3*x^5*y^2-x^5*y+2*x^4*y^2-2* x^4*y-x^3*y+x^2*y^2-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 5 4 3 2 12 10 9 8 7 (7 a - 7 a + 7 a - 7 a + 4 a - 2) (a - 3 a - 4 a - 8 a - 8 a 6 5 4 3 2 / - 10 a - 6 a + 2 a - 2 a + 5 a + 2 a + 1) / ( / 6 3 2 5 4 3 2 2 (a + a - 2 a + a - 1) (6 a + 5 a + 4 a + 3 a + 1) ) 7 3 2 where a is the root of the polynomial, x - x + x - 2 x + 1, and in decimals this is, 0.4184584030 BTW the ratio for words with, 500, letters is, 0.4203753684 ------------------------------------------------ "Theorem Number 76" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 17 12 ) | ) C(m, n) x y | = (x y - 6 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 17 11 16 12 18 9 17 10 16 11 + 14 x y - 14 x y + x y - 15 x y + 41 x y - 6 x y 18 8 17 9 16 10 15 11 18 7 17 8 + 5 x y - 65 x y + 13 x y - x y + 4 x y + 60 x y 16 9 15 10 14 11 18 6 17 7 16 8 - 7 x y + 9 x y - x y - 4 x y - 32 x y - 19 x y 15 9 14 10 18 5 17 6 16 7 15 8 - 24 x y + 7 x y + x y + 9 x y + 40 x y + 27 x y 14 9 13 10 17 5 16 6 15 7 14 8 - 18 x y - 2 x y - x y - 33 x y - 14 x y + 21 x y 13 9 16 5 15 6 14 7 13 8 12 9 + 4 x y + 13 x y + 7 x y - 11 x y - 6 x y - 8 x y 16 4 15 5 14 6 13 7 12 8 11 9 - 2 x y - 8 x y + 3 x y + 17 x y + 22 x y - x y 15 4 14 5 13 6 12 7 15 3 14 4 + 5 x y - 2 x y - 25 x y - 18 x y - x y + x y 13 5 12 6 11 7 10 8 13 4 12 5 + 15 x y + 5 x y + 8 x y + x y - 3 x y - 7 x y 11 6 10 7 9 8 12 4 11 5 9 7 12 3 - 14 x y - 2 x y + x y + 9 x y + 9 x y - x y - 3 x y 11 4 10 5 9 6 8 7 11 3 9 5 8 6 - 3 x y + 2 x y + 3 x y + 3 x y + 2 x y - 4 x y + 7 x y 11 2 10 3 9 4 8 5 7 6 9 3 8 4 7 5 - x y - x y + 2 x y - 10 x y + 8 x y - x y - 4 x y - x y 8 3 7 4 7 3 5 5 7 2 6 3 5 4 + 4 x y - 13 x y + 2 x y - x y + 3 x y - 7 x y - 2 x y 6 2 5 3 4 4 6 4 3 5 4 2 3 3 + 3 x y - 2 x y - 3 x y + x y - 5 x y + x y + 3 x y - 8 x y 3 2 3 2 2 2 / 15 8 15 7 + 2 x y + x y - 4 x y + x y - 2 x y - 1) / (x y - 5 x y / 15 6 15 5 13 7 15 4 13 6 15 3 13 5 + 10 x y - 10 x y - x y + 5 x y + 3 x y - x y - 3 x y 12 6 11 7 13 4 12 5 11 6 12 4 11 5 + x y + x y + x y - 3 x y - 2 x y + 3 x y + x y 10 6 12 3 11 4 10 5 9 6 11 3 10 4 + x y - x y - x y - 3 x y - x y + 2 x y + 3 x y 11 2 10 3 9 4 8 5 9 3 8 4 7 4 6 4 - x y - x y + 2 x y - x y - x y + x y - x y - x y 7 2 6 2 5 3 6 4 3 5 4 2 3 2 + x y + x y + x y + x y + x y + x y + x y + x y + x y - 1) and in Maple notation (x^18*y^12-6*x^18*y^11+2*x^17*y^12+14*x^18*y^10-14*x^17*y^11+x^16*y^12-15*x^18* y^9+41*x^17*y^10-6*x^16*y^11+5*x^18*y^8-65*x^17*y^9+13*x^16*y^10-x^15*y^11+4*x^ 18*y^7+60*x^17*y^8-7*x^16*y^9+9*x^15*y^10-x^14*y^11-4*x^18*y^6-32*x^17*y^7-19*x ^16*y^8-24*x^15*y^9+7*x^14*y^10+x^18*y^5+9*x^17*y^6+40*x^16*y^7+27*x^15*y^8-18* x^14*y^9-2*x^13*y^10-x^17*y^5-33*x^16*y^6-14*x^15*y^7+21*x^14*y^8+4*x^13*y^9+13 *x^16*y^5+7*x^15*y^6-11*x^14*y^7-6*x^13*y^8-8*x^12*y^9-2*x^16*y^4-8*x^15*y^5+3* x^14*y^6+17*x^13*y^7+22*x^12*y^8-x^11*y^9+5*x^15*y^4-2*x^14*y^5-25*x^13*y^6-18* x^12*y^7-x^15*y^3+x^14*y^4+15*x^13*y^5+5*x^12*y^6+8*x^11*y^7+x^10*y^8-3*x^13*y^ 4-7*x^12*y^5-14*x^11*y^6-2*x^10*y^7+x^9*y^8+9*x^12*y^4+9*x^11*y^5-x^9*y^7-3*x^ 12*y^3-3*x^11*y^4+2*x^10*y^5+3*x^9*y^6+3*x^8*y^7+2*x^11*y^3-4*x^9*y^5+7*x^8*y^6 -x^11*y^2-x^10*y^3+2*x^9*y^4-10*x^8*y^5+8*x^7*y^6-x^9*y^3-4*x^8*y^4-x^7*y^5+4*x ^8*y^3-13*x^7*y^4+2*x^7*y^3-x^5*y^5+3*x^7*y^2-7*x^6*y^3-2*x^5*y^4+3*x^6*y^2-2*x ^5*y^3-3*x^4*y^4+x^6*y-5*x^4*y^3+x^5*y+3*x^4*y^2-8*x^3*y^3+2*x^3*y^2+x^3*y-4*x^ 2*y^2+x^2*y-2*x*y-1)/(x^15*y^8-5*x^15*y^7+10*x^15*y^6-10*x^15*y^5-x^13*y^7+5*x^ 15*y^4+3*x^13*y^6-x^15*y^3-3*x^13*y^5+x^12*y^6+x^11*y^7+x^13*y^4-3*x^12*y^5-2*x ^11*y^6+3*x^12*y^4+x^11*y^5+x^10*y^6-x^12*y^3-x^11*y^4-3*x^10*y^5-x^9*y^6+2*x^ 11*y^3+3*x^10*y^4-x^11*y^2-x^10*y^3+2*x^9*y^4-x^8*y^5-x^9*y^3+x^8*y^4-x^7*y^4-x ^6*y^4+x^7*y^2+x^6*y^2+x^5*y^3+x^6*y+x^4*y^3+x^5*y+x^4*y^2+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 9 bors of a random word of length n tends to n times, 2 (a + 3 a - 4 a 8 7 6 5 4 3 2 - 8 a - 11 a - 11 a - 9 a + 2 a - a + 5 a + 2 a + 1) 4 3 2 / 2 2 3 2 (6 a + 10 a + 8 a + 3 a + 2) a / ((a + 1) (a + 1) (a + a + 1) / 4 3 2 2 (5 a + 4 a + 3 a + 1) ) 6 5 4 3 2 where a is the root of the polynomial, x + 2 x + 2 x + x + x - 1, and in decimals this is, 0.4081414416 BTW the ratio for words with, 500, letters is, 0.4104265801 ------------------------------------------------ "Theorem Number 77" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 12 18 11 17 12 ) | ) C(m, n) x y | = (x y - 6 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 17 11 16 12 18 9 17 10 16 11 + 14 x y - 14 x y + x y - 15 x y + 41 x y - 6 x y 18 8 17 9 16 10 15 11 18 7 17 8 + 5 x y - 65 x y + 13 x y - x y + 4 x y + 60 x y 16 9 15 10 14 11 18 6 17 7 16 8 - 7 x y + 9 x y - x y - 4 x y - 32 x y - 19 x y 15 9 14 10 18 5 17 6 16 7 15 8 - 24 x y + 7 x y + x y + 9 x y + 40 x y + 27 x y 14 9 13 10 17 5 16 6 15 7 14 8 - 18 x y - 2 x y - x y - 33 x y - 14 x y + 21 x y 13 9 16 5 15 6 14 7 13 8 12 9 + 4 x y + 13 x y + 7 x y - 11 x y - 6 x y - 8 x y 16 4 15 5 14 6 13 7 12 8 11 9 - 2 x y - 8 x y + 3 x y + 17 x y + 22 x y - x y 15 4 14 5 13 6 12 7 15 3 14 4 + 5 x y - 2 x y - 25 x y - 18 x y - x y + x y 13 5 12 6 11 7 10 8 13 4 12 5 + 15 x y + 5 x y + 8 x y + x y - 3 x y - 7 x y 11 6 10 7 9 8 12 4 11 5 9 7 12 3 - 14 x y - 2 x y + x y + 9 x y + 9 x y - x y - 3 x y 11 4 10 5 9 6 8 7 11 3 9 5 8 6 - 3 x y + 2 x y + 3 x y + 3 x y + 2 x y - 4 x y + 7 x y 11 2 10 3 9 4 8 5 7 6 9 3 8 4 7 5 - x y - x y + 2 x y - 10 x y + 8 x y - x y - 4 x y - x y 8 3 7 4 7 3 5 5 7 2 6 3 5 4 + 4 x y - 13 x y + 2 x y - x y + 3 x y - 7 x y - 2 x y 6 2 5 3 4 4 6 4 3 5 4 2 3 3 + 3 x y - 2 x y - 3 x y + x y - 5 x y + x y + 3 x y - 8 x y 3 2 3 2 2 2 / 15 8 15 7 + 2 x y + x y - 4 x y + x y - 2 x y - 1) / (x y - 5 x y / 15 6 15 5 13 7 15 4 13 6 15 3 13 5 + 10 x y - 10 x y - x y + 5 x y + 3 x y - x y - 3 x y 12 6 11 7 13 4 12 5 11 6 12 4 11 5 + x y + x y + x y - 3 x y - 2 x y + 3 x y + x y 10 6 12 3 11 4 10 5 9 6 11 3 10 4 + x y - x y - x y - 3 x y - x y + 2 x y + 3 x y 11 2 10 3 9 4 8 5 9 3 8 4 7 4 6 4 - x y - x y + 2 x y - x y - x y + x y - x y - x y 7 2 6 2 5 3 6 4 3 5 4 2 3 2 + x y + x y + x y + x y + x y + x y + x y + x y + x y - 1) and in Maple notation (x^18*y^12-6*x^18*y^11+2*x^17*y^12+14*x^18*y^10-14*x^17*y^11+x^16*y^12-15*x^18* y^9+41*x^17*y^10-6*x^16*y^11+5*x^18*y^8-65*x^17*y^9+13*x^16*y^10-x^15*y^11+4*x^ 18*y^7+60*x^17*y^8-7*x^16*y^9+9*x^15*y^10-x^14*y^11-4*x^18*y^6-32*x^17*y^7-19*x ^16*y^8-24*x^15*y^9+7*x^14*y^10+x^18*y^5+9*x^17*y^6+40*x^16*y^7+27*x^15*y^8-18* x^14*y^9-2*x^13*y^10-x^17*y^5-33*x^16*y^6-14*x^15*y^7+21*x^14*y^8+4*x^13*y^9+13 *x^16*y^5+7*x^15*y^6-11*x^14*y^7-6*x^13*y^8-8*x^12*y^9-2*x^16*y^4-8*x^15*y^5+3* x^14*y^6+17*x^13*y^7+22*x^12*y^8-x^11*y^9+5*x^15*y^4-2*x^14*y^5-25*x^13*y^6-18* x^12*y^7-x^15*y^3+x^14*y^4+15*x^13*y^5+5*x^12*y^6+8*x^11*y^7+x^10*y^8-3*x^13*y^ 4-7*x^12*y^5-14*x^11*y^6-2*x^10*y^7+x^9*y^8+9*x^12*y^4+9*x^11*y^5-x^9*y^7-3*x^ 12*y^3-3*x^11*y^4+2*x^10*y^5+3*x^9*y^6+3*x^8*y^7+2*x^11*y^3-4*x^9*y^5+7*x^8*y^6 -x^11*y^2-x^10*y^3+2*x^9*y^4-10*x^8*y^5+8*x^7*y^6-x^9*y^3-4*x^8*y^4-x^7*y^5+4*x ^8*y^3-13*x^7*y^4+2*x^7*y^3-x^5*y^5+3*x^7*y^2-7*x^6*y^3-2*x^5*y^4+3*x^6*y^2-2*x ^5*y^3-3*x^4*y^4+x^6*y-5*x^4*y^3+x^5*y+3*x^4*y^2-8*x^3*y^3+2*x^3*y^2+x^3*y-4*x^ 2*y^2+x^2*y-2*x*y-1)/(x^15*y^8-5*x^15*y^7+10*x^15*y^6-10*x^15*y^5-x^13*y^7+5*x^ 15*y^4+3*x^13*y^6-x^15*y^3-3*x^13*y^5+x^12*y^6+x^11*y^7+x^13*y^4-3*x^12*y^5-2*x ^11*y^6+3*x^12*y^4+x^11*y^5+x^10*y^6-x^12*y^3-x^11*y^4-3*x^10*y^5-x^9*y^6+2*x^ 11*y^3+3*x^10*y^4-x^11*y^2-x^10*y^3+2*x^9*y^4-x^8*y^5-x^9*y^3+x^8*y^4-x^7*y^4-x ^6*y^4+x^7*y^2+x^6*y^2+x^5*y^3+x^6*y+x^4*y^3+x^5*y+x^4*y^2+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 9 bors of a random word of length n tends to n times, 2 (a + 3 a - 4 a 8 7 6 5 4 3 2 - 8 a - 11 a - 11 a - 9 a + 2 a - a + 5 a + 2 a + 1) 4 3 2 / 2 2 3 2 (6 a + 10 a + 8 a + 3 a + 2) a / ((a + 1) (a + 1) (a + a + 1) / 4 3 2 2 (5 a + 4 a + 3 a + 1) ) 6 5 4 3 2 where a is the root of the polynomial, x + 2 x + 2 x + x + x - 1, and in decimals this is, 0.4081414416 BTW the ratio for words with, 500, letters is, 0.4104265801 ------------------------------------------------ "Theorem Number 78" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [1, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 6 10 7 11 5 10 6 ) | ) C(m, n) x y | = (2 x y - x y - 6 x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 7 11 4 10 5 9 6 8 7 11 3 10 4 - x y + 6 x y - 8 x y + 3 x y - x y - 2 x y + 4 x y 9 5 8 6 7 7 10 3 8 5 7 6 10 2 9 3 - 3 x y + x y - x y + x y + 5 x y - 2 x y - x y + x y 8 4 7 5 9 2 8 3 7 4 6 5 9 7 3 - 9 x y + 6 x y + x y + 4 x y - x y + x y - x y - 3 x y 6 4 5 5 7 2 6 3 5 4 6 2 5 3 4 4 + x y - x y + x y - 5 x y + 4 x y + x y - 2 x y - 2 x y 6 5 2 4 3 5 3 3 4 3 2 4 3 + 2 x y - 2 x y + 2 x y + x y + 4 x y - x y - 2 x y + x - x y 2 2 3 2 / 8 2 8 7 2 + 2 x y - x - x y + x y - x + 1) / ((x - 1) (x y - x y + x y / 7 6 5 2 5 4 2 3 3 - x y - x y - x y + x y - x y + x y + x + x y - 1)) and in Maple notation (2*x^11*y^6-x^10*y^7-6*x^11*y^5+5*x^10*y^6-x^9*y^7+6*x^11*y^4-8*x^10*y^5+3*x^9* y^6-x^8*y^7-2*x^11*y^3+4*x^10*y^4-3*x^9*y^5+x^8*y^6-x^7*y^7+x^10*y^3+5*x^8*y^5-\ 2*x^7*y^6-x^10*y^2+x^9*y^3-9*x^8*y^4+6*x^7*y^5+x^9*y^2+4*x^8*y^3-x^7*y^4+x^6*y^ 5-x^9*y-3*x^7*y^3+x^6*y^4-x^5*y^5+x^7*y^2-5*x^6*y^3+4*x^5*y^4+x^6*y^2-2*x^5*y^3 -2*x^4*y^4+2*x^6*y-2*x^5*y^2+2*x^4*y^3+x^5*y+4*x^3*y^3-x^4*y-2*x^3*y^2+x^4-x^3* y+2*x^2*y^2-x^3-x^2*y+x*y-x+1)/(x-1)/(x^8*y^2-x^8*y+x^7*y^2-x^7*y-x^6*y-x^5*y^2 +x^5*y-x^4*y^2+x^3*y+x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3912376418 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 79" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 6 10 7 11 5 10 6 ) | ) C(m, n) x y | = (2 x y - x y - 6 x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 7 11 4 10 5 9 6 8 7 11 3 10 4 - x y + 6 x y - 8 x y + 3 x y - x y - 2 x y + 4 x y 9 5 8 6 7 7 10 3 8 5 7 6 10 2 9 3 - 3 x y + x y - x y + x y + 5 x y - 2 x y - x y + x y 8 4 7 5 9 2 8 3 7 4 6 5 9 7 3 - 9 x y + 6 x y + x y + 4 x y - x y + x y - x y - 3 x y 6 4 5 5 7 2 6 3 5 4 6 2 5 3 4 4 + x y - x y + x y - 5 x y + 4 x y + x y - 2 x y - 2 x y 6 5 2 4 3 5 3 3 4 3 2 4 3 + 2 x y - 2 x y + 2 x y + x y + 4 x y - x y - 2 x y + x - x y 2 2 3 2 / + 2 x y - x - x y + x y - x + 1) / ( / 5 2 5 4 2 4 3 4 3 (x y - x y + x y - x y - x y - x y + 1) (x - x - x + 1)) and in Maple notation (2*x^11*y^6-x^10*y^7-6*x^11*y^5+5*x^10*y^6-x^9*y^7+6*x^11*y^4-8*x^10*y^5+3*x^9* y^6-x^8*y^7-2*x^11*y^3+4*x^10*y^4-3*x^9*y^5+x^8*y^6-x^7*y^7+x^10*y^3+5*x^8*y^5-\ 2*x^7*y^6-x^10*y^2+x^9*y^3-9*x^8*y^4+6*x^7*y^5+x^9*y^2+4*x^8*y^3-x^7*y^4+x^6*y^ 5-x^9*y-3*x^7*y^3+x^6*y^4-x^5*y^5+x^7*y^2-5*x^6*y^3+4*x^5*y^4+x^6*y^2-2*x^5*y^3 -2*x^4*y^4+2*x^6*y-2*x^5*y^2+2*x^4*y^3+x^5*y+4*x^3*y^3-x^4*y-2*x^3*y^2+x^4-x^3* y+2*x^2*y^2-x^3-x^2*y+x*y-x+1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1)/(x^4-x ^3-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3912376418 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 80" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 10 6 10 5 10 4 8 6 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 3 8 5 7 6 8 4 7 5 8 3 7 4 - x y - 4 x y + 2 x y + 5 x y - x y - 2 x y - 4 x y 6 5 7 3 6 4 7 2 6 3 5 4 6 2 5 3 + 2 x y + 4 x y - 4 x y - x y + 2 x y - 3 x y + x y + x y 4 4 6 5 2 4 3 4 2 3 3 3 2 3 + 3 x y - x y + x y - 2 x y - x y - 4 x y + 2 x y + x y 2 2 2 / - 2 x y + x y - x y + x - 1) / ( / 5 2 5 4 2 4 3 (x y - x y + x y - x y - x y - x y + 1) (x - 1)) and in Maple notation (x^10*y^6-3*x^10*y^5+3*x^10*y^4+x^8*y^6-x^10*y^3-4*x^8*y^5+2*x^7*y^6+5*x^8*y^4- x^7*y^5-2*x^8*y^3-4*x^7*y^4+2*x^6*y^5+4*x^7*y^3-4*x^6*y^4-x^7*y^2+2*x^6*y^3-3*x ^5*y^4+x^6*y^2+x^5*y^3+3*x^4*y^4-x^6*y+x^5*y^2-2*x^4*y^3-x^4*y^2-4*x^3*y^3+2*x^ 3*y^2+x^3*y-2*x^2*y^2+x^2*y-x*y+x-1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1)/ (x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3911031983 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 81" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 7 9 6 9 5 8 6 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 4 8 5 7 6 8 4 7 5 6 6 8 3 7 4 - x y + 3 x y - 2 x y - 3 x y + 5 x y - x y + x y - 3 x y 6 5 7 3 5 5 7 2 6 3 5 4 6 2 5 3 + 2 x y - x y - 2 x y + x y - x y + 3 x y - x y - x y 4 4 6 4 3 3 3 3 2 3 2 2 2 - 2 x y + x y + 2 x y + 4 x y - 2 x y - x y + 2 x y - x y / 6 2 6 4 2 3 2 + x y - x + 1) / (x y - x y - x y + x y - x y + x y + x - 1) / and in Maple notation -(x^9*y^7-3*x^9*y^6+3*x^9*y^5-x^8*y^6-x^9*y^4+3*x^8*y^5-2*x^7*y^6-3*x^8*y^4+5*x ^7*y^5-x^6*y^6+x^8*y^3-3*x^7*y^4+2*x^6*y^5-x^7*y^3-2*x^5*y^5+x^7*y^2-x^6*y^3+3* x^5*y^4-x^6*y^2-x^5*y^3-2*x^4*y^4+x^6*y+2*x^4*y^3+4*x^3*y^3-2*x^3*y^2-x^3*y+2*x ^2*y^2-x^2*y+x*y-x+1)/(x^6*y^2-x^6*y-x^4*y^2+x^3*y-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3910570206 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 82" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 5 7 4 6 5 7 3 6 4 ) | ) C(m, n) x y | = - (x y - x y + x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 5 7 2 5 4 6 2 5 3 4 4 6 5 2 - 3 x y + x y + 4 x y - x y - 2 x y - 3 x y + x y + x y 4 3 4 2 3 3 3 2 3 2 2 2 + 2 x y + x y + 4 x y - 2 x y - x y + 2 x y - x y + x y - x / 5 2 5 4 2 4 3 + 1) / ((x y - x y + x y - x y - x y - x y + 1) (x - 1)) / and in Maple notation -(x^7*y^5-x^7*y^4+x^6*y^5-x^7*y^3-x^6*y^4-3*x^5*y^5+x^7*y^2+4*x^5*y^4-x^6*y^2-2 *x^5*y^3-3*x^4*y^4+x^6*y+x^5*y^2+2*x^4*y^3+x^4*y^2+4*x^3*y^3-2*x^3*y^2-x^3*y+2* x^2*y^2-x^2*y+x*y-x+1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3902406929 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 83" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 7 11 6 11 5 9 7 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 4 9 6 9 5 8 6 8 5 8 4 7 5 6 6 - x y + 2 x y - x y - 2 x y + 2 x y + x y - x y + x y 8 3 7 4 6 5 6 4 5 5 6 3 5 4 6 2 - x y + x y - x y + 3 x y + 2 x y - 2 x y - 4 x y + x y 5 3 4 4 6 5 2 4 3 4 2 3 3 3 2 + 3 x y + 3 x y - x y - x y - 2 x y - x y - 4 x y + 2 x y 3 2 2 2 / + x y - 2 x y + x y - x y + x - 1) / ( / 6 2 6 4 2 3 2 x y - x y - x y + x y - x y + x y + x - 1) and in Maple notation (x^11*y^7-3*x^11*y^6+3*x^11*y^5-x^9*y^7-x^11*y^4+2*x^9*y^6-x^9*y^5-2*x^8*y^6+2* x^8*y^5+x^8*y^4-x^7*y^5+x^6*y^6-x^8*y^3+x^7*y^4-x^6*y^5+3*x^6*y^4+2*x^5*y^5-2*x ^6*y^3-4*x^5*y^4+x^6*y^2+3*x^5*y^3+3*x^4*y^4-x^6*y-x^5*y^2-2*x^4*y^3-x^4*y^2-4* x^3*y^3+2*x^3*y^2+x^3*y-2*x^2*y^2+x^2*y-x*y+x-1)/(x^6*y^2-x^6*y-x^4*y^2+x^3*y-x ^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 6 5 4 3 2 3 2 2 (3 a + 3 a + 8 a + a + a - 5 a - 2 a - 1) (4 a - 3 a + 2 a - 2) - ------------------------------------------------------------------------- 2 2 6 3 2 (3 a + 1) (a - a - a - 1) 4 3 2 where a is the root of the polynomial, x - x + x - 2 x + 1, and in decimals this is, 0.3885080074 BTW the ratio for words with, 500, letters is, 0.3902020734 ------------------------------------------------ "Theorem Number 84" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 5 9 4 8 5 9 3 ) | ) C(m, n) x y | = - (4 x y - 8 x y - x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 4 7 5 8 3 7 4 6 5 8 2 7 3 6 4 + 2 x y - 5 x y - 2 x y + 5 x y + x y + x y + x y + x y 5 5 7 2 6 3 5 4 7 6 2 5 3 4 4 + x y - 2 x y - x y - 7 x y + x y + x y + 6 x y + 5 x y 6 4 2 3 3 4 3 2 2 2 2 2 - x y - 3 x y - 4 x y - x y + 4 x y - 2 x y + 2 x y - x - x y / 7 2 7 6 2 6 5 2 4 2 4 + 2 x - 1) / (x y - x y - x y + x y - x y + x y + x y / 3 2 2 - 2 x y + 2 x y + x - x y - 2 x + 1) and in Maple notation -(4*x^9*y^5-8*x^9*y^4-x^8*y^5+4*x^9*y^3+2*x^8*y^4-5*x^7*y^5-2*x^8*y^3+5*x^7*y^4 +x^6*y^5+x^8*y^2+x^7*y^3+x^6*y^4+x^5*y^5-2*x^7*y^2-x^6*y^3-7*x^5*y^4+x^7*y+x^6* y^2+6*x^5*y^3+5*x^4*y^4-x^6*y-3*x^4*y^2-4*x^3*y^3-x^4*y+4*x^3*y^2-2*x^2*y^2+2*x ^2*y-x^2-x*y+2*x-1)/(x^7*y^2-x^7*y-x^6*y^2+x^6*y-x^5*y^2+x^4*y^2+x^4*y-2*x^3*y+ 2*x^2*y+x^2-x*y-2*x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 9 8 7 6 5 4 3 2 (2 a - 3 a + 4 a - 11 a + 8 a - 8 a + 10 a - 2 a - 1) 4 3 2 / 3 2 2 (5 a - 8 a + 6 a - 6 a + 3) / ((4 a - 3 a + 2 a - 2) / 6 4 2 (a + a - a + a - 1)) 4 3 2 where a is the root of the polynomial, (x - 1) (x - x + x - 2 x + 1), and in decimals this is, 0.3885080112 BTW the ratio for words with, 500, letters is, 0.3901872963 ------------------------------------------------ "Theorem Number 85" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 5 9 4 8 5 9 3 ) | ) C(m, n) x y | = - (4 x y - 8 x y - x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 4 7 5 8 3 7 4 6 5 8 2 7 3 6 4 + 2 x y - 5 x y - 2 x y + 5 x y + x y + x y + x y + x y 5 5 7 2 6 3 5 4 7 6 2 5 3 4 4 + x y - 2 x y - x y - 7 x y + x y + x y + 6 x y + 5 x y 6 4 2 3 3 4 3 2 2 2 2 2 - x y - 3 x y - 4 x y - x y + 4 x y - 2 x y + 2 x y - x - x y / 2 + 2 x - 1) / ((x - 2 x + 1) / 5 2 5 4 2 4 3 (x y - x y + x y - x y - x y - x y + 1)) and in Maple notation -(4*x^9*y^5-8*x^9*y^4-x^8*y^5+4*x^9*y^3+2*x^8*y^4-5*x^7*y^5-2*x^8*y^3+5*x^7*y^4 +x^6*y^5+x^8*y^2+x^7*y^3+x^6*y^4+x^5*y^5-2*x^7*y^2-x^6*y^3-7*x^5*y^4+x^7*y+x^6* y^2+6*x^5*y^3+5*x^4*y^4-x^6*y-3*x^4*y^2-4*x^3*y^3-x^4*y+4*x^3*y^2-2*x^2*y^2+2*x ^2*y-x^2-x*y+2*x-1)/(x^2-2*x+1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3901872963 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 86" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [1, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 5 11 6 10 7 ) | ) C(m, n) x y | = - (x y - 7 x y + 4 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 4 11 5 10 6 12 3 11 4 9 6 8 7 + 11 x y - 12 x y + x y - 6 x y + 12 x y - 2 x y - x y 12 2 11 3 10 4 9 5 8 6 7 7 10 3 + x y - 4 x y + x y + 5 x y - 2 x y - x y - x y 9 4 8 5 7 6 9 3 8 4 7 5 9 2 - 4 x y + 9 x y - 2 x y + 2 x y - 5 x y + 4 x y - 2 x y 8 3 7 4 6 5 9 8 2 7 3 5 5 8 - 3 x y + 5 x y + x y + x y + x y - 10 x y - x y + x y 7 2 6 3 5 4 7 6 2 5 3 4 4 + 2 x y - 2 x y + 2 x y + 2 x y - 2 x y + 2 x y - 2 x y 6 5 2 4 3 4 2 3 3 4 4 3 + 3 x y - 2 x y + 3 x y + 2 x y + 4 x y - x y - x - x y 2 2 3 / 4 3 + 2 x y - x + x y + 1) / ((x + x - 1) / 5 2 5 4 2 4 3 (x y - x y + x y - x y - x y - x y + 1)) and in Maple notation -(x^12*y^7-7*x^12*y^5+4*x^11*y^6-x^10*y^7+11*x^12*y^4-12*x^11*y^5+x^10*y^6-6*x^ 12*y^3+12*x^11*y^4-2*x^9*y^6-x^8*y^7+x^12*y^2-4*x^11*y^3+x^10*y^4+5*x^9*y^5-2*x ^8*y^6-x^7*y^7-x^10*y^3-4*x^9*y^4+9*x^8*y^5-2*x^7*y^6+2*x^9*y^3-5*x^8*y^4+4*x^7 *y^5-2*x^9*y^2-3*x^8*y^3+5*x^7*y^4+x^6*y^5+x^9*y+x^8*y^2-10*x^7*y^3-x^5*y^5+x^8 *y+2*x^7*y^2-2*x^6*y^3+2*x^5*y^4+2*x^7*y-2*x^6*y^2+2*x^5*y^3-2*x^4*y^4+3*x^6*y-\ 2*x^5*y^2+3*x^4*y^3+2*x^4*y^2+4*x^3*y^3-x^4*y-x^4-x^3*y+2*x^2*y^2-x^3+x*y+1)/(x ^4+x^3-1)/(x^5*y^2-x^5*y+x^4*y^2-x^4*y-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3890428997 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 87" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 5 11 6 10 7 ) | ) C(m, n) x y | = - (x y - 7 x y + 4 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 4 11 5 10 6 12 3 11 4 9 6 8 7 + 11 x y - 12 x y + x y - 6 x y + 12 x y - 2 x y - x y 12 2 11 3 10 4 9 5 8 6 7 7 10 3 + x y - 4 x y + x y + 5 x y - 2 x y - x y - x y 9 4 8 5 7 6 9 3 8 4 7 5 9 2 - 4 x y + 9 x y - 2 x y + 2 x y - 5 x y + 4 x y - 2 x y 8 3 7 4 6 5 9 8 2 7 3 5 5 8 - 3 x y + 5 x y + x y + x y + x y - 10 x y - x y + x y 7 2 6 3 5 4 7 6 2 5 3 4 4 + 2 x y - 2 x y + 2 x y + 2 x y - 2 x y + 2 x y - 2 x y 6 5 2 4 3 4 2 3 3 4 4 3 + 3 x y - 2 x y + 3 x y + 2 x y + 4 x y - x y - x - x y 2 2 3 / 9 2 9 8 2 8 7 2 + 2 x y - x + x y + 1) / (x y - x y + 2 x y - 2 x y + x y / 7 6 5 2 4 2 4 3 3 - 2 x y - x y - x y - x y + x + x y + x + x y - 1) and in Maple notation -(x^12*y^7-7*x^12*y^5+4*x^11*y^6-x^10*y^7+11*x^12*y^4-12*x^11*y^5+x^10*y^6-6*x^ 12*y^3+12*x^11*y^4-2*x^9*y^6-x^8*y^7+x^12*y^2-4*x^11*y^3+x^10*y^4+5*x^9*y^5-2*x ^8*y^6-x^7*y^7-x^10*y^3-4*x^9*y^4+9*x^8*y^5-2*x^7*y^6+2*x^9*y^3-5*x^8*y^4+4*x^7 *y^5-2*x^9*y^2-3*x^8*y^3+5*x^7*y^4+x^6*y^5+x^9*y+x^8*y^2-10*x^7*y^3-x^5*y^5+x^8 *y+2*x^7*y^2-2*x^6*y^3+2*x^5*y^4+2*x^7*y-2*x^6*y^2+2*x^5*y^3-2*x^4*y^4+3*x^6*y-\ 2*x^5*y^2+3*x^4*y^3+2*x^4*y^2+4*x^3*y^3-x^4*y-x^4-x^3*y+2*x^2*y^2-x^3+x*y+1)/(x ^9*y^2-x^9*y+2*x^8*y^2-2*x^8*y+x^7*y^2-2*x^7*y-x^6*y-x^5*y^2-x^4*y^2+x^4+x^3*y+ x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3890428997 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 88" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 8 11 7 10 8 11 6 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 10 6 10 5 9 5 8 6 7 7 9 4 8 5 - x y - x y + x y - x y - x y - x y + x y - 2 x y 8 4 7 5 8 3 7 4 6 5 7 3 6 4 + 5 x y - 3 x y - 2 x y + 7 x y - x y - 3 x y + 6 x y 5 5 6 3 5 4 5 3 4 4 5 2 4 3 5 - x y - 4 x y + 3 x y - x y + 6 x y - 2 x y - 3 x y + x y 4 2 3 3 4 3 2 2 / 12 4 12 3 - 3 x y - 2 x y + x y + x y - x y - 1) / (x y - 2 x y / 11 4 12 2 11 3 10 4 11 2 10 3 9 4 + 2 x y + x y - 4 x y + 2 x y + 2 x y - 5 x y + x y 10 2 9 3 9 2 8 3 8 2 7 3 7 2 + 3 x y - 4 x y + 3 x y - 3 x y + 3 x y - x y + 3 x y 6 3 7 6 2 5 3 6 5 2 5 4 2 - 2 x y - x y + 3 x y - x y - x y + 4 x y - 2 x y + 2 x y 4 3 2 2 - 2 x y - x y + x y - 2 x y + 1) and in Maple notation -(x^11*y^8-2*x^11*y^7+x^10*y^8+x^11*y^6-x^10*y^7-x^10*y^6+x^10*y^5-x^9*y^5-x^8* y^6-x^7*y^7+x^9*y^4-2*x^8*y^5+5*x^8*y^4-3*x^7*y^5-2*x^8*y^3+7*x^7*y^4-x^6*y^5-3 *x^7*y^3+6*x^6*y^4-x^5*y^5-4*x^6*y^3+3*x^5*y^4-x^5*y^3+6*x^4*y^4-2*x^5*y^2-3*x^ 4*y^3+x^5*y-3*x^4*y^2-2*x^3*y^3+x^4*y+x^3*y-x^2*y^2-1)/(x^12*y^4-2*x^12*y^3+2*x ^11*y^4+x^12*y^2-4*x^11*y^3+2*x^10*y^4+2*x^11*y^2-5*x^10*y^3+x^9*y^4+3*x^10*y^2 -4*x^9*y^3+3*x^9*y^2-3*x^8*y^3+3*x^8*y^2-x^7*y^3+3*x^7*y^2-2*x^6*y^3-x^7*y+3*x^ 6*y^2-x^5*y^3-x^6*y+4*x^5*y^2-2*x^5*y+2*x^4*y^2-2*x^4*y-x^3*y+x^2*y^2-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 12 10 9 8 6 5 4 3 2 2 (2 a + 4 a - 2 a - a - 9 a + 9 a - 8 a + 10 a - 2 a - 1) - --------------------------------------------------------------------- 6 4 2 6 4 3 2 (7 a + 5 a - 3 a + 2 a - 2) (a + a - a - a - 1) 7 5 3 2 where a is the root of the polynomial, x + x - x + x - 2 x + 1, and in decimals this is, 0.3885080106 BTW the ratio for words with, 500, letters is, 0.3885216756 ------------------------------------------------ "Theorem Number 89" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 11 8 11 7 10 8 11 6 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 10 6 10 5 9 5 8 6 7 7 9 4 8 5 - x y - x y + x y - x y - x y - x y + x y - 2 x y 8 4 7 5 8 3 7 4 6 5 7 3 6 4 + 5 x y - 3 x y - 2 x y + 7 x y - x y - 3 x y + 6 x y 5 5 6 3 5 4 5 3 4 4 5 2 4 3 5 - x y - 4 x y + 3 x y - x y + 6 x y - 2 x y - 3 x y + x y 4 2 3 3 4 3 2 2 / 12 4 12 3 - 3 x y - 2 x y + x y + x y - x y - 1) / (x y - 2 x y / 11 4 12 2 11 3 10 4 11 2 10 3 9 4 + 2 x y + x y - 4 x y + 2 x y + 2 x y - 5 x y + x y 10 2 9 3 9 2 8 3 8 2 7 3 7 2 + 3 x y - 4 x y + 3 x y - 3 x y + 3 x y - x y + 3 x y 6 3 7 6 2 5 3 6 5 2 5 4 2 - 2 x y - x y + 3 x y - x y - x y + 4 x y - 2 x y + 2 x y 4 3 2 2 - 2 x y - x y + x y - 2 x y + 1) and in Maple notation -(x^11*y^8-2*x^11*y^7+x^10*y^8+x^11*y^6-x^10*y^7-x^10*y^6+x^10*y^5-x^9*y^5-x^8* y^6-x^7*y^7+x^9*y^4-2*x^8*y^5+5*x^8*y^4-3*x^7*y^5-2*x^8*y^3+7*x^7*y^4-x^6*y^5-3 *x^7*y^3+6*x^6*y^4-x^5*y^5-4*x^6*y^3+3*x^5*y^4-x^5*y^3+6*x^4*y^4-2*x^5*y^2-3*x^ 4*y^3+x^5*y-3*x^4*y^2-2*x^3*y^3+x^4*y+x^3*y-x^2*y^2-1)/(x^12*y^4-2*x^12*y^3+2*x ^11*y^4+x^12*y^2-4*x^11*y^3+2*x^10*y^4+2*x^11*y^2-5*x^10*y^3+x^9*y^4+3*x^10*y^2 -4*x^9*y^3+3*x^9*y^2-3*x^8*y^3+3*x^8*y^2-x^7*y^3+3*x^7*y^2-2*x^6*y^3-x^7*y+3*x^ 6*y^2-x^5*y^3-x^6*y+4*x^5*y^2-2*x^5*y+2*x^4*y^2-2*x^4*y-x^3*y+x^2*y^2-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 12 10 9 8 6 5 4 3 2 2 (2 a + 4 a - 2 a - a - 9 a + 9 a - 8 a + 10 a - 2 a - 1) - --------------------------------------------------------------------- 6 4 2 6 4 3 2 (7 a + 5 a - 3 a + 2 a - 2) (a + a - a - a - 1) 7 5 3 2 where a is the root of the polynomial, x + x - x + x - 2 x + 1, and in decimals this is, 0.3885080106 BTW the ratio for words with, 500, letters is, 0.3885216756 ------------------------------------------------ "Theorem Number 90" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 11 19 10 18 11 19 9 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 19 8 18 9 17 10 19 7 18 8 + 4 x y - 10 x y - 6 x y - 3 x y + 5 x y + 4 x y 17 9 16 10 19 6 18 7 17 8 16 9 15 10 + 13 x y - 3 x y - x y - x y - 22 x y + 9 x y - x y 17 7 16 8 15 9 17 6 16 7 15 8 14 9 + 18 x y - 6 x y + 2 x y - 7 x y - 7 x y - x y - x y 17 5 16 6 15 7 14 8 13 9 16 5 + x y + 12 x y + 2 x y + 3 x y - 2 x y - 6 x y 15 6 14 7 13 8 16 4 15 5 14 6 13 7 - 5 x y - 3 x y + 13 x y + x y + 4 x y + x y - 32 x y 12 8 15 4 13 6 12 7 11 8 13 5 + 3 x y - x y + 37 x y - 2 x y + x y - 20 x y 12 6 11 7 13 4 12 5 11 6 10 7 - 11 x y - 2 x y + 4 x y + 16 x y + 7 x y + 3 x y 12 4 11 5 10 6 9 7 11 4 10 5 9 6 - 6 x y - 13 x y + x y + 2 x y + 9 x y - 5 x y + x y 11 3 10 4 9 5 8 6 10 3 9 4 8 5 - 2 x y - 3 x y - 2 x y + 2 x y + 5 x y - 2 x y - 4 x y 7 6 10 2 7 5 9 2 8 3 7 4 6 5 7 3 - x y - x y + 7 x y + x y + x y - 13 x y - 7 x y + 7 x y 6 4 5 5 7 2 6 3 5 4 6 2 5 3 + 10 x y - 2 x y - x y - 6 x y - 5 x y + x y + 6 x y 4 4 5 2 4 2 3 3 4 3 2 3 2 2 - 3 x y - 2 x y + 2 x y - 8 x y - x y + 4 x y + x y - 4 x y 2 / 16 8 16 7 16 6 16 5 + 2 x y - 2 x y + x - 1) / (x y - 4 x y + 6 x y - 4 x y / 14 7 16 4 14 6 14 5 14 4 11 5 11 4 + x y + x y - 3 x y + 3 x y - x y - x y + 2 x y 11 3 10 4 10 3 10 2 9 3 9 2 7 3 - x y - 2 x y + 3 x y - x y - x y + x y + 3 x y 7 2 6 3 4 2 4 3 - 2 x y + x y + 2 x y - x y + x y + x - 1) and in Maple notation (x^19*y^11-5*x^19*y^10-x^18*y^11+10*x^19*y^9+4*x^18*y^10-10*x^19*y^8-6*x^18*y^9 -3*x^17*y^10+5*x^19*y^7+4*x^18*y^8+13*x^17*y^9-3*x^16*y^10-x^19*y^6-x^18*y^7-22 *x^17*y^8+9*x^16*y^9-x^15*y^10+18*x^17*y^7-6*x^16*y^8+2*x^15*y^9-7*x^17*y^6-7*x ^16*y^7-x^15*y^8-x^14*y^9+x^17*y^5+12*x^16*y^6+2*x^15*y^7+3*x^14*y^8-2*x^13*y^9 -6*x^16*y^5-5*x^15*y^6-3*x^14*y^7+13*x^13*y^8+x^16*y^4+4*x^15*y^5+x^14*y^6-32*x ^13*y^7+3*x^12*y^8-x^15*y^4+37*x^13*y^6-2*x^12*y^7+x^11*y^8-20*x^13*y^5-11*x^12 *y^6-2*x^11*y^7+4*x^13*y^4+16*x^12*y^5+7*x^11*y^6+3*x^10*y^7-6*x^12*y^4-13*x^11 *y^5+x^10*y^6+2*x^9*y^7+9*x^11*y^4-5*x^10*y^5+x^9*y^6-2*x^11*y^3-3*x^10*y^4-2*x ^9*y^5+2*x^8*y^6+5*x^10*y^3-2*x^9*y^4-4*x^8*y^5-x^7*y^6-x^10*y^2+7*x^7*y^5+x^9* y^2+x^8*y^3-13*x^7*y^4-7*x^6*y^5+7*x^7*y^3+10*x^6*y^4-2*x^5*y^5-x^7*y^2-6*x^6*y ^3-5*x^5*y^4+x^6*y^2+6*x^5*y^3-3*x^4*y^4-2*x^5*y^2+2*x^4*y^2-8*x^3*y^3-x^4*y+4* x^3*y^2+x^3*y-4*x^2*y^2+2*x^2*y-2*x*y+x-1)/(x^16*y^8-4*x^16*y^7+6*x^16*y^6-4*x^ 16*y^5+x^14*y^7+x^16*y^4-3*x^14*y^6+3*x^14*y^5-x^14*y^4-x^11*y^5+2*x^11*y^4-x^ 11*y^3-2*x^10*y^4+3*x^10*y^3-x^10*y^2-x^9*y^3+x^9*y^2+3*x^7*y^3-2*x^7*y^2+x^6*y ^3+2*x^4*y^2-x^4*y+x^3*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 13 bors of a random word of length n tends to n times, 2 (a + 2 a - a 11 10 9 8 7 6 5 4 3 2 - 4 a - 6 a - 6 a - 14 a - 6 a - 7 a - 7 a + 2 a - 2 a + 5 a / 2 2 2 2 + 2 a + 1) / ((a + 1) (a - a + 1) (a + 1) / 6 5 3 2 (7 a + 6 a + 4 a + 3 a + 1)) 7 6 4 3 where a is the root of the polynomial, x + x + x + x + x - 1, and in decimals this is, 0.3836760632 BTW the ratio for words with, 500, letters is, 0.3859175881 ------------------------------------------------ "Theorem Number 91" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 11 19 10 18 11 19 9 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 19 8 18 9 17 10 19 7 18 8 + 4 x y - 10 x y - 6 x y - 3 x y + 5 x y + 4 x y 17 9 16 10 19 6 18 7 17 8 16 9 15 10 + 13 x y - 3 x y - x y - x y - 22 x y + 9 x y - x y 17 7 16 8 15 9 17 6 16 7 15 8 14 9 + 18 x y - 6 x y + 2 x y - 7 x y - 7 x y - x y - x y 17 5 16 6 15 7 14 8 13 9 16 5 + x y + 12 x y + 2 x y + 3 x y - 2 x y - 6 x y 15 6 14 7 13 8 16 4 15 5 14 6 13 7 - 5 x y - 3 x y + 13 x y + x y + 4 x y + x y - 32 x y 12 8 15 4 13 6 12 7 11 8 13 5 + 3 x y - x y + 37 x y - 2 x y + x y - 20 x y 12 6 11 7 13 4 12 5 11 6 10 7 - 11 x y - 2 x y + 4 x y + 16 x y + 7 x y + 3 x y 12 4 11 5 10 6 9 7 11 4 10 5 9 6 - 6 x y - 13 x y + x y + 2 x y + 9 x y - 5 x y + x y 11 3 10 4 9 5 8 6 10 3 9 4 8 5 - 2 x y - 3 x y - 2 x y + 2 x y + 5 x y - 2 x y - 4 x y 7 6 10 2 7 5 9 2 8 3 7 4 6 5 7 3 - x y - x y + 7 x y + x y + x y - 13 x y - 7 x y + 7 x y 6 4 5 5 7 2 6 3 5 4 6 2 5 3 + 10 x y - 2 x y - x y - 6 x y - 5 x y + x y + 6 x y 4 4 5 2 4 2 3 3 4 3 2 3 2 2 - 3 x y - 2 x y + 2 x y - 8 x y - x y + 4 x y + x y - 4 x y 2 / 16 8 16 7 16 6 16 5 + 2 x y - 2 x y + x - 1) / (x y - 4 x y + 6 x y - 4 x y / 14 7 16 4 14 6 14 5 14 4 11 5 11 4 + x y + x y - 3 x y + 3 x y - x y - x y + 2 x y 11 3 10 4 10 3 10 2 9 3 9 2 7 3 - x y - 2 x y + 3 x y - x y - x y + x y + 3 x y 7 2 6 3 4 2 4 3 - 2 x y + x y + 2 x y - x y + x y + x - 1) and in Maple notation (x^19*y^11-5*x^19*y^10-x^18*y^11+10*x^19*y^9+4*x^18*y^10-10*x^19*y^8-6*x^18*y^9 -3*x^17*y^10+5*x^19*y^7+4*x^18*y^8+13*x^17*y^9-3*x^16*y^10-x^19*y^6-x^18*y^7-22 *x^17*y^8+9*x^16*y^9-x^15*y^10+18*x^17*y^7-6*x^16*y^8+2*x^15*y^9-7*x^17*y^6-7*x ^16*y^7-x^15*y^8-x^14*y^9+x^17*y^5+12*x^16*y^6+2*x^15*y^7+3*x^14*y^8-2*x^13*y^9 -6*x^16*y^5-5*x^15*y^6-3*x^14*y^7+13*x^13*y^8+x^16*y^4+4*x^15*y^5+x^14*y^6-32*x ^13*y^7+3*x^12*y^8-x^15*y^4+37*x^13*y^6-2*x^12*y^7+x^11*y^8-20*x^13*y^5-11*x^12 *y^6-2*x^11*y^7+4*x^13*y^4+16*x^12*y^5+7*x^11*y^6+3*x^10*y^7-6*x^12*y^4-13*x^11 *y^5+x^10*y^6+2*x^9*y^7+9*x^11*y^4-5*x^10*y^5+x^9*y^6-2*x^11*y^3-3*x^10*y^4-2*x ^9*y^5+2*x^8*y^6+5*x^10*y^3-2*x^9*y^4-4*x^8*y^5-x^7*y^6-x^10*y^2+7*x^7*y^5+x^9* y^2+x^8*y^3-13*x^7*y^4-7*x^6*y^5+7*x^7*y^3+10*x^6*y^4-2*x^5*y^5-x^7*y^2-6*x^6*y ^3-5*x^5*y^4+x^6*y^2+6*x^5*y^3-3*x^4*y^4-2*x^5*y^2+2*x^4*y^2-8*x^3*y^3-x^4*y+4* x^3*y^2+x^3*y-4*x^2*y^2+2*x^2*y-2*x*y+x-1)/(x^16*y^8-4*x^16*y^7+6*x^16*y^6-4*x^ 16*y^5+x^14*y^7+x^16*y^4-3*x^14*y^6+3*x^14*y^5-x^14*y^4-x^11*y^5+2*x^11*y^4-x^ 11*y^3-2*x^10*y^4+3*x^10*y^3-x^10*y^2-x^9*y^3+x^9*y^2+3*x^7*y^3-2*x^7*y^2+x^6*y ^3+2*x^4*y^2-x^4*y+x^3*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 13 bors of a random word of length n tends to n times, 2 (a + 2 a - a 11 10 9 8 7 6 5 4 3 2 - 4 a - 6 a - 6 a - 14 a - 6 a - 7 a - 7 a + 2 a - 2 a + 5 a / 2 2 2 2 + 2 a + 1) / ((a + 1) (a - a + 1) (a + 1) / 6 5 3 2 (7 a + 6 a + 4 a + 3 a + 1)) 7 6 4 3 where a is the root of the polynomial, x + x + x + x + x - 1, and in decimals this is, 0.3836760632 BTW the ratio for words with, 500, letters is, 0.3859175881 ------------------------------------------------ "Theorem Number 92" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 8 12 7 12 6 11 7 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 5 11 6 10 7 12 4 11 5 10 6 9 7 - 4 x y - 4 x y - 2 x y + x y + 6 x y + 7 x y + 2 x y 11 4 10 5 9 6 8 7 11 3 10 4 9 5 - 4 x y - 10 x y - 9 x y + x y + x y + 8 x y + 15 x y 8 6 10 3 9 4 8 5 7 6 10 2 9 3 + x y - 4 x y - 12 x y - 10 x y + x y + x y + 5 x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 + 13 x y - 2 x y + 2 x y - x y - 5 x y - x y + 2 x y 7 3 6 4 5 5 7 2 6 3 5 4 6 2 + 4 x y - 8 x y + 4 x y - 2 x y + 2 x y - 3 x y + 2 x y 5 3 4 4 5 2 4 3 5 3 3 4 3 2 - 4 x y + 5 x y + 2 x y - 6 x y + x y + 8 x y + x y - 6 x y 3 2 2 2 / 10 4 10 3 9 4 - x y + 4 x y - 3 x y + 2 x y - x + 1) / (x y - 2 x y - x y / 10 2 9 3 9 2 7 3 7 2 6 3 6 2 5 4 + x y + 2 x y - x y + x y - x y - x y + x y + x y + x y 3 2 - x y - x y - x + 1) and in Maple notation (x^12*y^8-4*x^12*y^7+6*x^12*y^6+x^11*y^7-4*x^12*y^5-4*x^11*y^6-2*x^10*y^7+x^12* y^4+6*x^11*y^5+7*x^10*y^6+2*x^9*y^7-4*x^11*y^4-10*x^10*y^5-9*x^9*y^6+x^8*y^7+x^ 11*y^3+8*x^10*y^4+15*x^9*y^5+x^8*y^6-4*x^10*y^3-12*x^9*y^4-10*x^8*y^5+x^7*y^6+x ^10*y^2+5*x^9*y^3+13*x^8*y^4-2*x^7*y^5+2*x^6*y^6-x^9*y^2-5*x^8*y^3-x^7*y^4+2*x^ 6*y^5+4*x^7*y^3-8*x^6*y^4+4*x^5*y^5-2*x^7*y^2+2*x^6*y^3-3*x^5*y^4+2*x^6*y^2-4*x ^5*y^3+5*x^4*y^4+2*x^5*y^2-6*x^4*y^3+x^5*y+8*x^3*y^3+x^4*y-6*x^3*y^2-x^3*y+4*x^ 2*y^2-3*x^2*y+2*x*y-x+1)/(x^10*y^4-2*x^10*y^3-x^9*y^4+x^10*y^2+2*x^9*y^3-x^9*y^ 2+x^7*y^3-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y+x^4*y-x^3*y-x^2*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3848546611 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 93" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 8 12 7 12 6 11 7 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 5 11 6 10 7 12 4 11 5 10 6 9 7 - 4 x y - 4 x y - 2 x y + x y + 6 x y + 7 x y + 2 x y 11 4 10 5 9 6 8 7 11 3 10 4 9 5 - 4 x y - 10 x y - 9 x y + x y + x y + 8 x y + 15 x y 8 6 10 3 9 4 8 5 7 6 10 2 9 3 + x y - 4 x y - 12 x y - 10 x y + x y + x y + 5 x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 + 13 x y - 2 x y + 2 x y - x y - 5 x y - x y + 2 x y 7 3 6 4 5 5 7 2 6 3 5 4 6 2 + 4 x y - 8 x y + 4 x y - 2 x y + 2 x y - 3 x y + 2 x y 5 3 4 4 5 2 4 3 5 3 3 4 3 2 - 4 x y + 5 x y + 2 x y - 6 x y + x y + 8 x y + x y - 6 x y 3 2 2 2 / - x y + 4 x y - 3 x y + 2 x y - x + 1) / ((x - 1) / 9 4 9 3 9 2 6 3 6 2 4 3 2 (x y - 2 x y + x y + x y - x y + x y + 2 x y + x y - 1)) and in Maple notation (x^12*y^8-4*x^12*y^7+6*x^12*y^6+x^11*y^7-4*x^12*y^5-4*x^11*y^6-2*x^10*y^7+x^12* y^4+6*x^11*y^5+7*x^10*y^6+2*x^9*y^7-4*x^11*y^4-10*x^10*y^5-9*x^9*y^6+x^8*y^7+x^ 11*y^3+8*x^10*y^4+15*x^9*y^5+x^8*y^6-4*x^10*y^3-12*x^9*y^4-10*x^8*y^5+x^7*y^6+x ^10*y^2+5*x^9*y^3+13*x^8*y^4-2*x^7*y^5+2*x^6*y^6-x^9*y^2-5*x^8*y^3-x^7*y^4+2*x^ 6*y^5+4*x^7*y^3-8*x^6*y^4+4*x^5*y^5-2*x^7*y^2+2*x^6*y^3-3*x^5*y^4+2*x^6*y^2-4*x ^5*y^3+5*x^4*y^4+2*x^5*y^2-6*x^4*y^3+x^5*y+8*x^3*y^3+x^4*y-6*x^3*y^2-x^3*y+4*x^ 2*y^2-3*x^2*y+2*x*y-x+1)/(x-1)/(x^9*y^4-2*x^9*y^3+x^9*y^2+x^6*y^3-x^6*y^2+x^4*y +2*x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3848546611 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 94" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 14 7 13 8 14 6 ) | ) C(m, n) x y | = (x y - 4 x y - 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 14 5 13 6 12 7 14 4 13 5 + 8 x y + x y - 4 x y - 12 x y - 6 x y + x y + 8 x y 12 6 11 7 13 4 12 5 11 6 11 5 10 6 + 11 x y + x y - 2 x y - 7 x y - 3 x y + 3 x y + x y 9 7 12 3 11 4 10 5 9 6 10 4 9 5 + x y + x y - x y - 4 x y - 5 x y + 5 x y + 6 x y 8 6 10 3 9 4 8 5 9 3 8 4 7 5 + 3 x y - 2 x y - 2 x y - 8 x y + 2 x y + 11 x y + 4 x y 9 2 8 3 7 4 6 5 7 3 6 4 7 2 6 3 - 2 x y - 6 x y - 9 x y - x y + 5 x y - 5 x y - x y + x y 5 4 6 2 5 3 4 4 5 2 4 3 5 - 2 x y + 3 x y - 2 x y + 3 x y + 2 x y - 4 x y + x y 3 3 4 3 2 3 2 2 2 / + 8 x y + x y - 6 x y - x y + 4 x y - 3 x y + 2 x y - x + 1) / / 10 4 10 3 9 4 10 2 9 3 9 2 7 3 7 2 (x y - 2 x y - x y + x y + 2 x y - x y + x y - x y 6 3 6 2 5 4 3 2 - x y + x y + x y + x y - x y - x y - x + 1) and in Maple notation (x^14*y^8-4*x^14*y^7-2*x^13*y^8+6*x^14*y^6+8*x^13*y^7+x^12*y^8-4*x^14*y^5-12*x^ 13*y^6-6*x^12*y^7+x^14*y^4+8*x^13*y^5+11*x^12*y^6+x^11*y^7-2*x^13*y^4-7*x^12*y^ 5-3*x^11*y^6+3*x^11*y^5+x^10*y^6+x^9*y^7+x^12*y^3-x^11*y^4-4*x^10*y^5-5*x^9*y^6 +5*x^10*y^4+6*x^9*y^5+3*x^8*y^6-2*x^10*y^3-2*x^9*y^4-8*x^8*y^5+2*x^9*y^3+11*x^8 *y^4+4*x^7*y^5-2*x^9*y^2-6*x^8*y^3-9*x^7*y^4-x^6*y^5+5*x^7*y^3-5*x^6*y^4-x^7*y^ 2+x^6*y^3-2*x^5*y^4+3*x^6*y^2-2*x^5*y^3+3*x^4*y^4+2*x^5*y^2-4*x^4*y^3+x^5*y+8*x ^3*y^3+x^4*y-6*x^3*y^2-x^3*y+4*x^2*y^2-3*x^2*y+2*x*y-x+1)/(x^10*y^4-2*x^10*y^3- x^9*y^4+x^10*y^2+2*x^9*y^3-x^9*y^2+x^7*y^3-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y+x^4*y- x^3*y-x^2*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3838639149 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 95" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 14 7 13 8 14 6 ) | ) C(m, n) x y | = (x y - 4 x y - 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 14 5 13 6 12 7 14 4 13 5 + 8 x y + x y - 4 x y - 12 x y - 6 x y + x y + 8 x y 12 6 11 7 13 4 12 5 11 6 11 5 10 6 + 11 x y + x y - 2 x y - 7 x y - 3 x y + 3 x y + x y 9 7 12 3 11 4 10 5 9 6 10 4 9 5 + x y + x y - x y - 4 x y - 5 x y + 5 x y + 6 x y 8 6 10 3 9 4 8 5 9 3 8 4 7 5 + 3 x y - 2 x y - 2 x y - 8 x y + 2 x y + 11 x y + 4 x y 9 2 8 3 7 4 6 5 7 3 6 4 7 2 6 3 - 2 x y - 6 x y - 9 x y - x y + 5 x y - 5 x y - x y + x y 5 4 6 2 5 3 4 4 5 2 4 3 5 - 2 x y + 3 x y - 2 x y + 3 x y + 2 x y - 4 x y + x y 3 3 4 3 2 3 2 2 2 / + 8 x y + x y - 6 x y - x y + 4 x y - 3 x y + 2 x y - x + 1) / / 10 4 10 3 9 4 10 2 9 3 9 2 7 3 7 2 (x y - 2 x y - x y + x y + 2 x y - x y + x y - x y 6 3 6 2 5 4 3 2 - x y + x y + x y + x y - x y - x y - x + 1) and in Maple notation (x^14*y^8-4*x^14*y^7-2*x^13*y^8+6*x^14*y^6+8*x^13*y^7+x^12*y^8-4*x^14*y^5-12*x^ 13*y^6-6*x^12*y^7+x^14*y^4+8*x^13*y^5+11*x^12*y^6+x^11*y^7-2*x^13*y^4-7*x^12*y^ 5-3*x^11*y^6+3*x^11*y^5+x^10*y^6+x^9*y^7+x^12*y^3-x^11*y^4-4*x^10*y^5-5*x^9*y^6 +5*x^10*y^4+6*x^9*y^5+3*x^8*y^6-2*x^10*y^3-2*x^9*y^4-8*x^8*y^5+2*x^9*y^3+11*x^8 *y^4+4*x^7*y^5-2*x^9*y^2-6*x^8*y^3-9*x^7*y^4-x^6*y^5+5*x^7*y^3-5*x^6*y^4-x^7*y^ 2+x^6*y^3-2*x^5*y^4+3*x^6*y^2-2*x^5*y^3+3*x^4*y^4+2*x^5*y^2-4*x^4*y^3+x^5*y+8*x ^3*y^3+x^4*y-6*x^3*y^2-x^3*y+4*x^2*y^2-3*x^2*y+2*x*y-x+1)/(x^10*y^4-2*x^10*y^3- x^9*y^4+x^10*y^2+2*x^9*y^3-x^9*y^2+x^7*y^3-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y+x^4*y- x^3*y-x^2*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3838639149 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 96" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 9 12 8 11 9 12 7 ) | ) C(m, n) x y | = (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 8 12 6 11 7 12 5 11 6 10 7 12 4 - 2 x y - 2 x y - x y + 3 x y + 4 x y - 3 x y - 3 x y 10 6 9 7 8 8 12 3 11 4 10 5 9 6 + 7 x y - 2 x y - x y + x y - 5 x y - 2 x y + 4 x y 11 3 10 4 8 6 7 7 11 2 9 4 7 6 + 4 x y - 4 x y + 2 x y - x y - x y - 4 x y + x y 10 2 9 3 7 5 6 6 10 8 3 7 4 6 5 + 3 x y + 2 x y + x y - x y - x y - 3 x y - x y + 3 x y 8 2 5 5 8 6 3 5 4 6 2 5 3 4 4 + 3 x y + 5 x y - x y - 2 x y - 3 x y - x y - x y + 4 x y 6 5 2 4 3 5 4 2 3 3 5 4 2 2 + x y - 3 x y - 3 x y + x y - 2 x y - 3 x y + x + x y - x y 3 / 13 7 13 6 12 7 13 5 12 6 + x - x y - 1) / (x y - 3 x y + x y + 3 x y - 3 x y / 13 4 12 5 11 6 12 4 11 5 11 4 10 5 - x y + 3 x y - x y - x y + 2 x y - x y - x y 10 4 9 4 10 2 9 3 7 5 10 9 2 7 4 + x y + x y + x y - 2 x y + x y - x y + x y - x y 8 2 7 3 6 4 8 7 2 6 3 6 2 5 3 5 2 + x y + x y - x y - x y - x y + 2 x y - x y - x y - 2 x y 5 4 2 3 3 5 2 2 3 + x y + x y - x y + x + x y + x + x y - 1) and in Maple notation (x^12*y^9-3*x^12*y^8+x^11*y^9+3*x^12*y^7-2*x^11*y^8-2*x^12*y^6-x^11*y^7+3*x^12* y^5+4*x^11*y^6-3*x^10*y^7-3*x^12*y^4+7*x^10*y^6-2*x^9*y^7-x^8*y^8+x^12*y^3-5*x^ 11*y^4-2*x^10*y^5+4*x^9*y^6+4*x^11*y^3-4*x^10*y^4+2*x^8*y^6-x^7*y^7-x^11*y^2-4* x^9*y^4+x^7*y^6+3*x^10*y^2+2*x^9*y^3+x^7*y^5-x^6*y^6-x^10*y-3*x^8*y^3-x^7*y^4+3 *x^6*y^5+3*x^8*y^2+5*x^5*y^5-x^8*y-2*x^6*y^3-3*x^5*y^4-x^6*y^2-x^5*y^3+4*x^4*y^ 4+x^6*y-3*x^5*y^2-3*x^4*y^3+x^5*y-2*x^4*y^2-3*x^3*y^3+x^5+x^4*y-x^2*y^2+x^3-x*y -1)/(x^13*y^7-3*x^13*y^6+x^12*y^7+3*x^13*y^5-3*x^12*y^6-x^13*y^4+3*x^12*y^5-x^ 11*y^6-x^12*y^4+2*x^11*y^5-x^11*y^4-x^10*y^5+x^10*y^4+x^9*y^4+x^10*y^2-2*x^9*y^ 3+x^7*y^5-x^10*y+x^9*y^2-x^7*y^4+x^8*y^2+x^7*y^3-x^6*y^4-x^8*y-x^7*y^2+2*x^6*y^ 3-x^6*y^2-x^5*y^3-2*x^5*y^2+x^5*y+x^4*y^2-x^3*y^3+x^5+x^2*y^2+x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 9 8 7 5 4 3 2 2 (a + 2 a + 3 a - 2 a - 7 a - 6 a + 3 a + 2 a + 1) - ---------------------------------------------------------- 4 3 3 2 (5 a - 4 a - 2 a - 1) (2 a + a + a + 1) 5 4 2 where a is the root of the polynomial, x - x - x - x + 1, and in decimals this is, 0.3790633662 BTW the ratio for words with, 500, letters is, 0.3813785896 ------------------------------------------------ "Theorem Number 97" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 9 12 8 11 9 12 7 ) | ) C(m, n) x y | = (x y - 3 x y + x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 8 12 6 11 7 12 5 11 6 10 7 12 4 - 2 x y - 2 x y - x y + 3 x y + 4 x y - 3 x y - 3 x y 10 6 9 7 8 8 12 3 11 4 10 5 9 6 + 7 x y - 2 x y - x y + x y - 5 x y - 2 x y + 4 x y 11 3 10 4 8 6 7 7 11 2 9 4 7 6 + 4 x y - 4 x y + 2 x y - x y - x y - 4 x y + x y 10 2 9 3 7 5 6 6 10 8 3 7 4 6 5 + 3 x y + 2 x y + x y - x y - x y - 3 x y - x y + 3 x y 8 2 5 5 8 6 3 5 4 6 2 5 3 4 4 + 3 x y + 5 x y - x y - 2 x y - 3 x y - x y - x y + 4 x y 6 5 2 4 3 5 4 2 3 3 5 4 2 2 + x y - 3 x y - 3 x y + x y - 2 x y - 3 x y + x + x y - x y 3 / 13 7 13 6 12 7 13 5 12 6 + x - x y - 1) / (x y - 3 x y + x y + 3 x y - 3 x y / 13 4 12 5 11 6 12 4 11 5 11 4 10 5 - x y + 3 x y - x y - x y + 2 x y - x y - x y 10 4 9 4 10 2 9 3 7 5 10 9 2 7 4 + x y + x y + x y - 2 x y + x y - x y + x y - x y 8 2 7 3 6 4 8 7 2 6 3 6 2 5 3 5 2 + x y + x y - x y - x y - x y + 2 x y - x y - x y - 2 x y 5 4 2 3 3 5 2 2 3 + x y + x y - x y + x + x y + x + x y - 1) and in Maple notation (x^12*y^9-3*x^12*y^8+x^11*y^9+3*x^12*y^7-2*x^11*y^8-2*x^12*y^6-x^11*y^7+3*x^12* y^5+4*x^11*y^6-3*x^10*y^7-3*x^12*y^4+7*x^10*y^6-2*x^9*y^7-x^8*y^8+x^12*y^3-5*x^ 11*y^4-2*x^10*y^5+4*x^9*y^6+4*x^11*y^3-4*x^10*y^4+2*x^8*y^6-x^7*y^7-x^11*y^2-4* x^9*y^4+x^7*y^6+3*x^10*y^2+2*x^9*y^3+x^7*y^5-x^6*y^6-x^10*y-3*x^8*y^3-x^7*y^4+3 *x^6*y^5+3*x^8*y^2+5*x^5*y^5-x^8*y-2*x^6*y^3-3*x^5*y^4-x^6*y^2-x^5*y^3+4*x^4*y^ 4+x^6*y-3*x^5*y^2-3*x^4*y^3+x^5*y-2*x^4*y^2-3*x^3*y^3+x^5+x^4*y-x^2*y^2+x^3-x*y -1)/(x^13*y^7-3*x^13*y^6+x^12*y^7+3*x^13*y^5-3*x^12*y^6-x^13*y^4+3*x^12*y^5-x^ 11*y^6-x^12*y^4+2*x^11*y^5-x^11*y^4-x^10*y^5+x^10*y^4+x^9*y^4+x^10*y^2-2*x^9*y^ 3+x^7*y^5-x^10*y+x^9*y^2-x^7*y^4+x^8*y^2+x^7*y^3-x^6*y^4-x^8*y-x^7*y^2+2*x^6*y^ 3-x^6*y^2-x^5*y^3-2*x^5*y^2+x^5*y+x^4*y^2-x^3*y^3+x^5+x^2*y^2+x^3+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 9 8 7 5 4 3 2 2 (a + 2 a + 3 a - 2 a - 7 a - 6 a + 3 a + 2 a + 1) - ---------------------------------------------------------- 4 3 3 2 (5 a - 4 a - 2 a - 1) (2 a + a + a + 1) 5 4 2 where a is the root of the polynomial, x - x - x - x + 1, and in decimals this is, 0.3790633662 BTW the ratio for words with, 500, letters is, 0.3813785896 ------------------------------------------------ "Theorem Number 98" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 9 13 8 12 9 13 7 ) | ) C(m, n) x y | = (x y - 2 x y - 2 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 8 13 6 12 7 11 8 13 5 12 6 + 5 x y + 5 x y - 2 x y - 2 x y - 4 x y - 2 x y 11 7 10 8 13 4 12 5 11 6 10 7 9 8 + 8 x y - x y + x y - x y - 15 x y - 2 x y + 2 x y 12 4 11 5 10 6 9 7 12 3 11 4 10 5 + 3 x y + 17 x y + 4 x y - x y - x y - 11 x y - 2 x y 9 6 8 7 11 3 10 4 9 5 8 6 10 3 + 3 x y + 2 x y + 3 x y + 5 x y - 12 x y - 4 x y - 6 x y 9 4 8 5 7 6 10 2 9 3 8 4 7 5 + 11 x y + 4 x y + 3 x y + 2 x y - x y - 5 x y + 3 x y 6 6 9 2 8 3 7 4 6 5 9 8 2 - 2 x y - 3 x y + 5 x y - 4 x y - 4 x y + x y - 2 x y 7 3 6 4 5 5 7 2 6 3 5 4 7 6 2 - 4 x y + x y - 2 x y + 4 x y + 5 x y + 3 x y - x y - 3 x y 5 3 4 4 6 5 2 4 3 4 2 3 3 3 2 - 2 x y - 3 x y + x y + x y + x y + 2 x y + 4 x y - 3 x y 3 2 2 2 / 8 3 7 4 - x y + 2 x y - x y + x y - x + 1) / ((x - 1) (x y + x y / 8 2 7 3 8 7 2 6 3 7 6 2 5 2 4 3 - 2 x y - x y + x y - x y - x y + x y + x y - x y - x y 5 4 2 4 3 2 3 + x y - x y + x y + x y + x y + x y - 1)) and in Maple notation (x^13*y^9-2*x^13*y^8-2*x^12*y^9-x^13*y^7+5*x^12*y^8+5*x^13*y^6-2*x^12*y^7-2*x^ 11*y^8-4*x^13*y^5-2*x^12*y^6+8*x^11*y^7-x^10*y^8+x^13*y^4-x^12*y^5-15*x^11*y^6-\ 2*x^10*y^7+2*x^9*y^8+3*x^12*y^4+17*x^11*y^5+4*x^10*y^6-x^9*y^7-x^12*y^3-11*x^11 *y^4-2*x^10*y^5+3*x^9*y^6+2*x^8*y^7+3*x^11*y^3+5*x^10*y^4-12*x^9*y^5-4*x^8*y^6-\ 6*x^10*y^3+11*x^9*y^4+4*x^8*y^5+3*x^7*y^6+2*x^10*y^2-x^9*y^3-5*x^8*y^4+3*x^7*y^ 5-2*x^6*y^6-3*x^9*y^2+5*x^8*y^3-4*x^7*y^4-4*x^6*y^5+x^9*y-2*x^8*y^2-4*x^7*y^3+x ^6*y^4-2*x^5*y^5+4*x^7*y^2+5*x^6*y^3+3*x^5*y^4-x^7*y-3*x^6*y^2-2*x^5*y^3-3*x^4* y^4+x^6*y+x^5*y^2+x^4*y^3+2*x^4*y^2+4*x^3*y^3-3*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x *y-x+1)/(x-1)/(x^8*y^3+x^7*y^4-2*x^8*y^2-x^7*y^3+x^8*y-x^7*y^2-x^6*y^3+x^7*y+x^ 6*y^2-x^5*y^2-x^4*y^3+x^5*y-x^4*y^2+x^4*y+x^3*y^2+x^3*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 9 bors of a random word of length n tends to n times, 2 (a - 2 a - 5 a 8 7 6 5 4 3 2 + 13 a - 2 a + 4 a - 17 a - 3 a - 3 a + 5 a + 2 a + 1) 4 3 2 / 3 2 2 (5 a - 12 a + 6 a - 2 a + 2) / ((4 a - 6 a - 1) / 7 6 2 (a - 2 a + a + 1)) 4 3 where a is the root of the polynomial, (x - 1) (x - 2 x - x + 1), and in decimals this is, 0.3732988050 BTW the ratio for words with, 500, letters is, 0.3754194989 ------------------------------------------------ "Theorem Number 99" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 9 14 8 14 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 6 12 8 14 5 12 7 11 8 14 4 12 6 - 10 x y - x y + 5 x y + 4 x y + x y - x y - 5 x y 11 7 11 6 12 4 11 5 10 6 9 7 12 3 - 3 x y + 2 x y + 5 x y + 3 x y + x y - 3 x y - 4 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 - 7 x y - 4 x y + 6 x y + x y + x y + 7 x y + 6 x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 + 2 x y - 2 x y - 4 x y - 5 x y - 10 x y - 2 x y - x y 11 10 2 9 3 8 4 7 5 6 6 10 + x y + 3 x y + 5 x y + 6 x y + 4 x y + 2 x y - x y 8 3 7 4 6 5 7 3 6 4 5 5 7 2 - 3 x y - 6 x y - 5 x y + 2 x y + 3 x y + 2 x y + 2 x y 6 3 5 4 7 6 2 5 3 4 4 6 6 5 + x y - 4 x y - x y - x y + 2 x y + 3 x y + x y - x - x y 4 2 3 3 5 4 3 2 4 3 2 2 3 - 2 x y - 4 x y + x + x y + 3 x y - x - x y - 2 x y + x 2 / 12 6 12 5 12 4 + 2 x y - x y + x - 1) / ((x - 1) (x y - 3 x y + 3 x y / 11 5 12 3 11 4 9 6 11 3 9 5 11 2 - x y - x y + 3 x y - x y - 3 x y + 2 x y + x y 10 3 9 4 8 5 10 2 8 4 10 8 3 7 4 - x y - x y + x y + 2 x y - 2 x y - x y + x y - x y 7 3 7 2 6 3 5 3 6 4 2 5 4 3 2 3 + 2 x y - x y + x y - x y - x y + x y + x - x y - x y + x 2 + x y + x y - 1)) and in Maple notation -(x^14*y^9-5*x^14*y^8+10*x^14*y^7-10*x^14*y^6-x^12*y^8+5*x^14*y^5+4*x^12*y^7+x^ 11*y^8-x^14*y^4-5*x^12*y^6-3*x^11*y^7+2*x^11*y^6+5*x^12*y^4+3*x^11*y^5+x^10*y^6 -3*x^9*y^7-4*x^12*y^3-7*x^11*y^4-4*x^10*y^5+6*x^9*y^6+x^8*y^7+x^12*y^2+7*x^11*y ^3+6*x^10*y^4+2*x^9*y^5-2*x^8*y^6-4*x^11*y^2-5*x^10*y^3-10*x^9*y^4-2*x^8*y^5-x^ 7*y^6+x^11*y+3*x^10*y^2+5*x^9*y^3+6*x^8*y^4+4*x^7*y^5+2*x^6*y^6-x^10*y-3*x^8*y^ 3-6*x^7*y^4-5*x^6*y^5+2*x^7*y^3+3*x^6*y^4+2*x^5*y^5+2*x^7*y^2+x^6*y^3-4*x^5*y^4 -x^7*y-x^6*y^2+2*x^5*y^3+3*x^4*y^4+x^6*y-x^6-x^5*y-2*x^4*y^2-4*x^3*y^3+x^5+x^4* y+3*x^3*y^2-x^4-x^3*y-2*x^2*y^2+x^3+2*x^2*y-x*y+x-1)/(x-1)/(x^12*y^6-3*x^12*y^5 +3*x^12*y^4-x^11*y^5-x^12*y^3+3*x^11*y^4-x^9*y^6-3*x^11*y^3+2*x^9*y^5+x^11*y^2- x^10*y^3-x^9*y^4+x^8*y^5+2*x^10*y^2-2*x^8*y^4-x^10*y+x^8*y^3-x^7*y^4+2*x^7*y^3- x^7*y^2+x^6*y^3-x^5*y^3-x^6*y+x^4*y^2+x^5-x^4*y-x^3*y^2+x^3+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3750419081 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 100" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 9 14 8 14 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 6 12 8 14 5 12 7 11 8 14 4 12 6 - 10 x y - x y + 5 x y + 4 x y + x y - x y - 5 x y 11 7 11 6 12 4 11 5 10 6 9 7 12 3 - 3 x y + 2 x y + 5 x y + 3 x y + x y - 3 x y - 4 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 - 7 x y - 4 x y + 6 x y + x y + x y + 7 x y + 6 x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 + 2 x y - 2 x y - 4 x y - 5 x y - 10 x y - 2 x y - x y 11 10 2 9 3 8 4 7 5 6 6 10 + x y + 3 x y + 5 x y + 6 x y + 4 x y + 2 x y - x y 8 3 7 4 6 5 7 3 6 4 5 5 7 2 - 3 x y - 6 x y - 5 x y + 2 x y + 3 x y + 2 x y + 2 x y 6 3 5 4 7 6 2 5 3 4 4 6 6 5 + x y - 4 x y - x y - x y + 2 x y + 3 x y + x y - x - x y 4 2 3 3 5 4 3 2 4 3 2 2 3 - 2 x y - 4 x y + x + x y + 3 x y - x - x y - 2 x y + x 2 / 12 6 12 5 12 4 + 2 x y - x y + x - 1) / ((x - 1) (x y - 3 x y + 3 x y / 11 5 12 3 11 4 9 6 11 3 9 5 11 2 - x y - x y + 3 x y - x y - 3 x y + 2 x y + x y 10 3 9 4 8 5 10 2 8 4 10 8 3 7 4 - x y - x y + x y + 2 x y - 2 x y - x y + x y - x y 7 3 7 2 6 3 5 3 6 4 2 5 4 3 2 3 + 2 x y - x y + x y - x y - x y + x y + x - x y - x y + x 2 + x y + x y - 1)) and in Maple notation -(x^14*y^9-5*x^14*y^8+10*x^14*y^7-10*x^14*y^6-x^12*y^8+5*x^14*y^5+4*x^12*y^7+x^ 11*y^8-x^14*y^4-5*x^12*y^6-3*x^11*y^7+2*x^11*y^6+5*x^12*y^4+3*x^11*y^5+x^10*y^6 -3*x^9*y^7-4*x^12*y^3-7*x^11*y^4-4*x^10*y^5+6*x^9*y^6+x^8*y^7+x^12*y^2+7*x^11*y ^3+6*x^10*y^4+2*x^9*y^5-2*x^8*y^6-4*x^11*y^2-5*x^10*y^3-10*x^9*y^4-2*x^8*y^5-x^ 7*y^6+x^11*y+3*x^10*y^2+5*x^9*y^3+6*x^8*y^4+4*x^7*y^5+2*x^6*y^6-x^10*y-3*x^8*y^ 3-6*x^7*y^4-5*x^6*y^5+2*x^7*y^3+3*x^6*y^4+2*x^5*y^5+2*x^7*y^2+x^6*y^3-4*x^5*y^4 -x^7*y-x^6*y^2+2*x^5*y^3+3*x^4*y^4+x^6*y-x^6-x^5*y-2*x^4*y^2-4*x^3*y^3+x^5+x^4* y+3*x^3*y^2-x^4-x^3*y-2*x^2*y^2+x^3+2*x^2*y-x*y+x-1)/(x-1)/(x^12*y^6-3*x^12*y^5 +3*x^12*y^4-x^11*y^5-x^12*y^3+3*x^11*y^4-x^9*y^6-3*x^11*y^3+2*x^9*y^5+x^11*y^2- x^10*y^3-x^9*y^4+x^8*y^5+2*x^10*y^2-2*x^8*y^4-x^10*y+x^8*y^3-x^7*y^4+2*x^7*y^3- x^7*y^2+x^6*y^3-x^5*y^3-x^6*y+x^4*y^2+x^5-x^4*y-x^3*y^2+x^3+x^2*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3750419081 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 101" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 9 14 7 13 8 12 9 ) | ) C(m, n) x y | = (x y + x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 6 13 7 12 8 14 5 13 6 12 7 - 4 x y - 3 x y + 4 x y + 6 x y + 3 x y - 9 x y 11 8 14 4 13 5 12 6 11 7 10 8 14 3 - 3 x y - 4 x y + 4 x y + 14 x y + 8 x y - x y + x y 13 4 12 5 11 6 10 7 9 8 13 3 - 6 x y - 14 x y - 14 x y - 4 x y + x y + 2 x y 12 4 11 5 10 6 9 7 12 3 11 4 + 9 x y + 21 x y + 7 x y - 2 x y - 4 x y - 17 x y 10 5 9 6 8 7 12 2 11 3 10 4 9 5 + 3 x y + 8 x y + 3 x y + x y + 5 x y - 4 x y - 15 x y 8 6 10 3 9 4 8 5 7 6 10 2 9 3 - 2 x y - 3 x y + 13 x y + 5 x y + 4 x y + 2 x y - 4 x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 - 11 x y + 2 x y - x y - 2 x y + 6 x y - 6 x y - 3 x y 9 8 2 7 3 6 4 5 5 7 2 6 3 7 + x y - x y - 4 x y - 2 x y - 3 x y + 5 x y + 5 x y - x y 6 2 5 3 4 4 6 5 2 4 3 4 2 3 3 - 2 x y + x y - 3 x y + x y + x y + x y + 2 x y + 4 x y 3 2 3 2 2 2 / 9 3 8 4 - 3 x y - x y + 2 x y - x y + x y - x + 1) / (x y + x y / 9 2 8 3 7 4 9 8 2 7 2 6 3 7 - 2 x y - 2 x y - x y + x y + x y + 2 x y + x y - x y 6 2 5 3 6 4 3 4 2 3 2 3 2 - 2 x y - x y + x y + x y + 2 x y - x y - x y + x y - x y - x + 1) and in Maple notation (x^13*y^9+x^14*y^7-x^13*y^8-x^12*y^9-4*x^14*y^6-3*x^13*y^7+4*x^12*y^8+6*x^14*y^ 5+3*x^13*y^6-9*x^12*y^7-3*x^11*y^8-4*x^14*y^4+4*x^13*y^5+14*x^12*y^6+8*x^11*y^7 -x^10*y^8+x^14*y^3-6*x^13*y^4-14*x^12*y^5-14*x^11*y^6-4*x^10*y^7+x^9*y^8+2*x^13 *y^3+9*x^12*y^4+21*x^11*y^5+7*x^10*y^6-2*x^9*y^7-4*x^12*y^3-17*x^11*y^4+3*x^10* y^5+8*x^9*y^6+3*x^8*y^7+x^12*y^2+5*x^11*y^3-4*x^10*y^4-15*x^9*y^5-2*x^8*y^6-3*x ^10*y^3+13*x^9*y^4+5*x^8*y^5+4*x^7*y^6+2*x^10*y^2-4*x^9*y^3-11*x^8*y^4+2*x^7*y^ 5-x^6*y^6-2*x^9*y^2+6*x^8*y^3-6*x^7*y^4-3*x^6*y^5+x^9*y-x^8*y^2-4*x^7*y^3-2*x^6 *y^4-3*x^5*y^5+5*x^7*y^2+5*x^6*y^3-x^7*y-2*x^6*y^2+x^5*y^3-3*x^4*y^4+x^6*y+x^5* y^2+x^4*y^3+2*x^4*y^2+4*x^3*y^3-3*x^3*y^2-x^3*y+2*x^2*y^2-x^2*y+x*y-x+1)/(x^9*y ^3+x^8*y^4-2*x^9*y^2-2*x^8*y^3-x^7*y^4+x^9*y+x^8*y^2+2*x^7*y^2+x^6*y^3-x^7*y-2* x^6*y^2-x^5*y^3+x^6*y+x^4*y^3+2*x^4*y^2-x^3*y^2-x^3*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3748033263 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 102" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 11 17 10 17 9 16 10 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 9 15 10 17 7 16 8 17 6 16 7 - 8 x y + x y - 5 x y + 12 x y + 4 x y - 8 x y 15 8 14 9 13 10 17 5 16 6 15 7 - 12 x y + 2 x y - x y - x y + 2 x y + 28 x y 13 9 15 6 14 7 13 8 12 9 15 5 + 3 x y - 27 x y - 14 x y - x y - 6 x y + 12 x y 14 6 13 7 12 8 15 4 14 5 13 6 + 20 x y + 2 x y + 16 x y - 2 x y - 6 x y - 10 x y 12 7 11 8 14 4 13 5 12 6 11 7 - 18 x y - 8 x y - 4 x y + 4 x y + 25 x y + 16 x y 10 8 14 3 13 4 12 5 11 6 10 7 13 3 + x y + 2 x y + 8 x y - 33 x y - 17 x y - x y - 5 x y 12 4 11 5 10 6 9 7 12 3 11 4 + 19 x y + 31 x y - 7 x y + 2 x y - 3 x y - 33 x y 10 5 9 6 8 7 11 3 10 4 9 5 8 6 + 3 x y + 2 x y + x y + 9 x y + 14 x y - 16 x y + 5 x y 11 2 10 3 9 4 8 5 7 6 10 2 9 3 + 2 x y - 8 x y + 7 x y - 2 x y + x y - 4 x y + 12 x y 8 4 7 5 6 6 10 9 2 8 3 7 4 - 10 x y + 4 x y + x y + 2 x y - 4 x y - 3 x y - 3 x y 6 5 9 8 2 7 3 6 4 5 5 9 7 2 + 2 x y - 4 x y + 11 x y - 4 x y + 2 x y + 2 x y + x - 2 x y 6 3 5 4 8 7 6 2 5 3 4 4 6 + 4 x y - 3 x y - 2 x + 4 x y - 7 x y - 2 x y + 5 x y - 2 x y 5 2 4 3 6 5 4 2 3 3 5 4 + 6 x y - 10 x y + 2 x - 2 x y + 2 x y + 8 x y - x + 4 x y 3 2 4 2 2 3 2 / 9 6 - 8 x y - x + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / (-x y / 10 4 9 5 10 3 9 4 8 5 10 2 9 3 + 2 x y + 2 x y - 4 x y - 3 x y - 2 x y + 2 x y + 6 x y 8 4 9 2 7 4 8 2 7 3 9 7 2 8 + 2 x y - 5 x y - x y + 2 x y - 2 x y + x + 3 x y - 2 x 6 2 6 5 4 3 - 2 x y + 2 x - x - x + 2 x - 2 x + 1) and in Maple notation (x^17*y^11-4*x^17*y^10+5*x^17*y^9+2*x^16*y^10-8*x^16*y^9+x^15*y^10-5*x^17*y^7+ 12*x^16*y^8+4*x^17*y^6-8*x^16*y^7-12*x^15*y^8+2*x^14*y^9-x^13*y^10-x^17*y^5+2*x ^16*y^6+28*x^15*y^7+3*x^13*y^9-27*x^15*y^6-14*x^14*y^7-x^13*y^8-6*x^12*y^9+12*x ^15*y^5+20*x^14*y^6+2*x^13*y^7+16*x^12*y^8-2*x^15*y^4-6*x^14*y^5-10*x^13*y^6-18 *x^12*y^7-8*x^11*y^8-4*x^14*y^4+4*x^13*y^5+25*x^12*y^6+16*x^11*y^7+x^10*y^8+2*x ^14*y^3+8*x^13*y^4-33*x^12*y^5-17*x^11*y^6-x^10*y^7-5*x^13*y^3+19*x^12*y^4+31*x ^11*y^5-7*x^10*y^6+2*x^9*y^7-3*x^12*y^3-33*x^11*y^4+3*x^10*y^5+2*x^9*y^6+x^8*y^ 7+9*x^11*y^3+14*x^10*y^4-16*x^9*y^5+5*x^8*y^6+2*x^11*y^2-8*x^10*y^3+7*x^9*y^4-2 *x^8*y^5+x^7*y^6-4*x^10*y^2+12*x^9*y^3-10*x^8*y^4+4*x^7*y^5+x^6*y^6+2*x^10*y-4* x^9*y^2-3*x^8*y^3-3*x^7*y^4+2*x^6*y^5-4*x^9*y+11*x^8*y^2-4*x^7*y^3+2*x^6*y^4+2* x^5*y^5+x^9-2*x^7*y^2+4*x^6*y^3-3*x^5*y^4-2*x^8+4*x^7*y-7*x^6*y^2-2*x^5*y^3+5*x ^4*y^4-2*x^6*y+6*x^5*y^2-10*x^4*y^3+2*x^6-2*x^5*y+2*x^4*y^2+8*x^3*y^3-x^5+4*x^4 *y-8*x^3*y^2-x^4+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(-x^9*y^6+2*x^10*y^4+2*x^ 9*y^5-4*x^10*y^3-3*x^9*y^4-2*x^8*y^5+2*x^10*y^2+6*x^9*y^3+2*x^8*y^4-5*x^9*y^2-x ^7*y^4+2*x^8*y^2-2*x^7*y^3+x^9+3*x^7*y^2-2*x^8-2*x^6*y^2+2*x^6-x^5-x^4+2*x^3-2* x+1) As the length of the word goes to infinity, the average number of good neigh\ 13 12 11 bors of a random word of length n tends to n times, - 2 (a - 5 a - a 10 9 8 7 6 5 4 3 / + 13 a + 3 a - 23 a + 4 a + 25 a - 15 a - 6 a + 7 a - 1) / ( / 4 3 2 6 3 (5 a + 4 a - 6 a + 2) (2 a + 2 a + 1)) 5 4 3 where a is the root of the polynomial, x + x - 2 x + 2 x - 1, and in decimals this is, 0.3724465204 BTW the ratio for words with, 500, letters is, 0.3746942415 ------------------------------------------------ "Theorem Number 103" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 10 16 9 15 10 16 8 ) | ) C(m, n) x y | = (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 9 16 7 15 8 14 9 16 6 15 7 - 6 x y - 10 x y + 14 x y - x y + 5 x y - 16 x y 14 8 16 5 15 6 14 7 13 8 12 9 15 5 + 4 x y - x y + 9 x y - 7 x y + x y + 3 x y - 2 x y 14 6 13 7 12 8 14 5 13 6 12 7 11 8 + 7 x y - 2 x y - 9 x y - 4 x y + x y + 7 x y - 3 x y 14 4 13 5 12 6 11 7 13 4 12 5 11 6 + x y - x y + 2 x y + 13 x y + 2 x y - 2 x y - 17 x y 10 7 13 3 12 4 11 5 10 6 9 7 12 3 + x y - x y - 4 x y + 6 x y - 4 x y + 3 x y + 4 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 + 5 x y + 7 x y - 6 x y - 2 x y - x y - 7 x y - 7 x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 - 4 x y + 6 x y + 4 x y + 5 x y + 16 x y - 4 x y + x y 11 10 2 9 3 8 4 7 5 10 9 2 - x y - 3 x y - 10 x y - 8 x y - 7 x y + x y + x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 + 9 x y + 13 x y + 5 x y - 2 x y - 7 x y - 5 x y - 2 x y 7 2 6 3 7 6 2 5 3 4 4 6 5 2 - x y - 2 x y + x y + 2 x y + 2 x y - 5 x y - x y - x y 4 3 6 5 4 2 3 3 5 4 3 2 4 + 2 x y + x + x y + 2 x y + 4 x y - x - x y - 3 x y + x 3 2 2 3 2 / 13 6 13 5 + x y + 2 x y - x - 2 x y + x y - x + 1) / (x y - 3 x y / 12 6 13 4 12 5 13 3 11 5 10 6 12 3 - x y + 3 x y + 2 x y - x y + x y - x y - 2 x y 11 4 10 5 9 6 12 2 11 3 10 4 9 5 - 3 x y + 2 x y + x y + x y + 2 x y - x y - x y 11 2 10 3 9 4 8 5 11 10 2 9 3 8 4 + x y + x y - x y - x y - x y - 2 x y + x y + x y 10 8 3 7 4 8 2 7 3 7 2 6 3 7 5 3 + x y + x y + x y - x y - x y + x y - 2 x y - x y + x y 6 5 2 6 5 4 2 5 4 3 2 4 3 3 + x y + x y + x - x y - 2 x y - x + x y + x y + x + x y - x - x y - x + 1) and in Maple notation (x^16*y^10-5*x^16*y^9+x^15*y^10+10*x^16*y^8-6*x^15*y^9-10*x^16*y^7+14*x^15*y^8- x^14*y^9+5*x^16*y^6-16*x^15*y^7+4*x^14*y^8-x^16*y^5+9*x^15*y^6-7*x^14*y^7+x^13* y^8+3*x^12*y^9-2*x^15*y^5+7*x^14*y^6-2*x^13*y^7-9*x^12*y^8-4*x^14*y^5+x^13*y^6+ 7*x^12*y^7-3*x^11*y^8+x^14*y^4-x^13*y^5+2*x^12*y^6+13*x^11*y^7+2*x^13*y^4-2*x^ 12*y^5-17*x^11*y^6+x^10*y^7-x^13*y^3-4*x^12*y^4+6*x^11*y^5-4*x^10*y^6+3*x^9*y^7 +4*x^12*y^3+5*x^11*y^4+7*x^10*y^5-6*x^9*y^6-2*x^8*y^7-x^12*y^2-7*x^11*y^3-7*x^ 10*y^4-4*x^9*y^5+6*x^8*y^6+4*x^11*y^2+5*x^10*y^3+16*x^9*y^4-4*x^8*y^5+x^7*y^6-x ^11*y-3*x^10*y^2-10*x^9*y^3-8*x^8*y^4-7*x^7*y^5+x^10*y+x^9*y^2+9*x^8*y^3+13*x^7 *y^4+5*x^6*y^5-2*x^8*y^2-7*x^7*y^3-5*x^6*y^4-2*x^5*y^5-x^7*y^2-2*x^6*y^3+x^7*y+ 2*x^6*y^2+2*x^5*y^3-5*x^4*y^4-x^6*y-x^5*y^2+2*x^4*y^3+x^6+x^5*y+2*x^4*y^2+4*x^3 *y^3-x^5-x^4*y-3*x^3*y^2+x^4+x^3*y+2*x^2*y^2-x^3-2*x^2*y+x*y-x+1)/(x^13*y^6-3*x ^13*y^5-x^12*y^6+3*x^13*y^4+2*x^12*y^5-x^13*y^3+x^11*y^5-x^10*y^6-2*x^12*y^3-3* x^11*y^4+2*x^10*y^5+x^9*y^6+x^12*y^2+2*x^11*y^3-x^10*y^4-x^9*y^5+x^11*y^2+x^10* y^3-x^9*y^4-x^8*y^5-x^11*y-2*x^10*y^2+x^9*y^3+x^8*y^4+x^10*y+x^8*y^3+x^7*y^4-x^ 8*y^2-x^7*y^3+x^7*y^2-2*x^6*y^3-x^7*y+x^5*y^3+x^6*y+x^5*y^2+x^6-x^5*y-2*x^4*y^2 -x^5+x^4*y+x^3*y^2+x^4+x^3*y-x^3-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3742530029 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 104" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 10 16 9 15 10 16 8 ) | ) C(m, n) x y | = (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 9 16 7 15 8 14 9 16 6 15 7 - 6 x y - 10 x y + 14 x y - x y + 5 x y - 16 x y 14 8 16 5 15 6 14 7 13 8 12 9 15 5 + 4 x y - x y + 9 x y - 7 x y + x y + 3 x y - 2 x y 14 6 13 7 12 8 14 5 13 6 12 7 11 8 + 7 x y - 2 x y - 9 x y - 4 x y + x y + 7 x y - 3 x y 14 4 13 5 12 6 11 7 13 4 12 5 11 6 + x y - x y + 2 x y + 13 x y + 2 x y - 2 x y - 17 x y 10 7 13 3 12 4 11 5 10 6 9 7 12 3 + x y - x y - 4 x y + 6 x y - 4 x y + 3 x y + 4 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 + 5 x y + 7 x y - 6 x y - 2 x y - x y - 7 x y - 7 x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 - 4 x y + 6 x y + 4 x y + 5 x y + 16 x y - 4 x y + x y 11 10 2 9 3 8 4 7 5 10 9 2 - x y - 3 x y - 10 x y - 8 x y - 7 x y + x y + x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 + 9 x y + 13 x y + 5 x y - 2 x y - 7 x y - 5 x y - 2 x y 7 2 6 3 7 6 2 5 3 4 4 6 5 2 - x y - 2 x y + x y + 2 x y + 2 x y - 5 x y - x y - x y 4 3 6 5 4 2 3 3 5 4 3 2 4 + 2 x y + x + x y + 2 x y + 4 x y - x - x y - 3 x y + x 3 2 2 3 2 / 12 6 12 5 + x y + 2 x y - x - 2 x y + x y - x + 1) / ((x y - 3 x y / 12 4 11 5 12 3 11 4 9 6 11 3 9 5 + 3 x y - x y - x y + 3 x y - x y - 3 x y + 2 x y 11 2 10 3 9 4 8 5 10 2 8 4 10 8 3 + x y - x y - x y + x y + 2 x y - 2 x y - x y + x y 7 4 7 3 7 2 6 3 5 3 6 4 2 5 4 - x y + 2 x y - x y + x y - x y - x y + x y + x - x y 3 2 3 2 - x y + x + x y + x y - 1) (x - 1)) and in Maple notation (x^16*y^10-5*x^16*y^9+x^15*y^10+10*x^16*y^8-6*x^15*y^9-10*x^16*y^7+14*x^15*y^8- x^14*y^9+5*x^16*y^6-16*x^15*y^7+4*x^14*y^8-x^16*y^5+9*x^15*y^6-7*x^14*y^7+x^13* y^8+3*x^12*y^9-2*x^15*y^5+7*x^14*y^6-2*x^13*y^7-9*x^12*y^8-4*x^14*y^5+x^13*y^6+ 7*x^12*y^7-3*x^11*y^8+x^14*y^4-x^13*y^5+2*x^12*y^6+13*x^11*y^7+2*x^13*y^4-2*x^ 12*y^5-17*x^11*y^6+x^10*y^7-x^13*y^3-4*x^12*y^4+6*x^11*y^5-4*x^10*y^6+3*x^9*y^7 +4*x^12*y^3+5*x^11*y^4+7*x^10*y^5-6*x^9*y^6-2*x^8*y^7-x^12*y^2-7*x^11*y^3-7*x^ 10*y^4-4*x^9*y^5+6*x^8*y^6+4*x^11*y^2+5*x^10*y^3+16*x^9*y^4-4*x^8*y^5+x^7*y^6-x ^11*y-3*x^10*y^2-10*x^9*y^3-8*x^8*y^4-7*x^7*y^5+x^10*y+x^9*y^2+9*x^8*y^3+13*x^7 *y^4+5*x^6*y^5-2*x^8*y^2-7*x^7*y^3-5*x^6*y^4-2*x^5*y^5-x^7*y^2-2*x^6*y^3+x^7*y+ 2*x^6*y^2+2*x^5*y^3-5*x^4*y^4-x^6*y-x^5*y^2+2*x^4*y^3+x^6+x^5*y+2*x^4*y^2+4*x^3 *y^3-x^5-x^4*y-3*x^3*y^2+x^4+x^3*y+2*x^2*y^2-x^3-2*x^2*y+x*y-x+1)/(x^12*y^6-3*x ^12*y^5+3*x^12*y^4-x^11*y^5-x^12*y^3+3*x^11*y^4-x^9*y^6-3*x^11*y^3+2*x^9*y^5+x^ 11*y^2-x^10*y^3-x^9*y^4+x^8*y^5+2*x^10*y^2-2*x^8*y^4-x^10*y+x^8*y^3-x^7*y^4+2*x ^7*y^3-x^7*y^2+x^6*y^3-x^5*y^3-x^6*y+x^4*y^2+x^5-x^4*y-x^3*y^2+x^3+x^2*y+x*y-1) /(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3742530029 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 105" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 14 9 16 6 ) | ) C(m, n) x y | = (x y - 4 x y + 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 8 16 5 14 7 13 8 16 4 14 6 13 7 - 8 x y - 4 x y + 12 x y + x y + x y - 9 x y - 4 x y 12 8 14 5 13 6 12 7 11 8 14 4 + 3 x y + 5 x y + 7 x y - 11 x y - 3 x y - 3 x y 13 5 12 6 11 7 14 3 13 4 12 5 - 7 x y + 11 x y + 15 x y + x y + 4 x y + 3 x y 11 6 10 7 13 3 12 4 11 5 10 6 - 23 x y - 8 x y - x y - 10 x y + 7 x y + 21 x y 12 3 11 4 10 5 9 6 11 3 10 4 + 4 x y + 10 x y - 15 x y - 6 x y - 6 x y - 4 x y 9 5 8 6 10 3 9 4 8 5 7 6 10 2 + 14 x y - 4 x y + 9 x y - 10 x y + 6 x y + 3 x y - 3 x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 + 7 x y - 15 x y + x y + 4 x y - 14 x y + 17 x y + 6 x y 9 8 2 7 3 6 4 5 5 8 7 2 - 2 x y + 3 x y - x y - 18 x y + 2 x y + 2 x y - 6 x y 6 3 5 4 7 5 3 4 4 6 5 2 + 12 x y + 4 x y + 2 x y - 10 x y + 5 x y - 2 x y + 6 x y 4 3 6 4 2 3 3 5 4 3 2 3 - 2 x y + x - 4 x y + 8 x y - x + 2 x y - 4 x y - 2 x y 2 2 2 / 12 5 12 4 11 5 + 4 x y - 2 x y + 2 x y - x + 1) / (x y - 2 x y - x y / 12 3 11 4 11 3 9 4 9 3 8 4 9 2 8 3 + x y + 2 x y - x y - x y + 2 x y + 2 x y + x y - 4 x y 7 4 9 7 3 8 7 2 5 2 6 4 2 5 - x y - 2 x y + 2 x y + 2 x y - x y + 2 x y + x - 2 x y - x 4 3 + 2 x y - 2 x y - x + 1) and in Maple notation (x^16*y^8-4*x^16*y^7+2*x^14*y^9+6*x^16*y^6-8*x^14*y^8-4*x^16*y^5+12*x^14*y^7+x^ 13*y^8+x^16*y^4-9*x^14*y^6-4*x^13*y^7+3*x^12*y^8+5*x^14*y^5+7*x^13*y^6-11*x^12* y^7-3*x^11*y^8-3*x^14*y^4-7*x^13*y^5+11*x^12*y^6+15*x^11*y^7+x^14*y^3+4*x^13*y^ 4+3*x^12*y^5-23*x^11*y^6-8*x^10*y^7-x^13*y^3-10*x^12*y^4+7*x^11*y^5+21*x^10*y^6 +4*x^12*y^3+10*x^11*y^4-15*x^10*y^5-6*x^9*y^6-6*x^11*y^3-4*x^10*y^4+14*x^9*y^5-\ 4*x^8*y^6+9*x^10*y^3-10*x^9*y^4+6*x^8*y^5+3*x^7*y^6-3*x^10*y^2+7*x^8*y^4-15*x^7 *y^5+x^6*y^6+4*x^9*y^2-14*x^8*y^3+17*x^7*y^4+6*x^6*y^5-2*x^9*y+3*x^8*y^2-x^7*y^ 3-18*x^6*y^4+2*x^5*y^5+2*x^8*y-6*x^7*y^2+12*x^6*y^3+4*x^5*y^4+2*x^7*y-10*x^5*y^ 3+5*x^4*y^4-2*x^6*y+6*x^5*y^2-2*x^4*y^3+x^6-4*x^4*y^2+8*x^3*y^3-x^5+2*x^4*y-4*x ^3*y^2-2*x^3*y+4*x^2*y^2-2*x^2*y+2*x*y-x+1)/(x^12*y^5-2*x^12*y^4-x^11*y^5+x^12* y^3+2*x^11*y^4-x^11*y^3-x^9*y^4+2*x^9*y^3+2*x^8*y^4+x^9*y^2-4*x^8*y^3-x^7*y^4-2 *x^9*y+2*x^7*y^3+2*x^8*y-x^7*y^2+2*x^5*y^2+x^6-2*x^4*y^2-x^5+2*x^4*y-2*x^3*y-x+ 1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3653972420 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 106" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 14 9 16 6 ) | ) C(m, n) x y | = (x y - 4 x y + 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 8 16 5 14 7 13 8 16 4 14 6 13 7 - 8 x y - 4 x y + 12 x y + x y + x y - 9 x y - 4 x y 12 8 14 5 13 6 12 7 11 8 14 4 + 3 x y + 5 x y + 7 x y - 11 x y - 3 x y - 3 x y 13 5 12 6 11 7 14 3 13 4 12 5 - 7 x y + 11 x y + 15 x y + x y + 4 x y + 3 x y 11 6 10 7 13 3 12 4 11 5 10 6 - 23 x y - 8 x y - x y - 10 x y + 7 x y + 21 x y 12 3 11 4 10 5 9 6 11 3 10 4 + 4 x y + 10 x y - 15 x y - 6 x y - 6 x y - 4 x y 9 5 8 6 10 3 9 4 8 5 7 6 10 2 + 14 x y - 4 x y + 9 x y - 10 x y + 6 x y + 3 x y - 3 x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 + 7 x y - 15 x y + x y + 4 x y - 14 x y + 17 x y + 6 x y 9 8 2 7 3 6 4 5 5 8 7 2 - 2 x y + 3 x y - x y - 18 x y + 2 x y + 2 x y - 6 x y 6 3 5 4 7 5 3 4 4 6 5 2 + 12 x y + 4 x y + 2 x y - 10 x y + 5 x y - 2 x y + 6 x y 4 3 6 4 2 3 3 5 4 3 2 3 - 2 x y + x - 4 x y + 8 x y - x + 2 x y - 4 x y - 2 x y 2 2 2 / 11 5 11 4 + 4 x y - 2 x y + 2 x y - x + 1) / ((x - 1) (x y - 2 x y / 11 3 8 4 8 3 7 4 8 2 7 3 8 7 2 + x y - x y + 2 x y + x y + x y - 2 x y - 2 x y + x y 4 2 5 3 + 2 x y + x + 2 x y - 1)) and in Maple notation (x^16*y^8-4*x^16*y^7+2*x^14*y^9+6*x^16*y^6-8*x^14*y^8-4*x^16*y^5+12*x^14*y^7+x^ 13*y^8+x^16*y^4-9*x^14*y^6-4*x^13*y^7+3*x^12*y^8+5*x^14*y^5+7*x^13*y^6-11*x^12* y^7-3*x^11*y^8-3*x^14*y^4-7*x^13*y^5+11*x^12*y^6+15*x^11*y^7+x^14*y^3+4*x^13*y^ 4+3*x^12*y^5-23*x^11*y^6-8*x^10*y^7-x^13*y^3-10*x^12*y^4+7*x^11*y^5+21*x^10*y^6 +4*x^12*y^3+10*x^11*y^4-15*x^10*y^5-6*x^9*y^6-6*x^11*y^3-4*x^10*y^4+14*x^9*y^5-\ 4*x^8*y^6+9*x^10*y^3-10*x^9*y^4+6*x^8*y^5+3*x^7*y^6-3*x^10*y^2+7*x^8*y^4-15*x^7 *y^5+x^6*y^6+4*x^9*y^2-14*x^8*y^3+17*x^7*y^4+6*x^6*y^5-2*x^9*y+3*x^8*y^2-x^7*y^ 3-18*x^6*y^4+2*x^5*y^5+2*x^8*y-6*x^7*y^2+12*x^6*y^3+4*x^5*y^4+2*x^7*y-10*x^5*y^ 3+5*x^4*y^4-2*x^6*y+6*x^5*y^2-2*x^4*y^3+x^6-4*x^4*y^2+8*x^3*y^3-x^5+2*x^4*y-4*x ^3*y^2-2*x^3*y+4*x^2*y^2-2*x^2*y+2*x*y-x+1)/(x-1)/(x^11*y^5-2*x^11*y^4+x^11*y^3 -x^8*y^4+2*x^8*y^3+x^7*y^4+x^8*y^2-2*x^7*y^3-2*x^8*y+x^7*y^2+2*x^4*y^2+x^5+2*x^ 3*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3653972420 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 107" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 15 9 16 7 15 8 ) | ) C(m, n) x y | = (x y - x y - 4 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 6 15 7 14 8 16 5 15 6 14 7 + 6 x y - 13 x y - 5 x y - 4 x y + 12 x y + 17 x y 13 8 16 4 15 5 14 6 13 7 12 8 + 2 x y + x y - 3 x y - 24 x y - 8 x y + 2 x y 15 4 14 5 13 6 12 7 15 3 14 4 - 2 x y + 20 x y + 13 x y - 6 x y + x y - 11 x y 13 5 12 6 11 7 14 3 13 4 12 5 - 11 x y + 5 x y + 10 x y + 3 x y + 6 x y + 3 x y 11 6 10 7 13 3 12 4 11 5 10 6 - 30 x y - 5 x y - 3 x y - 7 x y + 23 x y + 20 x y 13 2 12 3 11 4 10 5 9 6 11 3 + x y + 3 x y + 4 x y - 26 x y - 3 x y - 7 x y 10 4 9 5 8 6 10 3 9 4 8 5 10 2 + 10 x y + 10 x y - 2 x y + 5 x y - 7 x y + 4 x y - 4 x y 9 3 8 4 7 5 9 2 8 3 7 4 6 5 - 2 x y + x y - 11 x y + 4 x y - 8 x y + 14 x y + 2 x y 9 8 2 7 3 6 4 5 5 8 7 2 - 2 x y + 2 x y - x y - 16 x y + x y + 2 x y - 5 x y 6 3 5 4 7 5 3 4 4 6 5 2 6 + 14 x y + 4 x y + 2 x y - 9 x y + 3 x y - 2 x y + 6 x y + x 4 2 3 3 5 4 3 2 3 2 2 2 - 4 x y + 8 x y - x + 2 x y - 4 x y - 2 x y + 4 x y - 2 x y / 12 5 12 4 11 5 12 3 11 4 + 2 x y - x + 1) / (x y - 2 x y - x y + x y + 2 x y / 11 3 9 4 9 3 8 4 9 2 8 3 7 4 9 - x y - x y + 2 x y + 2 x y + x y - 4 x y - x y - 2 x y 7 3 8 7 2 5 2 6 4 2 5 4 3 + 2 x y + 2 x y - x y + 2 x y + x - 2 x y - x + 2 x y - 2 x y - x + 1) and in Maple notation (x^16*y^8-x^15*y^9-4*x^16*y^7+6*x^15*y^8+6*x^16*y^6-13*x^15*y^7-5*x^14*y^8-4*x^ 16*y^5+12*x^15*y^6+17*x^14*y^7+2*x^13*y^8+x^16*y^4-3*x^15*y^5-24*x^14*y^6-8*x^ 13*y^7+2*x^12*y^8-2*x^15*y^4+20*x^14*y^5+13*x^13*y^6-6*x^12*y^7+x^15*y^3-11*x^ 14*y^4-11*x^13*y^5+5*x^12*y^6+10*x^11*y^7+3*x^14*y^3+6*x^13*y^4+3*x^12*y^5-30*x ^11*y^6-5*x^10*y^7-3*x^13*y^3-7*x^12*y^4+23*x^11*y^5+20*x^10*y^6+x^13*y^2+3*x^ 12*y^3+4*x^11*y^4-26*x^10*y^5-3*x^9*y^6-7*x^11*y^3+10*x^10*y^4+10*x^9*y^5-2*x^8 *y^6+5*x^10*y^3-7*x^9*y^4+4*x^8*y^5-4*x^10*y^2-2*x^9*y^3+x^8*y^4-11*x^7*y^5+4*x ^9*y^2-8*x^8*y^3+14*x^7*y^4+2*x^6*y^5-2*x^9*y+2*x^8*y^2-x^7*y^3-16*x^6*y^4+x^5* y^5+2*x^8*y-5*x^7*y^2+14*x^6*y^3+4*x^5*y^4+2*x^7*y-9*x^5*y^3+3*x^4*y^4-2*x^6*y+ 6*x^5*y^2+x^6-4*x^4*y^2+8*x^3*y^3-x^5+2*x^4*y-4*x^3*y^2-2*x^3*y+4*x^2*y^2-2*x^2 *y+2*x*y-x+1)/(x^12*y^5-2*x^12*y^4-x^11*y^5+x^12*y^3+2*x^11*y^4-x^11*y^3-x^9*y^ 4+2*x^9*y^3+2*x^8*y^4+x^9*y^2-4*x^8*y^3-x^7*y^4-2*x^9*y+2*x^7*y^3+2*x^8*y-x^7*y ^2+2*x^5*y^2+x^6-2*x^4*y^2-x^5+2*x^4*y-2*x^3*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ 3 2 11 bors of a random word of length n tends to n times, 2 (a - a - 1) (a 10 9 8 7 6 5 4 3 2 + 4 a + 12 a + 23 a + 31 a + 25 a + 9 a - 8 a - 13 a - 11 a 5 4 2 / 4 2 2 - 4 a - 1) (6 a + 5 a - 6 a - 1) / (a (5 a + 8 a + 6) / 8 7 6 5 4 3 2 (a + a + a - a - a - 2 a - 2 a - a - 1)) 6 5 3 where a is the root of the polynomial, x + x - 2 x - x + 1, and in decimals this is, 0.3628185126 BTW the ratio for words with, 500, letters is, 0.3646492923 ------------------------------------------------ "Theorem Number 108" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 15 9 16 7 15 8 ) | ) C(m, n) x y | = (x y - x y - 4 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 6 15 7 14 8 16 5 15 6 14 7 + 6 x y - 13 x y - 5 x y - 4 x y + 12 x y + 17 x y 13 8 16 4 15 5 14 6 13 7 12 8 + 2 x y + x y - 3 x y - 24 x y - 8 x y + 2 x y 15 4 14 5 13 6 12 7 15 3 14 4 - 2 x y + 20 x y + 13 x y - 6 x y + x y - 11 x y 13 5 12 6 11 7 14 3 13 4 12 5 - 11 x y + 5 x y + 10 x y + 3 x y + 6 x y + 3 x y 11 6 10 7 13 3 12 4 11 5 10 6 - 30 x y - 5 x y - 3 x y - 7 x y + 23 x y + 20 x y 13 2 12 3 11 4 10 5 9 6 11 3 + x y + 3 x y + 4 x y - 26 x y - 3 x y - 7 x y 10 4 9 5 8 6 10 3 9 4 8 5 10 2 + 10 x y + 10 x y - 2 x y + 5 x y - 7 x y + 4 x y - 4 x y 9 3 8 4 7 5 9 2 8 3 7 4 6 5 - 2 x y + x y - 11 x y + 4 x y - 8 x y + 14 x y + 2 x y 9 8 2 7 3 6 4 5 5 8 7 2 - 2 x y + 2 x y - x y - 16 x y + x y + 2 x y - 5 x y 6 3 5 4 7 5 3 4 4 6 5 2 6 + 14 x y + 4 x y + 2 x y - 9 x y + 3 x y - 2 x y + 6 x y + x 4 2 3 3 5 4 3 2 3 2 2 2 - 4 x y + 8 x y - x + 2 x y - 4 x y - 2 x y + 4 x y - 2 x y / 11 5 11 4 11 3 8 4 8 3 + 2 x y - x + 1) / ((x y - 2 x y + x y - x y + 2 x y / 7 4 8 2 7 3 8 7 2 4 2 5 3 + x y + x y - 2 x y - 2 x y + x y + 2 x y + x + 2 x y - 1) (x - 1)) and in Maple notation (x^16*y^8-x^15*y^9-4*x^16*y^7+6*x^15*y^8+6*x^16*y^6-13*x^15*y^7-5*x^14*y^8-4*x^ 16*y^5+12*x^15*y^6+17*x^14*y^7+2*x^13*y^8+x^16*y^4-3*x^15*y^5-24*x^14*y^6-8*x^ 13*y^7+2*x^12*y^8-2*x^15*y^4+20*x^14*y^5+13*x^13*y^6-6*x^12*y^7+x^15*y^3-11*x^ 14*y^4-11*x^13*y^5+5*x^12*y^6+10*x^11*y^7+3*x^14*y^3+6*x^13*y^4+3*x^12*y^5-30*x ^11*y^6-5*x^10*y^7-3*x^13*y^3-7*x^12*y^4+23*x^11*y^5+20*x^10*y^6+x^13*y^2+3*x^ 12*y^3+4*x^11*y^4-26*x^10*y^5-3*x^9*y^6-7*x^11*y^3+10*x^10*y^4+10*x^9*y^5-2*x^8 *y^6+5*x^10*y^3-7*x^9*y^4+4*x^8*y^5-4*x^10*y^2-2*x^9*y^3+x^8*y^4-11*x^7*y^5+4*x ^9*y^2-8*x^8*y^3+14*x^7*y^4+2*x^6*y^5-2*x^9*y+2*x^8*y^2-x^7*y^3-16*x^6*y^4+x^5* y^5+2*x^8*y-5*x^7*y^2+14*x^6*y^3+4*x^5*y^4+2*x^7*y-9*x^5*y^3+3*x^4*y^4-2*x^6*y+ 6*x^5*y^2+x^6-4*x^4*y^2+8*x^3*y^3-x^5+2*x^4*y-4*x^3*y^2-2*x^3*y+4*x^2*y^2-2*x^2 *y+2*x*y-x+1)/(x^11*y^5-2*x^11*y^4+x^11*y^3-x^8*y^4+2*x^8*y^3+x^7*y^4+x^8*y^2-2 *x^7*y^3-2*x^8*y+x^7*y^2+2*x^4*y^2+x^5+2*x^3*y-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ 3 2 11 bors of a random word of length n tends to n times, 2 (a - a - 1) (a 10 9 8 7 6 5 4 3 2 + 4 a + 12 a + 23 a + 31 a + 25 a + 9 a - 8 a - 13 a - 11 a 5 4 2 / 4 2 2 - 4 a - 1) (6 a + 5 a - 6 a - 1) / (a (5 a + 8 a + 6) / 8 7 6 5 4 3 2 (a + a + a - a - a - 2 a - 2 a - a - 1)) 5 4 3 where a is the root of the polynomial, (x - 1) (x + 2 x + 2 x - 1), and in decimals this is, 0.3628185126 BTW the ratio for words with, 500, letters is, 0.3646492923 ------------------------------------------------ "Theorem Number 109" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = - (x y - 4 x y + x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 14 9 16 6 15 7 14 8 13 9 16 5 - 3 x y + 2 x y - 4 x y + 2 x y - 10 x y - x y + x y 15 6 14 7 15 5 14 6 13 7 12 8 + 2 x y + 20 x y - 3 x y - 20 x y + 8 x y - x y 11 9 15 4 14 5 13 6 12 7 11 8 - 2 x y + x y + 10 x y - 15 x y + 2 x y + 6 x y 14 4 13 5 11 7 10 8 13 4 12 5 - 2 x y + 12 x y - 6 x y + x y - 5 x y - 2 x y 11 6 10 7 13 3 12 4 10 6 9 7 9 6 + 2 x y - 3 x y + x y + x y + 2 x y + x y - 4 x y 10 4 9 5 8 6 7 7 10 3 9 4 8 5 7 6 + x y + 3 x y + x y + x y - x y + x y - 4 x y - x y 9 3 8 4 6 6 9 2 8 3 7 4 6 5 9 + x y + 7 x y + x y - 3 x y - 2 x y + 2 x y + 2 x y + x y 8 2 7 3 6 4 5 5 7 2 6 3 5 4 - 2 x y + 2 x y - 2 x y + 3 x y - 3 x y + 4 x y - 2 x y 7 6 2 5 3 4 4 6 5 2 4 3 4 2 - x y - 4 x y - 2 x y + 2 x y - x y - x y - 3 x y - 3 x y 3 3 4 4 2 2 3 / 17 8 17 7 - 4 x y + x y + x - 2 x y + x - x y - 1) / (x y - 4 x y / 16 8 17 6 16 7 17 5 16 6 15 7 14 8 - x y + 6 x y + 4 x y - 4 x y - 6 x y + x y - x y 17 4 16 5 15 6 14 7 16 4 15 5 14 6 + x y + 4 x y - 4 x y + 3 x y - x y + 6 x y - 3 x y 15 4 14 5 15 3 12 6 12 5 11 6 12 4 - 4 x y + x y + x y - x y + 2 x y - x y - x y 11 5 11 4 11 3 10 4 8 6 10 3 9 4 + 3 x y - 3 x y + x y + x y + x y - 2 x y + 2 x y 8 5 10 2 9 3 9 2 7 4 9 7 2 6 2 5 3 - x y + x y - 2 x y + x y + x y - x y - x y - x y - x y 5 4 3 + x y - x - x - x y + 1) and in Maple notation -(x^16*y^9-4*x^16*y^8+x^15*y^9+6*x^16*y^7-3*x^15*y^8+2*x^14*y^9-4*x^16*y^6+2*x^ 15*y^7-10*x^14*y^8-x^13*y^9+x^16*y^5+2*x^15*y^6+20*x^14*y^7-3*x^15*y^5-20*x^14* y^6+8*x^13*y^7-x^12*y^8-2*x^11*y^9+x^15*y^4+10*x^14*y^5-15*x^13*y^6+2*x^12*y^7+ 6*x^11*y^8-2*x^14*y^4+12*x^13*y^5-6*x^11*y^7+x^10*y^8-5*x^13*y^4-2*x^12*y^5+2*x ^11*y^6-3*x^10*y^7+x^13*y^3+x^12*y^4+2*x^10*y^6+x^9*y^7-4*x^9*y^6+x^10*y^4+3*x^ 9*y^5+x^8*y^6+x^7*y^7-x^10*y^3+x^9*y^4-4*x^8*y^5-x^7*y^6+x^9*y^3+7*x^8*y^4+x^6* y^6-3*x^9*y^2-2*x^8*y^3+2*x^7*y^4+2*x^6*y^5+x^9*y-2*x^8*y^2+2*x^7*y^3-2*x^6*y^4 +3*x^5*y^5-3*x^7*y^2+4*x^6*y^3-2*x^5*y^4-x^7*y-4*x^6*y^2-2*x^5*y^3+2*x^4*y^4-x^ 6*y-x^5*y^2-3*x^4*y^3-3*x^4*y^2-4*x^3*y^3+x^4*y+x^4-2*x^2*y^2+x^3-x*y-1)/(x^17* y^8-4*x^17*y^7-x^16*y^8+6*x^17*y^6+4*x^16*y^7-4*x^17*y^5-6*x^16*y^6+x^15*y^7-x^ 14*y^8+x^17*y^4+4*x^16*y^5-4*x^15*y^6+3*x^14*y^7-x^16*y^4+6*x^15*y^5-3*x^14*y^6 -4*x^15*y^4+x^14*y^5+x^15*y^3-x^12*y^6+2*x^12*y^5-x^11*y^6-x^12*y^4+3*x^11*y^5-\ 3*x^11*y^4+x^11*y^3+x^10*y^4+x^8*y^6-2*x^10*y^3+2*x^9*y^4-x^8*y^5+x^10*y^2-2*x^ 9*y^3+x^9*y^2+x^7*y^4-x^9*y-x^7*y^2-x^6*y^2-x^5*y^3+x^5*y-x^4-x^3-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 13 10 9 8 6 5 4 3 2 2 (a - a - 5 a - 2 a + 6 a + 11 a + 3 a + 2 a - 5 a - 2 a - 1) - ------------------------------------------------------------------------- 5 3 2 5 4 3 2 (6 a + 4 a + 3 a + 1) (2 a + 2 a + 3 a + 2 a + a + 1) 6 4 3 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.3623371898 BTW the ratio for words with, 500, letters is, 0.3642637837 ------------------------------------------------ "Theorem Number 110" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = - (x y - 4 x y + x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 14 9 16 6 15 7 14 8 13 9 16 5 - 3 x y + 2 x y - 4 x y + 2 x y - 10 x y - x y + x y 15 6 14 7 15 5 14 6 13 7 12 8 + 2 x y + 20 x y - 3 x y - 20 x y + 8 x y - x y 11 9 15 4 14 5 13 6 12 7 11 8 - 2 x y + x y + 10 x y - 15 x y + 2 x y + 6 x y 14 4 13 5 11 7 10 8 13 4 12 5 - 2 x y + 12 x y - 6 x y + x y - 5 x y - 2 x y 11 6 10 7 13 3 12 4 10 6 9 7 9 6 + 2 x y - 3 x y + x y + x y + 2 x y + x y - 4 x y 10 4 9 5 8 6 7 7 10 3 9 4 8 5 7 6 + x y + 3 x y + x y + x y - x y + x y - 4 x y - x y 9 3 8 4 6 6 9 2 8 3 7 4 6 5 9 + x y + 7 x y + x y - 3 x y - 2 x y + 2 x y + 2 x y + x y 8 2 7 3 6 4 5 5 7 2 6 3 5 4 - 2 x y + 2 x y - 2 x y + 3 x y - 3 x y + 4 x y - 2 x y 7 6 2 5 3 4 4 6 5 2 4 3 4 2 - x y - 4 x y - 2 x y + 2 x y - x y - x y - 3 x y - 3 x y 3 3 4 4 2 2 3 / 17 8 17 7 - 4 x y + x y + x - 2 x y + x - x y - 1) / (x y - 4 x y / 16 8 17 6 16 7 17 5 16 6 15 7 14 8 - x y + 6 x y + 4 x y - 4 x y - 6 x y + x y - x y 17 4 16 5 15 6 14 7 16 4 15 5 14 6 + x y + 4 x y - 4 x y + 3 x y - x y + 6 x y - 3 x y 15 4 14 5 15 3 12 6 12 5 11 6 12 4 - 4 x y + x y + x y - x y + 2 x y - x y - x y 11 5 11 4 11 3 10 4 8 6 10 3 9 4 + 3 x y - 3 x y + x y + x y + x y - 2 x y + 2 x y 8 5 10 2 9 3 9 2 7 4 9 7 2 6 2 5 3 - x y + x y - 2 x y + x y + x y - x y - x y - x y - x y 5 4 3 + x y - x - x - x y + 1) and in Maple notation -(x^16*y^9-4*x^16*y^8+x^15*y^9+6*x^16*y^7-3*x^15*y^8+2*x^14*y^9-4*x^16*y^6+2*x^ 15*y^7-10*x^14*y^8-x^13*y^9+x^16*y^5+2*x^15*y^6+20*x^14*y^7-3*x^15*y^5-20*x^14* y^6+8*x^13*y^7-x^12*y^8-2*x^11*y^9+x^15*y^4+10*x^14*y^5-15*x^13*y^6+2*x^12*y^7+ 6*x^11*y^8-2*x^14*y^4+12*x^13*y^5-6*x^11*y^7+x^10*y^8-5*x^13*y^4-2*x^12*y^5+2*x ^11*y^6-3*x^10*y^7+x^13*y^3+x^12*y^4+2*x^10*y^6+x^9*y^7-4*x^9*y^6+x^10*y^4+3*x^ 9*y^5+x^8*y^6+x^7*y^7-x^10*y^3+x^9*y^4-4*x^8*y^5-x^7*y^6+x^9*y^3+7*x^8*y^4+x^6* y^6-3*x^9*y^2-2*x^8*y^3+2*x^7*y^4+2*x^6*y^5+x^9*y-2*x^8*y^2+2*x^7*y^3-2*x^6*y^4 +3*x^5*y^5-3*x^7*y^2+4*x^6*y^3-2*x^5*y^4-x^7*y-4*x^6*y^2-2*x^5*y^3+2*x^4*y^4-x^ 6*y-x^5*y^2-3*x^4*y^3-3*x^4*y^2-4*x^3*y^3+x^4*y+x^4-2*x^2*y^2+x^3-x*y-1)/(x^17* y^8-4*x^17*y^7-x^16*y^8+6*x^17*y^6+4*x^16*y^7-4*x^17*y^5-6*x^16*y^6+x^15*y^7-x^ 14*y^8+x^17*y^4+4*x^16*y^5-4*x^15*y^6+3*x^14*y^7-x^16*y^4+6*x^15*y^5-3*x^14*y^6 -4*x^15*y^4+x^14*y^5+x^15*y^3-x^12*y^6+2*x^12*y^5-x^11*y^6-x^12*y^4+3*x^11*y^5-\ 3*x^11*y^4+x^11*y^3+x^10*y^4+x^8*y^6-2*x^10*y^3+2*x^9*y^4-x^8*y^5+x^10*y^2-2*x^ 9*y^3+x^9*y^2+x^7*y^4-x^9*y-x^7*y^2-x^6*y^2-x^5*y^3+x^5*y-x^4-x^3-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 13 10 9 8 6 5 4 3 2 2 (a - a - 5 a - 2 a + 6 a + 11 a + 3 a + 2 a - 5 a - 2 a - 1) - ------------------------------------------------------------------------- 5 3 2 5 4 3 2 (6 a + 4 a + 3 a + 1) (2 a + 2 a + 3 a + 2 a + a + 1) 6 4 3 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.3623371898 BTW the ratio for words with, 500, letters is, 0.3642637837 ------------------------------------------------ "Theorem Number 111" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [1, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = - (x y + x y - 3 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 5 13 6 14 4 13 5 12 6 11 7 10 8 + 3 x y + 5 x y - x y - x y - x y - 2 x y - x y 13 4 12 5 11 6 10 7 13 3 12 4 11 5 - 2 x y + 3 x y + 5 x y + 5 x y + x y - 3 x y - 5 x y 10 6 9 7 12 3 11 4 10 5 9 6 11 3 - 6 x y + x y + x y + 3 x y - 2 x y - x y - x y 10 4 9 5 8 6 10 3 9 4 8 5 9 3 + 7 x y - 2 x y - x y - 3 x y + 2 x y + 5 x y + x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 7 3 - 6 x y + x y - x y - x y + 3 x y - x y + x y + x y 6 4 5 5 8 7 2 6 3 5 4 7 6 2 + 2 x y - 3 x y - x y - 2 x y - 5 x y + 3 x y + x y + 3 x y 5 3 4 4 5 2 4 3 5 4 2 3 3 - x y - 3 x y - 2 x y + 2 x y + 2 x y + 2 x y + 4 x y 4 3 2 4 2 2 3 2 / 15 6 - 2 x y - 2 x y + x + 2 x y - x - x y + x y - x + 1) / (x y / 15 5 14 6 15 4 14 5 15 3 13 5 12 6 - 3 x y - x y + 3 x y + 2 x y - x y + x y - x y 14 3 13 4 12 5 11 6 14 2 13 3 12 4 - 2 x y - 3 x y + 2 x y + x y + x y + 3 x y - 2 x y 11 5 13 2 12 3 11 4 10 5 12 2 11 3 - x y - x y + 2 x y - x y - x y - x y + x y 10 4 10 3 9 4 10 2 9 3 10 9 2 8 3 + 2 x y - 2 x y + 2 x y + 2 x y - x y - x y - x y - x y 7 3 6 4 8 7 2 6 3 6 2 5 3 4 3 2 + x y - x y + x y - x y - x y + x y + x y - x + x - x y + x y + x - 1) and in Maple notation -(x^14*y^7+x^13*y^8-3*x^14*y^6-4*x^13*y^7+3*x^14*y^5+5*x^13*y^6-x^14*y^4-x^13*y ^5-x^12*y^6-2*x^11*y^7-x^10*y^8-2*x^13*y^4+3*x^12*y^5+5*x^11*y^6+5*x^10*y^7+x^ 13*y^3-3*x^12*y^4-5*x^11*y^5-6*x^10*y^6+x^9*y^7+x^12*y^3+3*x^11*y^4-2*x^10*y^5- x^9*y^6-x^11*y^3+7*x^10*y^4-2*x^9*y^5-x^8*y^6-3*x^10*y^3+2*x^9*y^4+5*x^8*y^5+x^ 9*y^3-6*x^8*y^4+x^7*y^5-x^6*y^6-x^9*y^2+3*x^8*y^3-x^7*y^4+x^6*y^5+x^7*y^3+2*x^6 *y^4-3*x^5*y^5-x^8*y-2*x^7*y^2-5*x^6*y^3+3*x^5*y^4+x^7*y+3*x^6*y^2-x^5*y^3-3*x^ 4*y^4-2*x^5*y^2+2*x^4*y^3+2*x^5*y+2*x^4*y^2+4*x^3*y^3-2*x^4*y-2*x^3*y^2+x^4+2*x ^2*y^2-x^3-x^2*y+x*y-x+1)/(x^15*y^6-3*x^15*y^5-x^14*y^6+3*x^15*y^4+2*x^14*y^5-x ^15*y^3+x^13*y^5-x^12*y^6-2*x^14*y^3-3*x^13*y^4+2*x^12*y^5+x^11*y^6+x^14*y^2+3* x^13*y^3-2*x^12*y^4-x^11*y^5-x^13*y^2+2*x^12*y^3-x^11*y^4-x^10*y^5-x^12*y^2+x^ 11*y^3+2*x^10*y^4-2*x^10*y^3+2*x^9*y^4+2*x^10*y^2-x^9*y^3-x^10*y-x^9*y^2-x^8*y^ 3+x^7*y^3-x^6*y^4+x^8*y-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y^3-x^4+x^3-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3607566758 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 112" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = - (x y + x y - 3 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 5 13 6 14 4 13 5 12 6 11 7 10 8 + 3 x y + 5 x y - x y - x y - x y - 2 x y - x y 13 4 12 5 11 6 10 7 13 3 12 4 11 5 - 2 x y + 3 x y + 5 x y + 5 x y + x y - 3 x y - 5 x y 10 6 9 7 12 3 11 4 10 5 9 6 11 3 - 6 x y + x y + x y + 3 x y - 2 x y - x y - x y 10 4 9 5 8 6 10 3 9 4 8 5 9 3 + 7 x y - 2 x y - x y - 3 x y + 2 x y + 5 x y + x y 8 4 7 5 6 6 9 2 8 3 7 4 6 5 7 3 - 6 x y + x y - x y - x y + 3 x y - x y + x y + x y 6 4 5 5 8 7 2 6 3 5 4 7 6 2 + 2 x y - 3 x y - x y - 2 x y - 5 x y + 3 x y + x y + 3 x y 5 3 4 4 5 2 4 3 5 4 2 3 3 - x y - 3 x y - 2 x y + 2 x y + 2 x y + 2 x y + 4 x y 4 3 2 4 2 2 3 2 / 15 6 - 2 x y - 2 x y + x + 2 x y - x - x y + x y - x + 1) / (x y / 15 5 14 6 15 4 14 5 15 3 13 5 12 6 - 3 x y - x y + 3 x y + 2 x y - x y + x y - x y 14 3 13 4 12 5 11 6 14 2 13 3 12 4 - 2 x y - 3 x y + 2 x y + x y + x y + 3 x y - 2 x y 11 5 13 2 12 3 11 4 10 5 12 2 11 3 - x y - x y + 2 x y - x y - x y - x y + x y 10 4 10 3 9 4 10 2 9 3 10 9 2 8 3 + 2 x y - 2 x y + 2 x y + 2 x y - x y - x y - x y - x y 7 3 6 4 8 7 2 6 3 6 2 5 3 4 3 2 + x y - x y + x y - x y - x y + x y + x y - x + x - x y + x y + x - 1) and in Maple notation -(x^14*y^7+x^13*y^8-3*x^14*y^6-4*x^13*y^7+3*x^14*y^5+5*x^13*y^6-x^14*y^4-x^13*y ^5-x^12*y^6-2*x^11*y^7-x^10*y^8-2*x^13*y^4+3*x^12*y^5+5*x^11*y^6+5*x^10*y^7+x^ 13*y^3-3*x^12*y^4-5*x^11*y^5-6*x^10*y^6+x^9*y^7+x^12*y^3+3*x^11*y^4-2*x^10*y^5- x^9*y^6-x^11*y^3+7*x^10*y^4-2*x^9*y^5-x^8*y^6-3*x^10*y^3+2*x^9*y^4+5*x^8*y^5+x^ 9*y^3-6*x^8*y^4+x^7*y^5-x^6*y^6-x^9*y^2+3*x^8*y^3-x^7*y^4+x^6*y^5+x^7*y^3+2*x^6 *y^4-3*x^5*y^5-x^8*y-2*x^7*y^2-5*x^6*y^3+3*x^5*y^4+x^7*y+3*x^6*y^2-x^5*y^3-3*x^ 4*y^4-2*x^5*y^2+2*x^4*y^3+2*x^5*y+2*x^4*y^2+4*x^3*y^3-2*x^4*y-2*x^3*y^2+x^4+2*x ^2*y^2-x^3-x^2*y+x*y-x+1)/(x^15*y^6-3*x^15*y^5-x^14*y^6+3*x^15*y^4+2*x^14*y^5-x ^15*y^3+x^13*y^5-x^12*y^6-2*x^14*y^3-3*x^13*y^4+2*x^12*y^5+x^11*y^6+x^14*y^2+3* x^13*y^3-2*x^12*y^4-x^11*y^5-x^13*y^2+2*x^12*y^3-x^11*y^4-x^10*y^5-x^12*y^2+x^ 11*y^3+2*x^10*y^4-2*x^10*y^3+2*x^9*y^4+2*x^10*y^2-x^9*y^3-x^10*y-x^9*y^2-x^8*y^ 3+x^7*y^3-x^6*y^4+x^8*y-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y^3-x^4+x^3-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3607566758 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 113" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 16 8 15 9 18 5 + 3 x y - 10 x y - 2 x y + x y - 2 x y + 5 x y 17 6 16 7 15 8 14 9 18 4 17 5 16 6 - 2 x y - 4 x y + 8 x y + 2 x y - x y + 3 x y + 6 x y 15 7 14 8 17 4 16 5 15 6 13 8 16 4 - 13 x y - 4 x y - x y - 4 x y + 11 x y - 3 x y + x y 15 5 14 6 13 7 15 4 14 5 13 6 12 7 - 5 x y + 4 x y + 9 x y + x y - 3 x y - 9 x y + 3 x y 14 4 13 5 12 6 11 7 14 3 12 5 11 6 + 3 x y + 3 x y - 4 x y - x y - 3 x y - 3 x y + 2 x y 14 2 12 4 10 6 9 7 12 3 11 4 10 5 + x y + 6 x y - x y - 2 x y - 2 x y - 3 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 11 2 10 3 - 2 x y + 3 x y - x y + 8 x y + 3 x y - x y - 4 x y 9 4 8 5 10 2 9 3 8 4 7 5 10 6 5 - 5 x y - 7 x y + 4 x y + x y + 4 x y - 2 x y - x y + x y 8 2 7 3 6 4 5 5 8 7 2 6 3 5 4 - x y + 4 x y - x y + 2 x y + x y - 2 x y + 5 x y - x y 6 2 4 4 5 2 4 3 5 4 2 3 3 - 4 x y + 5 x y + 2 x y - 4 x y - 2 x y - 2 x y - 4 x y 4 3 2 4 2 2 3 2 / 15 6 + 2 x y + 2 x y - x - 2 x y + x + x y - x y + x - 1) / (x y / 15 5 14 6 15 4 14 5 15 3 13 5 12 6 - 3 x y - x y + 3 x y + 2 x y - x y + x y - x y 14 3 13 4 12 5 11 6 14 2 13 3 12 4 - 2 x y - 3 x y + 2 x y + x y + x y + 3 x y - 2 x y 11 5 13 2 12 3 11 4 10 5 12 2 11 3 - x y - x y + 2 x y - x y - x y - x y + x y 10 4 10 3 9 4 10 2 9 3 10 9 2 8 3 + 2 x y - 2 x y + 2 x y + 2 x y - x y - x y - x y - x y 7 3 6 4 8 7 2 6 3 6 2 5 3 4 3 2 + x y - x y + x y - x y - x y + x y + x y - x + x - x y + x y + x - 1) and in Maple notation (x^18*y^9-5*x^18*y^8-x^17*y^9+10*x^18*y^7+3*x^17*y^8-10*x^18*y^6-2*x^17*y^7+x^ 16*y^8-2*x^15*y^9+5*x^18*y^5-2*x^17*y^6-4*x^16*y^7+8*x^15*y^8+2*x^14*y^9-x^18*y ^4+3*x^17*y^5+6*x^16*y^6-13*x^15*y^7-4*x^14*y^8-x^17*y^4-4*x^16*y^5+11*x^15*y^6 -3*x^13*y^8+x^16*y^4-5*x^15*y^5+4*x^14*y^6+9*x^13*y^7+x^15*y^4-3*x^14*y^5-9*x^ 13*y^6+3*x^12*y^7+3*x^14*y^4+3*x^13*y^5-4*x^12*y^6-x^11*y^7-3*x^14*y^3-3*x^12*y ^5+2*x^11*y^6+x^14*y^2+6*x^12*y^4-x^10*y^6-2*x^9*y^7-2*x^12*y^3-3*x^11*y^4+3*x^ 10*y^5-2*x^9*y^6+3*x^11*y^3-x^10*y^4+8*x^9*y^5+3*x^8*y^6-x^11*y^2-4*x^10*y^3-5* x^9*y^4-7*x^8*y^5+4*x^10*y^2+x^9*y^3+4*x^8*y^4-2*x^7*y^5-x^10*y+x^6*y^5-x^8*y^2 +4*x^7*y^3-x^6*y^4+2*x^5*y^5+x^8*y-2*x^7*y^2+5*x^6*y^3-x^5*y^4-4*x^6*y^2+5*x^4* y^4+2*x^5*y^2-4*x^4*y^3-2*x^5*y-2*x^4*y^2-4*x^3*y^3+2*x^4*y+2*x^3*y^2-x^4-2*x^2 *y^2+x^3+x^2*y-x*y+x-1)/(x^15*y^6-3*x^15*y^5-x^14*y^6+3*x^15*y^4+2*x^14*y^5-x^ 15*y^3+x^13*y^5-x^12*y^6-2*x^14*y^3-3*x^13*y^4+2*x^12*y^5+x^11*y^6+x^14*y^2+3*x ^13*y^3-2*x^12*y^4-x^11*y^5-x^13*y^2+2*x^12*y^3-x^11*y^4-x^10*y^5-x^12*y^2+x^11 *y^3+2*x^10*y^4-2*x^10*y^3+2*x^9*y^4+2*x^10*y^2-x^9*y^3-x^10*y-x^9*y^2-x^8*y^3+ x^7*y^3-x^6*y^4+x^8*y-x^7*y^2-x^6*y^3+x^6*y^2+x^5*y^3-x^4+x^3-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3601792852 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 114" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 16 8 15 9 18 5 + 3 x y - 10 x y - 2 x y + x y - 2 x y + 5 x y 17 6 16 7 15 8 14 9 18 4 17 5 16 6 - 2 x y - 4 x y + 8 x y + 2 x y - x y + 3 x y + 6 x y 15 7 14 8 17 4 16 5 15 6 13 8 16 4 - 13 x y - 4 x y - x y - 4 x y + 11 x y - 3 x y + x y 15 5 14 6 13 7 15 4 14 5 13 6 12 7 - 5 x y + 4 x y + 9 x y + x y - 3 x y - 9 x y + 3 x y 14 4 13 5 12 6 11 7 14 3 12 5 11 6 + 3 x y + 3 x y - 4 x y - x y - 3 x y - 3 x y + 2 x y 14 2 12 4 10 6 9 7 12 3 11 4 10 5 + x y + 6 x y - x y - 2 x y - 2 x y - 3 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 11 2 10 3 - 2 x y + 3 x y - x y + 8 x y + 3 x y - x y - 4 x y 9 4 8 5 10 2 9 3 8 4 7 5 10 6 5 - 5 x y - 7 x y + 4 x y + x y + 4 x y - 2 x y - x y + x y 8 2 7 3 6 4 5 5 8 7 2 6 3 5 4 - x y + 4 x y - x y + 2 x y + x y - 2 x y + 5 x y - x y 6 2 4 4 5 2 4 3 5 4 2 3 3 - 4 x y + 5 x y + 2 x y - 4 x y - 2 x y - 2 x y - 4 x y 4 3 2 4 2 2 3 2 / 14 6 + 2 x y + 2 x y - x - 2 x y + x + x y - x y + x - 1) / ((x y / 14 5 14 4 13 5 14 3 13 4 11 6 13 3 - 3 x y + 3 x y - x y - x y + 3 x y - x y - 3 x y 11 5 13 2 11 4 10 5 11 3 10 4 11 2 + 2 x y + x y - 2 x y + x y + 2 x y - 3 x y - x y 10 3 9 4 10 2 9 3 8 4 9 2 7 4 9 7 3 + 3 x y - x y - x y + x y + x y + x y + x y - x y - x y 6 4 8 6 2 5 3 3 + x y - x y - x y - x y - x - x y + 1) (x - 1)) and in Maple notation (x^18*y^9-5*x^18*y^8-x^17*y^9+10*x^18*y^7+3*x^17*y^8-10*x^18*y^6-2*x^17*y^7+x^ 16*y^8-2*x^15*y^9+5*x^18*y^5-2*x^17*y^6-4*x^16*y^7+8*x^15*y^8+2*x^14*y^9-x^18*y ^4+3*x^17*y^5+6*x^16*y^6-13*x^15*y^7-4*x^14*y^8-x^17*y^4-4*x^16*y^5+11*x^15*y^6 -3*x^13*y^8+x^16*y^4-5*x^15*y^5+4*x^14*y^6+9*x^13*y^7+x^15*y^4-3*x^14*y^5-9*x^ 13*y^6+3*x^12*y^7+3*x^14*y^4+3*x^13*y^5-4*x^12*y^6-x^11*y^7-3*x^14*y^3-3*x^12*y ^5+2*x^11*y^6+x^14*y^2+6*x^12*y^4-x^10*y^6-2*x^9*y^7-2*x^12*y^3-3*x^11*y^4+3*x^ 10*y^5-2*x^9*y^6+3*x^11*y^3-x^10*y^4+8*x^9*y^5+3*x^8*y^6-x^11*y^2-4*x^10*y^3-5* x^9*y^4-7*x^8*y^5+4*x^10*y^2+x^9*y^3+4*x^8*y^4-2*x^7*y^5-x^10*y+x^6*y^5-x^8*y^2 +4*x^7*y^3-x^6*y^4+2*x^5*y^5+x^8*y-2*x^7*y^2+5*x^6*y^3-x^5*y^4-4*x^6*y^2+5*x^4* y^4+2*x^5*y^2-4*x^4*y^3-2*x^5*y-2*x^4*y^2-4*x^3*y^3+2*x^4*y+2*x^3*y^2-x^4-2*x^2 *y^2+x^3+x^2*y-x*y+x-1)/(x^14*y^6-3*x^14*y^5+3*x^14*y^4-x^13*y^5-x^14*y^3+3*x^ 13*y^4-x^11*y^6-3*x^13*y^3+2*x^11*y^5+x^13*y^2-2*x^11*y^4+x^10*y^5+2*x^11*y^3-3 *x^10*y^4-x^11*y^2+3*x^10*y^3-x^9*y^4-x^10*y^2+x^9*y^3+x^8*y^4+x^9*y^2+x^7*y^4- x^9*y-x^7*y^3+x^6*y^4-x^8*y-x^6*y^2-x^5*y^3-x^3-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3601792852 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 115" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 9 15 8 15 7 14 8 ) | ) C(m, n) x y | = (x y - 5 x y + 10 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 6 14 7 13 8 12 9 15 5 14 6 13 7 - 10 x y + 4 x y - x y - x y + 5 x y - 6 x y + 5 x y 12 8 15 4 14 5 13 6 12 7 11 8 14 4 + 2 x y - x y + 4 x y - 10 x y + x y - x y - x y 13 5 12 6 11 7 13 4 12 5 11 6 + 10 x y - 5 x y + 6 x y - 5 x y + 4 x y - 10 x y 10 7 13 3 12 4 11 5 10 6 9 7 11 4 + x y + x y - x y + 3 x y - 3 x y + x y + 5 x y 10 5 9 6 8 7 11 3 10 4 8 6 11 2 + 5 x y - x y + x y - 2 x y - 5 x y + 2 x y - 2 x y 10 3 9 4 8 5 11 8 4 7 5 9 2 8 3 + 2 x y - x y - 6 x y + x y + x y - x y + 2 x y + 4 x y 7 4 6 5 9 8 2 7 3 5 5 7 2 6 3 + x y - 4 x y - x y - 2 x y - x y + 3 x y + 2 x y + 2 x y 5 4 7 6 2 5 3 4 4 4 3 6 4 2 - 2 x y - x y + 2 x y - 2 x y + 3 x y - 2 x y - x - x y 3 3 5 3 2 3 2 2 2 / - 4 x y + x + 2 x y + x y - 2 x y + x y - x y + x - 1) / ( / 11 6 11 5 11 4 10 5 11 3 10 4 11 2 x y - 3 x y + 2 x y - x y + 2 x y + 3 x y - 3 x y 10 3 8 5 11 10 2 9 3 8 4 9 2 8 3 - 3 x y - x y + x y + x y - x y + x y + 2 x y + x y 7 4 9 8 2 7 3 7 6 2 6 6 4 2 + x y - x y - x y - 2 x y + x y + 3 x y - 2 x y - x - x y 5 3 2 + x + x y - x y + x y + x - 1) and in Maple notation (x^15*y^9-5*x^15*y^8+10*x^15*y^7-x^14*y^8-10*x^15*y^6+4*x^14*y^7-x^13*y^8-x^12* y^9+5*x^15*y^5-6*x^14*y^6+5*x^13*y^7+2*x^12*y^8-x^15*y^4+4*x^14*y^5-10*x^13*y^6 +x^12*y^7-x^11*y^8-x^14*y^4+10*x^13*y^5-5*x^12*y^6+6*x^11*y^7-5*x^13*y^4+4*x^12 *y^5-10*x^11*y^6+x^10*y^7+x^13*y^3-x^12*y^4+3*x^11*y^5-3*x^10*y^6+x^9*y^7+5*x^ 11*y^4+5*x^10*y^5-x^9*y^6+x^8*y^7-2*x^11*y^3-5*x^10*y^4+2*x^8*y^6-2*x^11*y^2+2* x^10*y^3-x^9*y^4-6*x^8*y^5+x^11*y+x^8*y^4-x^7*y^5+2*x^9*y^2+4*x^8*y^3+x^7*y^4-4 *x^6*y^5-x^9*y-2*x^8*y^2-x^7*y^3+3*x^5*y^5+2*x^7*y^2+2*x^6*y^3-2*x^5*y^4-x^7*y+ 2*x^6*y^2-2*x^5*y^3+3*x^4*y^4-2*x^4*y^3-x^6-x^4*y^2-4*x^3*y^3+x^5+2*x^3*y^2+x^3 *y-2*x^2*y^2+x^2*y-x*y+x-1)/(x^11*y^6-3*x^11*y^5+2*x^11*y^4-x^10*y^5+2*x^11*y^3 +3*x^10*y^4-3*x^11*y^2-3*x^10*y^3-x^8*y^5+x^11*y+x^10*y^2-x^9*y^3+x^8*y^4+2*x^9 *y^2+x^8*y^3+x^7*y^4-x^9*y-x^8*y^2-2*x^7*y^3+x^7*y+3*x^6*y^2-2*x^6*y-x^6-x^4*y^ 2+x^5+x^3*y-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 11 10 9 6 5 4 3 2 2 (a - 4 a + 3 a - 7 a + 3 a + 6 a - 9 a + 2 a + 1) ------------------------------------------------------------- 4 3 2 6 3 2 (5 a - 4 a + 3 a - 2 a + 2) (a + a + a + 1) 5 4 3 2 where a is the root of the polynomial, x - x + x - x + 2 x - 1, and in decimals this is, 0.3566159308 BTW the ratio for words with, 500, letters is, 0.3592420238 ------------------------------------------------ "Theorem Number 116" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 8 9 7 9 6 8 7 ) | ) C(m, n) x y | = (2 x y - 4 x y - 2 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 5 8 6 9 4 8 5 7 6 9 3 8 4 + 8 x y - 4 x y - 2 x y - 2 x y + 2 x y - 4 x y + 8 x y 7 5 6 6 9 2 8 3 7 4 6 5 8 2 - 6 x y - 2 x y + 2 x y - 2 x y + x y + 6 x y - 4 x y 7 3 6 4 5 5 8 7 2 6 3 5 4 + 8 x y - 5 x y - 4 x y + 2 x y - 4 x y - 4 x y + 3 x y 7 6 2 5 3 4 4 7 6 5 2 4 3 - 2 x y + 8 x y + 4 x y - 5 x y + x - 2 x y - 4 x y + 6 x y 6 5 3 3 5 4 3 2 4 3 2 2 - x + 2 x y - 8 x y - x - 2 x y + 6 x y + x + 2 x y - 4 x y 3 2 2 / 7 4 6 4 7 2 6 2 - x + 2 x y + x - 2 x y + x - 1) / (x y - x y - 2 x y + 2 x y / 7 6 4 2 5 3 2 4 3 2 + x - x - 2 x y - x + 2 x y + x - x + x + x - 1) and in Maple notation (2*x^9*y^8-4*x^9*y^7-2*x^9*y^6+2*x^8*y^7+8*x^9*y^5-4*x^8*y^6-2*x^9*y^4-2*x^8*y^ 5+2*x^7*y^6-4*x^9*y^3+8*x^8*y^4-6*x^7*y^5-2*x^6*y^6+2*x^9*y^2-2*x^8*y^3+x^7*y^4 +6*x^6*y^5-4*x^8*y^2+8*x^7*y^3-5*x^6*y^4-4*x^5*y^5+2*x^8*y-4*x^7*y^2-4*x^6*y^3+ 3*x^5*y^4-2*x^7*y+8*x^6*y^2+4*x^5*y^3-5*x^4*y^4+x^7-2*x^6*y-4*x^5*y^2+6*x^4*y^3 -x^6+2*x^5*y-8*x^3*y^3-x^5-2*x^4*y+6*x^3*y^2+x^4+2*x^3*y-4*x^2*y^2-x^3+2*x^2*y+ x^2-2*x*y+x-1)/(x^7*y^4-x^6*y^4-2*x^7*y^2+2*x^6*y^2+x^7-x^6-2*x^4*y^2-x^5+2*x^3 *y^2+x^4-x^3+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3451118637 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 117" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 5 6 6 7 4 6 5 ) | ) C(m, n) x y | = (2 x y + x y - 4 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 3 6 4 5 5 6 3 6 2 5 3 4 4 + 2 x y + x y + 2 x y + 6 x y - 4 x y - 8 x y + 3 x y 5 2 4 3 5 4 2 3 3 4 3 2 4 + 8 x y - 8 x y - 2 x y + x y + 8 x y + 4 x y - 9 x y - x 2 2 3 2 / + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ((x - 1) / 3 2 3 2 (x y - x + x + x - 1)) and in Maple notation (2*x^7*y^5+x^6*y^6-4*x^7*y^4-4*x^6*y^5+2*x^7*y^3+x^6*y^4+2*x^5*y^5+6*x^6*y^3-4* x^6*y^2-8*x^5*y^3+3*x^4*y^4+8*x^5*y^2-8*x^4*y^3-2*x^5*y+x^4*y^2+8*x^3*y^3+4*x^4 *y-9*x^3*y^2-x^4+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x-1)/(x^3*y^2-x^3+x^2+x-\ 1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3448417595 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 118" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 6 7 5 7 4 6 5 ) | ) C(m, n) x y | = (2 x y - 4 x y + 2 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 4 6 3 5 4 5 3 4 4 5 2 4 2 - 4 x y + 2 x y - 4 x y + 8 x y - 3 x y - 4 x y + 4 x y 3 3 4 3 2 3 2 2 3 2 2 - 8 x y - 2 x y + 5 x y + 2 x y - 4 x y - x + 2 x y + x - 2 x y / 3 2 3 2 + x - 1) / (x y - x + x + x - 1) / and in Maple notation (2*x^7*y^6-4*x^7*y^5+2*x^7*y^4+2*x^6*y^5-4*x^6*y^4+2*x^6*y^3-4*x^5*y^4+8*x^5*y^ 3-3*x^4*y^4-4*x^5*y^2+4*x^4*y^2-8*x^3*y^3-2*x^4*y+5*x^3*y^2+2*x^3*y-4*x^2*y^2-x ^3+2*x^2*y+x^2-2*x*y+x-1)/(x^3*y^2-x^3+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 2 (a - a - 8 a - 5 a + 3 a + 2 a + 1) ------------------------------------------ 5 4 3 2 2 a + 5 a + 4 a + 3 a + 3 a + 1 2 where a is the root of the polynomial, x + x - 1, and in decimals this is, 0.3416407896 BTW the ratio for words with, 500, letters is, 0.3446079199 ------------------------------------------------ "Theorem Number 119" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 1], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 12 20 10 19 11 ) | ) C(m, n) x y | = - (2 x y + x y - 8 x y / | / | ----- | ----- | m = 0 \ n = 0 / 20 9 19 10 20 8 19 9 18 10 16 12 - 4 x y + 10 x y + 4 x y - 2 x y - 4 x y - x y 20 7 19 8 18 9 17 10 16 11 20 6 + 4 x y - 4 x y + 14 x y - 4 x y + 4 x y - 10 x y 19 7 18 8 17 9 16 10 15 11 20 5 + 6 x y - 14 x y + 16 x y - 5 x y - 2 x y + 4 x y 19 6 17 8 16 9 15 10 20 4 19 5 - 12 x y - 20 x y - 2 x y + 6 x y + 4 x y + 10 x y 18 6 17 7 16 8 15 9 14 10 20 3 + x y + 4 x y + 16 x y - 8 x y + 2 x y - 4 x y 19 4 18 5 17 6 16 7 15 8 14 9 + 2 x y + 10 x y + 4 x y - 26 x y + 12 x y - 8 x y 20 2 19 3 18 4 17 5 16 6 15 7 + x y - 6 x y - 8 x y + 8 x y + 18 x y - 18 x y 14 8 13 9 19 2 17 4 16 5 15 6 + 14 x y + 4 x y + 2 x y - 12 x y + 2 x y + 16 x y 14 7 13 8 18 2 17 3 16 4 15 5 - 8 x y - 10 x y + x y + 4 x y - 11 x y - 10 x y 14 6 13 7 12 8 11 9 16 3 15 4 - 8 x y + 12 x y + 2 x y + 2 x y + 6 x y + 6 x y 14 5 13 6 12 7 16 2 15 3 14 4 + 8 x y - 12 x y + 6 x y - x y - 2 x y + 6 x y 13 5 12 6 11 7 10 8 14 3 13 4 + 4 x y - 22 x y - 6 x y + 2 x y - 8 x y + 6 x y 12 5 10 7 14 2 13 3 12 4 11 5 + 10 x y - 6 x y + 2 x y - 4 x y + 9 x y - 6 x y 10 6 9 7 12 3 11 4 10 5 9 6 - 4 x y - 4 x y - 2 x y + 22 x y + 8 x y - 2 x y 12 2 11 3 10 4 9 5 8 6 11 2 - 3 x y - 8 x y + 13 x y + 12 x y - 4 x y - 6 x y 10 3 8 5 11 10 2 9 3 8 4 7 5 - 16 x y + 10 x y + 2 x y + 3 x y - 8 x y - 2 x y + 4 x y 6 6 9 2 8 3 7 4 8 2 7 3 6 4 + 3 x y + 2 x y - 8 x y - 4 x y + 4 x y - 2 x y - x y 7 2 6 3 5 4 6 2 5 3 4 4 5 2 + 2 x y - 4 x y - 4 x y + 4 x y - 2 x y - 7 x y + 8 x y 4 3 6 5 4 2 3 3 4 2 2 / + 4 x y - x - 2 x y + 4 x y + 2 x y - 2 x y + x y + 1) / ( / 20 10 20 9 20 8 20 7 19 8 20 6 19 7 x y - 4 x y + 4 x y + 4 x y - 2 x y - 10 x y + 8 x y 18 8 20 5 19 6 18 7 16 9 20 4 - 2 x y + 4 x y - 10 x y + 8 x y - 2 x y + 4 x y 18 6 16 8 20 3 19 4 18 5 17 6 - 9 x y + 6 x y - 4 x y + 10 x y - 4 x y + 2 x y 16 7 20 2 19 3 18 4 17 5 16 6 - 4 x y + x y - 8 x y + 16 x y - 8 x y - 3 x y 15 7 19 2 18 3 17 4 16 5 15 6 + 2 x y + 2 x y - 12 x y + 12 x y + 2 x y - 6 x y 14 7 18 2 17 3 16 4 15 5 14 6 + 4 x y + 3 x y - 8 x y + 4 x y + 6 x y - 11 x y 12 8 17 2 16 3 15 4 14 5 13 6 12 7 + x y + 2 x y - 4 x y - 2 x y + 8 x y + 2 x y - 2 x y 16 2 14 4 13 5 12 6 14 3 13 4 12 5 + x y + 2 x y - 4 x y + 2 x y - 4 x y + 2 x y - 4 x y 14 2 13 3 12 4 10 6 13 2 12 3 11 4 + x y - 2 x y + 4 x y - 2 x y + 4 x y - 2 x y - 4 x y 10 5 13 12 2 11 3 10 4 12 10 3 + 4 x y - 2 x y + 3 x y + 6 x y - 5 x y - 2 x y + 6 x y 9 4 11 10 2 9 3 8 4 9 2 8 2 - 2 x y - 2 x y - 3 x y + 4 x y + x y - 2 x y - x y 7 3 7 2 6 3 6 2 6 5 2 6 5 + 2 x y - 2 x y + 2 x y - 4 x y + 2 x y - 4 x y + x + 2 x y 4 2 2 + 2 x y - x y + 2 x y - 1) and in Maple notation -(2*x^19*y^12+x^20*y^10-8*x^19*y^11-4*x^20*y^9+10*x^19*y^10+4*x^20*y^8-2*x^19*y ^9-4*x^18*y^10-x^16*y^12+4*x^20*y^7-4*x^19*y^8+14*x^18*y^9-4*x^17*y^10+4*x^16*y ^11-10*x^20*y^6+6*x^19*y^7-14*x^18*y^8+16*x^17*y^9-5*x^16*y^10-2*x^15*y^11+4*x^ 20*y^5-12*x^19*y^6-20*x^17*y^8-2*x^16*y^9+6*x^15*y^10+4*x^20*y^4+10*x^19*y^5+x^ 18*y^6+4*x^17*y^7+16*x^16*y^8-8*x^15*y^9+2*x^14*y^10-4*x^20*y^3+2*x^19*y^4+10*x ^18*y^5+4*x^17*y^6-26*x^16*y^7+12*x^15*y^8-8*x^14*y^9+x^20*y^2-6*x^19*y^3-8*x^ 18*y^4+8*x^17*y^5+18*x^16*y^6-18*x^15*y^7+14*x^14*y^8+4*x^13*y^9+2*x^19*y^2-12* x^17*y^4+2*x^16*y^5+16*x^15*y^6-8*x^14*y^7-10*x^13*y^8+x^18*y^2+4*x^17*y^3-11*x ^16*y^4-10*x^15*y^5-8*x^14*y^6+12*x^13*y^7+2*x^12*y^8+2*x^11*y^9+6*x^16*y^3+6*x ^15*y^4+8*x^14*y^5-12*x^13*y^6+6*x^12*y^7-x^16*y^2-2*x^15*y^3+6*x^14*y^4+4*x^13 *y^5-22*x^12*y^6-6*x^11*y^7+2*x^10*y^8-8*x^14*y^3+6*x^13*y^4+10*x^12*y^5-6*x^10 *y^7+2*x^14*y^2-4*x^13*y^3+9*x^12*y^4-6*x^11*y^5-4*x^10*y^6-4*x^9*y^7-2*x^12*y^ 3+22*x^11*y^4+8*x^10*y^5-2*x^9*y^6-3*x^12*y^2-8*x^11*y^3+13*x^10*y^4+12*x^9*y^5 -4*x^8*y^6-6*x^11*y^2-16*x^10*y^3+10*x^8*y^5+2*x^11*y+3*x^10*y^2-8*x^9*y^3-2*x^ 8*y^4+4*x^7*y^5+3*x^6*y^6+2*x^9*y^2-8*x^8*y^3-4*x^7*y^4+4*x^8*y^2-2*x^7*y^3-x^6 *y^4+2*x^7*y^2-4*x^6*y^3-4*x^5*y^4+4*x^6*y^2-2*x^5*y^3-7*x^4*y^4+8*x^5*y^2+4*x^ 4*y^3-x^6-2*x^5*y+4*x^4*y^2+2*x^3*y^3-2*x^4*y+x^2*y^2+1)/(x^20*y^10-4*x^20*y^9+ 4*x^20*y^8+4*x^20*y^7-2*x^19*y^8-10*x^20*y^6+8*x^19*y^7-2*x^18*y^8+4*x^20*y^5-\ 10*x^19*y^6+8*x^18*y^7-2*x^16*y^9+4*x^20*y^4-9*x^18*y^6+6*x^16*y^8-4*x^20*y^3+ 10*x^19*y^4-4*x^18*y^5+2*x^17*y^6-4*x^16*y^7+x^20*y^2-8*x^19*y^3+16*x^18*y^4-8* x^17*y^5-3*x^16*y^6+2*x^15*y^7+2*x^19*y^2-12*x^18*y^3+12*x^17*y^4+2*x^16*y^5-6* x^15*y^6+4*x^14*y^7+3*x^18*y^2-8*x^17*y^3+4*x^16*y^4+6*x^15*y^5-11*x^14*y^6+x^ 12*y^8+2*x^17*y^2-4*x^16*y^3-2*x^15*y^4+8*x^14*y^5+2*x^13*y^6-2*x^12*y^7+x^16*y ^2+2*x^14*y^4-4*x^13*y^5+2*x^12*y^6-4*x^14*y^3+2*x^13*y^4-4*x^12*y^5+x^14*y^2-2 *x^13*y^3+4*x^12*y^4-2*x^10*y^6+4*x^13*y^2-2*x^12*y^3-4*x^11*y^4+4*x^10*y^5-2*x ^13*y+3*x^12*y^2+6*x^11*y^3-5*x^10*y^4-2*x^12*y+6*x^10*y^3-2*x^9*y^4-2*x^11*y-3 *x^10*y^2+4*x^9*y^3+x^8*y^4-2*x^9*y^2-x^8*y^2+2*x^7*y^3-2*x^7*y^2+2*x^6*y^3-4*x ^6*y^2+2*x^6*y-4*x^5*y^2+x^6+2*x^5*y+2*x^4*y-x^2*y^2+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 10 bors of a random word of length n tends to n times, (2 a - 6 a + 10 a 9 8 7 6 5 4 3 2 / - 8 a - 2 a + 12 a - 16 a + 8 a + 3 a - 8 a + 2 a + 1) / ( / 5 4 3 6 4 3 2 (3 a - 5 a + 4 a - a + 1) (a - a + 2 a + a + 1)) 6 5 4 2 where a is the root of the polynomial, x - 2 x + 2 x - x + 2 x - 1, and in decimals this is, 0.3417070850 BTW the ratio for words with, 500, letters is, 0.3439863613 ------------------------------------------------ "Theorem Number 120" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 10 13 9 13 8 12 9 ) | ) C(m, n) x y | = (x y - 2 x y - x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 13 6 12 7 11 8 10 9 13 5 + 5 x y - 5 x y - 4 x y + x y + 2 x y - x y + x y 12 6 11 7 12 5 11 6 10 7 9 8 + 8 x y - 7 x y - 9 x y + 9 x y + 5 x y - 2 x y 12 4 11 5 10 6 9 7 11 4 10 5 9 6 + 3 x y - 2 x y - 7 x y + x y - 5 x y + 6 x y - x y 8 7 11 3 10 4 9 5 8 6 7 7 10 3 - 2 x y + 3 x y - 6 x y + 2 x y + 2 x y + x y + 2 x y 9 4 8 5 7 6 10 2 9 3 8 4 6 6 9 2 + x y - 5 x y - 2 x y + x y - 2 x y + 8 x y + 2 x y + x y 8 3 7 4 6 5 7 3 6 4 5 5 8 7 2 - 2 x y + 2 x y + 4 x y + 8 x y + x y + 2 x y - x y - 7 x y 6 3 7 5 3 4 4 6 5 2 4 3 4 2 - 5 x y - x y - x y + 2 x y - x y - 2 x y - 2 x y - 2 x y 3 3 3 2 4 3 2 2 / 7 4 7 3 - 4 x y + x y + x + x y - 2 x y - x y - 1) / (x y - x y / 6 3 6 2 5 2 4 3 4 2 4 3 2 4 3 - x y + x y - x y - x y - x y + x y + x y + x + x y + x y - 1) and in Maple notation (x^13*y^10-2*x^13*y^9-x^13*y^8+2*x^12*y^9+5*x^13*y^7-5*x^12*y^8-4*x^13*y^6+x^12 *y^7+2*x^11*y^8-x^10*y^9+x^13*y^5+8*x^12*y^6-7*x^11*y^7-9*x^12*y^5+9*x^11*y^6+5 *x^10*y^7-2*x^9*y^8+3*x^12*y^4-2*x^11*y^5-7*x^10*y^6+x^9*y^7-5*x^11*y^4+6*x^10* y^5-x^9*y^6-2*x^8*y^7+3*x^11*y^3-6*x^10*y^4+2*x^9*y^5+2*x^8*y^6+x^7*y^7+2*x^10* y^3+x^9*y^4-5*x^8*y^5-2*x^7*y^6+x^10*y^2-2*x^9*y^3+8*x^8*y^4+2*x^6*y^6+x^9*y^2-\ 2*x^8*y^3+2*x^7*y^4+4*x^6*y^5+8*x^7*y^3+x^6*y^4+2*x^5*y^5-x^8*y-7*x^7*y^2-5*x^6 *y^3-x^7*y-x^5*y^3+2*x^4*y^4-x^6*y-2*x^5*y^2-2*x^4*y^3-2*x^4*y^2-4*x^3*y^3+x^3* y^2+x^4+x^3*y-2*x^2*y^2-x*y-1)/(x^7*y^4-x^7*y^3-x^6*y^3+x^6*y^2-x^5*y^2-x^4*y^3 -x^4*y^2+x^4*y+x^3*y^2+x^4+x^3*y+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3438313502 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 121" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 6 10 8 11 6 ) | ) C(m, n) x y | = (x y - 2 x y + x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 12 4 11 5 9 7 12 3 11 4 - 2 x y - x y + 2 x y - 2 x y + 3 x y - x y + x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 + 3 x y - 2 x y - 2 x y - 3 x y - 2 x y + 5 x y + x y 9 4 7 6 8 4 7 5 6 6 9 2 8 3 + 3 x y - 4 x y - 7 x y + 7 x y + x y - 2 x y + 3 x y 7 4 6 5 9 8 2 7 3 6 4 5 5 8 + 3 x y - 9 x y + x y + 2 x y - 10 x y + 7 x y + 3 x y - x y 7 2 6 3 5 4 7 6 2 5 3 4 4 + 5 x y + 10 x y - 10 x y - x y - 10 x y + 3 x y + 5 x y 6 5 2 4 3 5 4 2 3 3 4 + x y + 5 x y - 14 x y - 2 x y + 5 x y + 8 x y + 4 x y 3 2 4 2 2 3 2 / - 9 x y - x + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ( / 4 2 3 2 4 3 x y - x y - x + 2 x - 2 x + 1) and in Maple notation (x^12*y^7-2*x^12*y^6+x^10*y^8+x^11*y^6-2*x^10*y^7-x^9*y^8+2*x^12*y^4-2*x^11*y^5 +3*x^9*y^7-x^12*y^3+x^11*y^4+3*x^10*y^5-2*x^9*y^6-2*x^8*y^7-3*x^10*y^4-2*x^9*y^ 5+5*x^8*y^6+x^10*y^3+3*x^9*y^4-4*x^7*y^6-7*x^8*y^4+7*x^7*y^5+x^6*y^6-2*x^9*y^2+ 3*x^8*y^3+3*x^7*y^4-9*x^6*y^5+x^9*y+2*x^8*y^2-10*x^7*y^3+7*x^6*y^4+3*x^5*y^5-x^ 8*y+5*x^7*y^2+10*x^6*y^3-10*x^5*y^4-x^7*y-10*x^6*y^2+3*x^5*y^3+5*x^4*y^4+x^6*y+ 5*x^5*y^2-14*x^4*y^3-2*x^5*y+5*x^4*y^2+8*x^3*y^3+4*x^4*y-9*x^3*y^2-x^4+4*x^2*y^ 2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^4*y^2-x^3*y^2-x^4+2*x^3-2*x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3438030252 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 122" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 6 10 8 11 6 ) | ) C(m, n) x y | = (x y - 2 x y + x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 12 4 11 5 9 7 12 3 11 4 - 2 x y - x y + 2 x y - 2 x y + 3 x y - x y + x y 10 5 9 6 8 7 10 4 9 5 8 6 10 3 + 3 x y - 2 x y - 2 x y - 3 x y - 2 x y + 5 x y + x y 9 4 7 6 8 4 7 5 6 6 9 2 8 3 + 3 x y - 4 x y - 7 x y + 7 x y + x y - 2 x y + 3 x y 7 4 6 5 9 8 2 7 3 6 4 5 5 8 + 3 x y - 9 x y + x y + 2 x y - 10 x y + 7 x y + 3 x y - x y 7 2 6 3 5 4 7 6 2 5 3 4 4 + 5 x y + 10 x y - 10 x y - x y - 10 x y + 3 x y + 5 x y 6 5 2 4 3 5 4 2 3 3 4 + x y + 5 x y - 14 x y - 2 x y + 5 x y + 8 x y + 4 x y 3 2 4 2 2 3 2 / - 9 x y - x + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ( / 3 2 3 2 (x y - x + x + x - 1) (x - 1)) and in Maple notation (x^12*y^7-2*x^12*y^6+x^10*y^8+x^11*y^6-2*x^10*y^7-x^9*y^8+2*x^12*y^4-2*x^11*y^5 +3*x^9*y^7-x^12*y^3+x^11*y^4+3*x^10*y^5-2*x^9*y^6-2*x^8*y^7-3*x^10*y^4-2*x^9*y^ 5+5*x^8*y^6+x^10*y^3+3*x^9*y^4-4*x^7*y^6-7*x^8*y^4+7*x^7*y^5+x^6*y^6-2*x^9*y^2+ 3*x^8*y^3+3*x^7*y^4-9*x^6*y^5+x^9*y+2*x^8*y^2-10*x^7*y^3+7*x^6*y^4+3*x^5*y^5-x^ 8*y+5*x^7*y^2+10*x^6*y^3-10*x^5*y^4-x^7*y-10*x^6*y^2+3*x^5*y^3+5*x^4*y^4+x^6*y+ 5*x^5*y^2-14*x^4*y^3-2*x^5*y+5*x^4*y^2+8*x^3*y^3+4*x^4*y-9*x^3*y^2-x^4+4*x^2*y^ 2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^3*y^2-x^3+x^2+x-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3438030252 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 123" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 21 11 21 10 20 11 ) | ) C(m, n) x y | = (3 x y - 18 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 21 9 20 10 21 8 20 9 19 10 21 7 + 45 x y + 17 x y - 60 x y - 40 x y - 8 x y + 45 x y 20 8 19 9 18 10 21 6 20 7 19 8 + 50 x y + 42 x y + 5 x y - 18 x y - 35 x y - 90 x y 18 9 17 10 21 5 20 6 19 7 18 8 - 26 x y + x y + 3 x y + 13 x y + 100 x y + 57 x y 17 9 16 10 20 5 19 6 18 7 17 8 - 2 x y - 4 x y - 2 x y - 60 x y - 69 x y - 5 x y 16 9 19 5 18 6 17 7 16 8 15 9 + 17 x y + 18 x y + 51 x y + 21 x y - 25 x y - 2 x y 14 10 19 4 18 5 17 6 16 7 15 8 + 4 x y - 2 x y - 24 x y - 29 x y + 9 x y + 16 x y 14 9 18 4 17 5 16 6 15 7 14 8 - x y + 7 x y + 20 x y + 15 x y - 36 x y - 26 x y 13 9 18 3 17 4 16 5 15 6 14 7 + 4 x y - x y - 7 x y - 20 x y + 30 x y + 41 x y 13 8 12 9 17 3 16 4 15 5 14 6 - 7 x y - 5 x y + x y + 10 x y - 4 x y - 17 x y 13 7 12 8 11 9 16 3 15 4 14 5 - 12 x y + 6 x y - x y - 2 x y - 6 x y - 5 x y 13 6 12 7 11 8 15 3 14 4 13 5 + 32 x y + 6 x y - 5 x y + 2 x y + 5 x y - 18 x y 12 6 11 7 10 8 14 3 13 4 12 5 11 6 - 10 x y + 14 x y - 2 x y - x y - x y + 3 x y - 4 x y 10 7 9 8 13 3 12 4 11 5 10 6 9 7 - 4 x y + x y + 2 x y + x y - 6 x y + 16 x y - 6 x y 12 3 11 4 10 5 9 6 8 7 12 2 11 3 - 2 x y - 2 x y - 15 x y - 6 x y + 3 x y + x y + 5 x y 10 4 9 5 8 6 11 2 9 4 8 5 7 6 + 5 x y + 20 x y + x y - x y - 12 x y - 6 x y + 8 x y 9 3 8 4 7 5 9 2 8 3 7 4 6 5 + 6 x y + 4 x y + 2 x y - 3 x y - 3 x y - 10 x y + 4 x y 8 2 6 4 5 5 8 7 2 6 3 5 4 7 + 3 x y - 5 x y - x y - x y + x y + 4 x y + 6 x y + x y 6 2 5 3 4 4 5 2 4 3 5 4 2 - 2 x y - 5 x y - 3 x y + 3 x y + 5 x y - 2 x y - 3 x y 3 3 4 3 2 4 3 2 2 3 2 - 8 x y + 2 x y + 6 x y - x - x y - 4 x y + x + 3 x y - 2 x y / 18 8 18 7 17 8 18 6 17 7 + x - 1) / (x y - 5 x y - x y + 10 x y + 5 x y / 18 5 17 6 16 7 18 4 17 5 16 6 - 10 x y - 10 x y - 2 x y + 5 x y + 10 x y + 8 x y 15 7 18 3 17 4 16 5 15 6 17 3 16 4 + x y - x y - 5 x y - 12 x y - 5 x y + x y + 8 x y 15 5 14 6 13 7 16 3 15 4 14 5 13 6 + 9 x y + x y - x y - 2 x y - 7 x y - 3 x y + 3 x y 15 3 14 4 13 5 12 6 11 7 14 3 13 4 + 2 x y + 3 x y - 2 x y - x y + x y - x y - x y 12 5 11 6 13 3 12 4 11 5 10 6 12 3 + 3 x y + x y + x y - 2 x y - 5 x y + x y - x y 11 4 10 5 9 6 12 2 11 3 10 4 11 2 + 2 x y + x y - x y + x y + 2 x y - 5 x y - x y 10 3 9 4 8 5 9 3 9 2 8 3 8 2 7 3 + 3 x y + 3 x y - x y - x y - x y + x y + x y - x y 6 4 8 7 2 6 3 7 6 2 4 3 4 2 4 3 - x y - x y - x y - x y + x y + x y + x y + x y - x - x y 3 2 + x + x y + x - 1) and in Maple notation (3*x^21*y^11-18*x^21*y^10-3*x^20*y^11+45*x^21*y^9+17*x^20*y^10-60*x^21*y^8-40*x ^20*y^9-8*x^19*y^10+45*x^21*y^7+50*x^20*y^8+42*x^19*y^9+5*x^18*y^10-18*x^21*y^6 -35*x^20*y^7-90*x^19*y^8-26*x^18*y^9+x^17*y^10+3*x^21*y^5+13*x^20*y^6+100*x^19* y^7+57*x^18*y^8-2*x^17*y^9-4*x^16*y^10-2*x^20*y^5-60*x^19*y^6-69*x^18*y^7-5*x^ 17*y^8+17*x^16*y^9+18*x^19*y^5+51*x^18*y^6+21*x^17*y^7-25*x^16*y^8-2*x^15*y^9+4 *x^14*y^10-2*x^19*y^4-24*x^18*y^5-29*x^17*y^6+9*x^16*y^7+16*x^15*y^8-x^14*y^9+7 *x^18*y^4+20*x^17*y^5+15*x^16*y^6-36*x^15*y^7-26*x^14*y^8+4*x^13*y^9-x^18*y^3-7 *x^17*y^4-20*x^16*y^5+30*x^15*y^6+41*x^14*y^7-7*x^13*y^8-5*x^12*y^9+x^17*y^3+10 *x^16*y^4-4*x^15*y^5-17*x^14*y^6-12*x^13*y^7+6*x^12*y^8-x^11*y^9-2*x^16*y^3-6*x ^15*y^4-5*x^14*y^5+32*x^13*y^6+6*x^12*y^7-5*x^11*y^8+2*x^15*y^3+5*x^14*y^4-18*x ^13*y^5-10*x^12*y^6+14*x^11*y^7-2*x^10*y^8-x^14*y^3-x^13*y^4+3*x^12*y^5-4*x^11* y^6-4*x^10*y^7+x^9*y^8+2*x^13*y^3+x^12*y^4-6*x^11*y^5+16*x^10*y^6-6*x^9*y^7-2*x ^12*y^3-2*x^11*y^4-15*x^10*y^5-6*x^9*y^6+3*x^8*y^7+x^12*y^2+5*x^11*y^3+5*x^10*y ^4+20*x^9*y^5+x^8*y^6-x^11*y^2-12*x^9*y^4-6*x^8*y^5+8*x^7*y^6+6*x^9*y^3+4*x^8*y ^4+2*x^7*y^5-3*x^9*y^2-3*x^8*y^3-10*x^7*y^4+4*x^6*y^5+3*x^8*y^2-5*x^6*y^4-x^5*y ^5-x^8*y+x^7*y^2+4*x^6*y^3+6*x^5*y^4+x^7*y-2*x^6*y^2-5*x^5*y^3-3*x^4*y^4+3*x^5* y^2+5*x^4*y^3-2*x^5*y-3*x^4*y^2-8*x^3*y^3+2*x^4*y+6*x^3*y^2-x^4-x^3*y-4*x^2*y^2 +x^3+3*x^2*y-2*x*y+x-1)/(x^18*y^8-5*x^18*y^7-x^17*y^8+10*x^18*y^6+5*x^17*y^7-10 *x^18*y^5-10*x^17*y^6-2*x^16*y^7+5*x^18*y^4+10*x^17*y^5+8*x^16*y^6+x^15*y^7-x^ 18*y^3-5*x^17*y^4-12*x^16*y^5-5*x^15*y^6+x^17*y^3+8*x^16*y^4+9*x^15*y^5+x^14*y^ 6-x^13*y^7-2*x^16*y^3-7*x^15*y^4-3*x^14*y^5+3*x^13*y^6+2*x^15*y^3+3*x^14*y^4-2* x^13*y^5-x^12*y^6+x^11*y^7-x^14*y^3-x^13*y^4+3*x^12*y^5+x^11*y^6+x^13*y^3-2*x^ 12*y^4-5*x^11*y^5+x^10*y^6-x^12*y^3+2*x^11*y^4+x^10*y^5-x^9*y^6+x^12*y^2+2*x^11 *y^3-5*x^10*y^4-x^11*y^2+3*x^10*y^3+3*x^9*y^4-x^8*y^5-x^9*y^3-x^9*y^2+x^8*y^3+x ^8*y^2-x^7*y^3-x^6*y^4-x^8*y-x^7*y^2-x^6*y^3+x^7*y+x^6*y^2+x^4*y^3+x^4*y^2-x^4- x^3*y+x^3+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 10 bors of a random word of length n tends to n times, 2 (a + 4 a + 3 a 9 8 7 6 5 3 2 - 2 a - 5 a - 10 a - 13 a - 10 a - 2 a + 5 a + 2 a + 1) 6 5 3 / 4 3 2 2 (7 a + 6 a - 4 a - 2 a - 1) / ((5 a + 4 a + 3 a + 1) / 8 7 6 5 3 2 (a + 2 a + a + a - 2 a - a - a - 1)) 7 6 4 2 where a is the root of the polynomial, x + x - x - x - x + 1, and in decimals this is, 0.3221144514 BTW the ratio for words with, 500, letters is, 0.3247642783 ------------------------------------------------ "Theorem Number 124" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 8 12 7 11 8 12 6 ) | ) C(m, n) x y | = (2 x y - 8 x y - 2 x y + 12 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 7 12 5 11 6 10 7 12 4 11 5 + 10 x y - 8 x y - 20 x y - 2 x y + 2 x y + 20 x y 10 6 9 7 8 8 11 4 10 5 9 6 11 3 + 9 x y + 2 x y - x y - 10 x y - 16 x y - 2 x y + 2 x y 10 4 9 5 8 6 7 7 10 3 9 4 8 5 + 14 x y - 4 x y + 3 x y - 2 x y - 6 x y + 4 x y + 4 x y 7 6 10 2 9 3 8 4 7 5 6 6 9 2 + 2 x y + x y + 2 x y - 15 x y + 8 x y - 4 x y - 2 x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 + 12 x y - 8 x y + 2 x y - 3 x y - 8 x y + 15 x y - 6 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 + 10 x y - 16 x y + 4 x y - 2 x y - x y + 12 x y - 6 x y 6 5 2 4 3 5 4 2 3 3 4 + 4 x y - 12 x y + 10 x y + 2 x y + 3 x y - 8 x y - 8 x y 3 2 4 3 2 2 3 2 / + 8 x y + x + 2 x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x - 1) / ( / 2 (x - 2 x + 1) 8 4 8 3 8 2 5 2 5 3 2 (x y - 2 x y + x y + 2 x y - 2 x y + 2 x y + x - 1)) and in Maple notation (2*x^12*y^8-8*x^12*y^7-2*x^11*y^8+12*x^12*y^6+10*x^11*y^7-8*x^12*y^5-20*x^11*y^ 6-2*x^10*y^7+2*x^12*y^4+20*x^11*y^5+9*x^10*y^6+2*x^9*y^7-x^8*y^8-10*x^11*y^4-16 *x^10*y^5-2*x^9*y^6+2*x^11*y^3+14*x^10*y^4-4*x^9*y^5+3*x^8*y^6-2*x^7*y^7-6*x^10 *y^3+4*x^9*y^4+4*x^8*y^5+2*x^7*y^6+x^10*y^2+2*x^9*y^3-15*x^8*y^4+8*x^7*y^5-4*x^ 6*y^6-2*x^9*y^2+12*x^8*y^3-8*x^7*y^4+2*x^6*y^5-3*x^8*y^2-8*x^7*y^3+15*x^6*y^4-6 *x^5*y^5+10*x^7*y^2-16*x^6*y^3+4*x^5*y^4-2*x^7*y-x^6*y^2+12*x^5*y^3-6*x^4*y^4+4 *x^6*y-12*x^5*y^2+10*x^4*y^3+2*x^5*y+3*x^4*y^2-8*x^3*y^3-8*x^4*y+8*x^3*y^2+x^4+ 2*x^3*y-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^2-2*x+1)/(x^8*y^4-2*x^8*y^3+x^8 *y^2+2*x^5*y^2-2*x^5*y+2*x^3*y+x^2-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3210591098 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 125" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 11 8 12 6 11 7 ) | ) C(m, n) x y | = (x y + x y - 4 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 5 11 6 10 7 12 4 11 5 10 6 12 3 + 6 x y + 8 x y + 2 x y - 4 x y - 4 x y - 7 x y + x y 11 4 10 5 9 6 11 3 10 4 9 5 8 6 - x y + 10 x y + x y + x y - 7 x y - 4 x y + 2 x y 10 3 9 4 8 5 7 6 9 3 8 4 7 5 + 2 x y + 6 x y - 12 x y + x y - 4 x y + 17 x y + 2 x y 9 2 8 3 7 4 6 5 8 2 7 3 6 4 5 5 + x y - 6 x y - 9 x y + 3 x y - x y + 10 x y - 9 x y + x y 7 2 6 3 5 4 6 2 5 3 4 4 6 - 4 x y + x y + 6 x y + 6 x y - 15 x y + 3 x y - 2 x y 5 2 4 3 5 4 2 3 3 4 3 2 + 6 x y + 2 x y + 2 x y - 8 x y + 8 x y + 4 x y - 4 x y 3 2 2 3 2 2 / - 4 x y + 4 x y + x - 2 x y - x + 2 x y - x + 1) / ((x - 1) / 8 4 8 3 8 2 5 2 5 3 2 (x y - 2 x y + x y + 2 x y - 2 x y + 2 x y + x - 1)) and in Maple notation (x^12*y^7+x^11*y^8-4*x^12*y^6-5*x^11*y^7+6*x^12*y^5+8*x^11*y^6+2*x^10*y^7-4*x^ 12*y^4-4*x^11*y^5-7*x^10*y^6+x^12*y^3-x^11*y^4+10*x^10*y^5+x^9*y^6+x^11*y^3-7*x ^10*y^4-4*x^9*y^5+2*x^8*y^6+2*x^10*y^3+6*x^9*y^4-12*x^8*y^5+x^7*y^6-4*x^9*y^3+ 17*x^8*y^4+2*x^7*y^5+x^9*y^2-6*x^8*y^3-9*x^7*y^4+3*x^6*y^5-x^8*y^2+10*x^7*y^3-9 *x^6*y^4+x^5*y^5-4*x^7*y^2+x^6*y^3+6*x^5*y^4+6*x^6*y^2-15*x^5*y^3+3*x^4*y^4-2*x ^6*y+6*x^5*y^2+2*x^4*y^3+2*x^5*y-8*x^4*y^2+8*x^3*y^3+4*x^4*y-4*x^3*y^2-4*x^3*y+ 4*x^2*y^2+x^3-2*x^2*y-x^2+2*x*y-x+1)/(x-1)/(x^8*y^4-2*x^8*y^3+x^8*y^2+2*x^5*y^2 -2*x^5*y+2*x^3*y+x^2-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3197741696 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 126" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 11 8 12 6 11 7 ) | ) C(m, n) x y | = (x y + x y - 4 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 5 11 6 10 7 12 4 11 5 10 6 12 3 + 6 x y + 8 x y + 2 x y - 4 x y - 4 x y - 7 x y + x y 11 4 10 5 9 6 11 3 10 4 9 5 8 6 - x y + 10 x y + x y + x y - 7 x y - 4 x y + 2 x y 10 3 9 4 8 5 7 6 9 3 8 4 7 5 + 2 x y + 6 x y - 12 x y + x y - 4 x y + 17 x y + 2 x y 9 2 8 3 7 4 6 5 8 2 7 3 6 4 5 5 + x y - 6 x y - 9 x y + 3 x y - x y + 10 x y - 9 x y + x y 7 2 6 3 5 4 6 2 5 3 4 4 6 - 4 x y + x y + 6 x y + 6 x y - 15 x y + 3 x y - 2 x y 5 2 4 3 5 4 2 3 3 4 3 2 + 6 x y + 2 x y + 2 x y - 8 x y + 8 x y + 4 x y - 4 x y 3 2 2 3 2 2 / 9 4 - 4 x y + 4 x y + x - 2 x y - x + 2 x y - x + 1) / (x y / 9 3 8 4 9 2 8 3 8 2 6 2 6 5 2 - 2 x y - x y + x y + 2 x y - x y + 2 x y - 2 x y - 2 x y 5 4 3 3 2 + 2 x y + 2 x y - 2 x y + x - x - x + 1) and in Maple notation (x^12*y^7+x^11*y^8-4*x^12*y^6-5*x^11*y^7+6*x^12*y^5+8*x^11*y^6+2*x^10*y^7-4*x^ 12*y^4-4*x^11*y^5-7*x^10*y^6+x^12*y^3-x^11*y^4+10*x^10*y^5+x^9*y^6+x^11*y^3-7*x ^10*y^4-4*x^9*y^5+2*x^8*y^6+2*x^10*y^3+6*x^9*y^4-12*x^8*y^5+x^7*y^6-4*x^9*y^3+ 17*x^8*y^4+2*x^7*y^5+x^9*y^2-6*x^8*y^3-9*x^7*y^4+3*x^6*y^5-x^8*y^2+10*x^7*y^3-9 *x^6*y^4+x^5*y^5-4*x^7*y^2+x^6*y^3+6*x^5*y^4+6*x^6*y^2-15*x^5*y^3+3*x^4*y^4-2*x ^6*y+6*x^5*y^2+2*x^4*y^3+2*x^5*y-8*x^4*y^2+8*x^3*y^3+4*x^4*y-4*x^3*y^2-4*x^3*y+ 4*x^2*y^2+x^3-2*x^2*y-x^2+2*x*y-x+1)/(x^9*y^4-2*x^9*y^3-x^8*y^4+x^9*y^2+2*x^8*y ^3-x^8*y^2+2*x^6*y^2-2*x^6*y-2*x^5*y^2+2*x^5*y+2*x^4*y-2*x^3*y+x^3-x^2-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3197741696 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 127" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 13 9 14 7 13 8 ) | ) C(m, n) x y | = (x y - 2 x y - 4 x y + 8 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 6 13 7 12 8 14 5 12 7 14 4 + 6 x y - 10 x y - 3 x y - 4 x y + 6 x y + x y 13 5 12 6 11 7 10 8 13 4 12 5 + 10 x y + x y + 4 x y - x y - 8 x y - 10 x y 11 6 13 3 12 4 11 5 10 6 9 7 - 14 x y + 2 x y + 9 x y + 18 x y + 11 x y - 2 x y 12 3 11 4 10 5 9 6 12 2 11 3 - 4 x y - 10 x y - 24 x y + 2 x y + x y + 2 x y 10 4 9 5 8 6 10 3 9 4 8 5 9 3 + 20 x y - 2 x y - 4 x y - 6 x y + 2 x y + 16 x y + 4 x y 8 4 7 5 9 2 8 3 7 4 8 2 7 3 - 24 x y - 4 x y - 4 x y + 14 x y + 4 x y - 3 x y - 12 x y 6 4 7 2 6 3 7 5 3 4 4 6 + 9 x y + 12 x y - 12 x y - 2 x y + 12 x y - 3 x y + 4 x y 5 2 4 3 5 4 2 3 3 4 3 2 4 - 12 x y + 8 x y + 2 x y + 2 x y - 8 x y - 8 x y + 8 x y + x 3 2 2 3 2 / 2 + 2 x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x - 1) / ((x - 2 x + 1) / 8 4 8 3 8 2 5 2 5 3 2 (x y - 2 x y + x y + 2 x y - 2 x y + 2 x y + x - 1)) and in Maple notation (x^14*y^8-2*x^13*y^9-4*x^14*y^7+8*x^13*y^8+6*x^14*y^6-10*x^13*y^7-3*x^12*y^8-4* x^14*y^5+6*x^12*y^7+x^14*y^4+10*x^13*y^5+x^12*y^6+4*x^11*y^7-x^10*y^8-8*x^13*y^ 4-10*x^12*y^5-14*x^11*y^6+2*x^13*y^3+9*x^12*y^4+18*x^11*y^5+11*x^10*y^6-2*x^9*y ^7-4*x^12*y^3-10*x^11*y^4-24*x^10*y^5+2*x^9*y^6+x^12*y^2+2*x^11*y^3+20*x^10*y^4 -2*x^9*y^5-4*x^8*y^6-6*x^10*y^3+2*x^9*y^4+16*x^8*y^5+4*x^9*y^3-24*x^8*y^4-4*x^7 *y^5-4*x^9*y^2+14*x^8*y^3+4*x^7*y^4-3*x^8*y^2-12*x^7*y^3+9*x^6*y^4+12*x^7*y^2-\ 12*x^6*y^3-2*x^7*y+12*x^5*y^3-3*x^4*y^4+4*x^6*y-12*x^5*y^2+8*x^4*y^3+2*x^5*y+2* x^4*y^2-8*x^3*y^3-8*x^4*y+8*x^3*y^2+x^4+2*x^3*y-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2 *x-1)/(x^2-2*x+1)/(x^8*y^4-2*x^8*y^3+x^8*y^2+2*x^5*y^2-2*x^5*y+2*x^3*y+x^2-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3192606483 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 128" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 16 6 15 7 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 8 13 9 16 5 15 6 14 7 13 8 15 5 - 3 x y + x y - x y - 3 x y + 11 x y - 4 x y + 3 x y 14 6 13 7 12 8 15 4 14 5 12 7 11 8 - 14 x y + 5 x y + 2 x y - x y + 6 x y - 7 x y - x y 14 4 13 5 12 6 11 7 14 3 13 4 12 5 + x y - 5 x y + 12 x y + x y - x y + 4 x y - 12 x y 11 6 10 7 13 3 12 4 11 5 10 6 9 7 + 2 x y - 3 x y - x y + 6 x y - 4 x y + 5 x y - x y 12 3 11 4 10 5 9 6 8 7 11 3 10 4 - x y + x y - 2 x y - 3 x y + 2 x y + 2 x y - x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 9 3 + 4 x y - 5 x y - x y + x y + x y + 10 x y + 5 x y - x y 8 4 7 5 8 3 7 4 6 5 7 3 6 4 - 8 x y - 5 x y + 2 x y + x y - 4 x y - 2 x y + 5 x y 5 5 7 2 6 3 5 4 6 2 4 4 6 5 2 - 2 x y + 2 x y + x y + 2 x y - 3 x y - 5 x y + x y + x y 4 3 4 2 3 3 4 3 2 2 2 2 + 3 x y + 3 x y + 4 x y - x y - 3 x y + 2 x y - x y + x y - x / 11 5 11 4 11 3 9 3 9 2 8 3 8 2 + 1) / (x y - 2 x y + x y - x y + x y - x y + x y / 6 2 5 3 5 2 4 3 4 2 4 3 2 2 + x y + x y - x y - x y - x y + x y + x y - x y + x y + x - 1) and in Maple notation -(x^16*y^8-3*x^16*y^7+3*x^16*y^6+x^15*y^7-3*x^14*y^8+x^13*y^9-x^16*y^5-3*x^15*y ^6+11*x^14*y^7-4*x^13*y^8+3*x^15*y^5-14*x^14*y^6+5*x^13*y^7+2*x^12*y^8-x^15*y^4 +6*x^14*y^5-7*x^12*y^7-x^11*y^8+x^14*y^4-5*x^13*y^5+12*x^12*y^6+x^11*y^7-x^14*y ^3+4*x^13*y^4-12*x^12*y^5+2*x^11*y^6-3*x^10*y^7-x^13*y^3+6*x^12*y^4-4*x^11*y^5+ 5*x^10*y^6-x^9*y^7-x^12*y^3+x^11*y^4-2*x^10*y^5-3*x^9*y^6+2*x^8*y^7+2*x^11*y^3- x^10*y^4+4*x^9*y^5-5*x^8*y^6-x^11*y^2+x^10*y^3+x^9*y^4+10*x^8*y^5+5*x^7*y^6-x^9 *y^3-8*x^8*y^4-5*x^7*y^5+2*x^8*y^3+x^7*y^4-4*x^6*y^5-2*x^7*y^3+5*x^6*y^4-2*x^5* y^5+2*x^7*y^2+x^6*y^3+2*x^5*y^4-3*x^6*y^2-5*x^4*y^4+x^6*y+x^5*y^2+3*x^4*y^3+3*x ^4*y^2+4*x^3*y^3-x^4*y-3*x^3*y^2+2*x^2*y^2-x^2*y+x*y-x+1)/(x^11*y^5-2*x^11*y^4+ x^11*y^3-x^9*y^3+x^9*y^2-x^8*y^3+x^8*y^2+x^6*y^2+x^5*y^3-x^5*y^2-x^4*y^3-x^4*y^ 2+x^4*y+x^3*y^2-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3163039121 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 129" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 16 6 15 7 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 8 13 9 16 5 15 6 14 7 13 8 15 5 - 3 x y + x y - x y - 3 x y + 11 x y - 4 x y + 3 x y 14 6 13 7 12 8 15 4 14 5 12 7 11 8 - 14 x y + 5 x y + 2 x y - x y + 6 x y - 7 x y - x y 14 4 13 5 12 6 11 7 14 3 13 4 12 5 + x y - 5 x y + 12 x y + x y - x y + 4 x y - 12 x y 11 6 10 7 13 3 12 4 11 5 10 6 9 7 + 2 x y - 3 x y - x y + 6 x y - 4 x y + 5 x y - x y 12 3 11 4 10 5 9 6 8 7 11 3 10 4 - x y + x y - 2 x y - 3 x y + 2 x y + 2 x y - x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 9 3 + 4 x y - 5 x y - x y + x y + x y + 10 x y + 5 x y - x y 8 4 7 5 8 3 7 4 6 5 7 3 6 4 - 8 x y - 5 x y + 2 x y + x y - 4 x y - 2 x y + 5 x y 5 5 7 2 6 3 5 4 6 2 4 4 6 5 2 - 2 x y + 2 x y + x y + 2 x y - 3 x y - 5 x y + x y + x y 4 3 4 2 3 3 4 3 2 2 2 2 + 3 x y + 3 x y + 4 x y - x y - 3 x y + 2 x y - x y + x y - x / 11 5 11 4 11 3 9 3 9 2 8 3 8 2 + 1) / (x y - 2 x y + x y - x y + x y - x y + x y / 6 2 5 3 5 2 4 3 4 2 4 3 2 2 + x y + x y - x y - x y - x y + x y + x y - x y + x y + x - 1) and in Maple notation -(x^16*y^8-3*x^16*y^7+3*x^16*y^6+x^15*y^7-3*x^14*y^8+x^13*y^9-x^16*y^5-3*x^15*y ^6+11*x^14*y^7-4*x^13*y^8+3*x^15*y^5-14*x^14*y^6+5*x^13*y^7+2*x^12*y^8-x^15*y^4 +6*x^14*y^5-7*x^12*y^7-x^11*y^8+x^14*y^4-5*x^13*y^5+12*x^12*y^6+x^11*y^7-x^14*y ^3+4*x^13*y^4-12*x^12*y^5+2*x^11*y^6-3*x^10*y^7-x^13*y^3+6*x^12*y^4-4*x^11*y^5+ 5*x^10*y^6-x^9*y^7-x^12*y^3+x^11*y^4-2*x^10*y^5-3*x^9*y^6+2*x^8*y^7+2*x^11*y^3- x^10*y^4+4*x^9*y^5-5*x^8*y^6-x^11*y^2+x^10*y^3+x^9*y^4+10*x^8*y^5+5*x^7*y^6-x^9 *y^3-8*x^8*y^4-5*x^7*y^5+2*x^8*y^3+x^7*y^4-4*x^6*y^5-2*x^7*y^3+5*x^6*y^4-2*x^5* y^5+2*x^7*y^2+x^6*y^3+2*x^5*y^4-3*x^6*y^2-5*x^4*y^4+x^6*y+x^5*y^2+3*x^4*y^3+3*x ^4*y^2+4*x^3*y^3-x^4*y-3*x^3*y^2+2*x^2*y^2-x^2*y+x*y-x+1)/(x^11*y^5-2*x^11*y^4+ x^11*y^3-x^9*y^3+x^9*y^2-x^8*y^3+x^8*y^2+x^6*y^2+x^5*y^3-x^5*y^2-x^4*y^3-x^4*y^ 2+x^4*y+x^3*y^2-x^2*y+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3163039121 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 130" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 11 19 10 19 9 ) | ) C(m, n) x y | = - (x y - 8 x y + 25 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 19 8 18 9 17 10 19 7 18 8 + x y - 40 x y - 6 x y - 2 x y + 35 x y + 14 x y 17 9 16 10 19 6 18 7 17 8 16 9 + 12 x y - x y - 16 x y - 16 x y - 29 x y + x y 19 5 18 6 17 7 16 8 15 9 18 5 + 3 x y + 9 x y + 36 x y + 7 x y + 2 x y - 2 x y 17 6 16 7 15 8 14 9 17 5 16 6 - 24 x y - 20 x y - 8 x y + x y + 8 x y + 24 x y 15 7 14 8 17 4 16 5 15 6 14 7 + 13 x y + 2 x y - x y - 16 x y - 12 x y - 8 x y 13 8 16 4 15 5 14 6 13 7 12 8 16 3 + 3 x y + 6 x y + 8 x y + 3 x y - x y + x y - x y 15 4 14 5 13 6 12 7 15 3 14 4 - 4 x y + 5 x y - 14 x y + 2 x y + x y - 3 x y 13 5 12 6 11 7 13 4 12 5 10 7 13 3 + 20 x y - 5 x y - x y - 9 x y - 2 x y + 4 x y + x y 12 4 11 5 10 6 12 3 11 4 10 5 + 6 x y + 7 x y - 9 x y - 2 x y - 9 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 11 2 10 3 + 5 x y + 2 x y + 9 x y - 11 x y - x y + x y - 10 x y 9 4 8 5 10 2 9 3 8 4 7 5 9 2 + 5 x y - 5 x y + 3 x y + 5 x y + 3 x y - 7 x y - 5 x y 8 3 7 4 6 5 9 6 4 8 6 3 5 4 + 3 x y + 6 x y - x y + x y - 9 x y - x y + 13 x y + 3 x y 6 2 5 3 4 4 6 5 2 4 3 5 - 3 x y - 8 x y - 3 x y - x y + 6 x y + 4 x y - 2 x y 4 2 3 3 4 3 2 4 3 2 2 3 2 - 4 x y - 8 x y + 3 x y + 6 x y - x - x y - 4 x y + x + 3 x y / 16 7 16 6 16 5 15 6 16 4 - 2 x y + x - 1) / (x y - 4 x y + 6 x y + x y - 4 x y / 15 5 16 3 15 4 13 6 15 3 13 5 13 4 - 3 x y + x y + 3 x y + x y - x y - 3 x y + 3 x y 12 5 13 3 12 4 12 3 11 4 11 3 10 4 + x y - x y - 2 x y + x y - x y + 2 x y - x y 11 2 10 3 9 4 10 2 9 3 8 4 9 2 8 3 - x y + 2 x y - x y - x y - x y + x y + 3 x y - 2 x y 9 8 6 3 6 2 6 4 4 3 3 2 - x y + x y - x y - x y + x y - x y + x + x y - x - x y - x + 1) and in Maple notation -(x^19*y^11-8*x^19*y^10+25*x^19*y^9+x^18*y^10-40*x^19*y^8-6*x^18*y^9-2*x^17*y^ 10+35*x^19*y^7+14*x^18*y^8+12*x^17*y^9-x^16*y^10-16*x^19*y^6-16*x^18*y^7-29*x^ 17*y^8+x^16*y^9+3*x^19*y^5+9*x^18*y^6+36*x^17*y^7+7*x^16*y^8+2*x^15*y^9-2*x^18* y^5-24*x^17*y^6-20*x^16*y^7-8*x^15*y^8+x^14*y^9+8*x^17*y^5+24*x^16*y^6+13*x^15* y^7+2*x^14*y^8-x^17*y^4-16*x^16*y^5-12*x^15*y^6-8*x^14*y^7+3*x^13*y^8+6*x^16*y^ 4+8*x^15*y^5+3*x^14*y^6-x^13*y^7+x^12*y^8-x^16*y^3-4*x^15*y^4+5*x^14*y^5-14*x^ 13*y^6+2*x^12*y^7+x^15*y^3-3*x^14*y^4+20*x^13*y^5-5*x^12*y^6-x^11*y^7-9*x^13*y^ 4-2*x^12*y^5+4*x^10*y^7+x^13*y^3+6*x^12*y^4+7*x^11*y^5-9*x^10*y^6-2*x^12*y^3-9* x^11*y^4+3*x^10*y^5+5*x^9*y^6+2*x^11*y^3+9*x^10*y^4-11*x^9*y^5-x^8*y^6+x^11*y^2 -10*x^10*y^3+5*x^9*y^4-5*x^8*y^5+3*x^10*y^2+5*x^9*y^3+3*x^8*y^4-7*x^7*y^5-5*x^9 *y^2+3*x^8*y^3+6*x^7*y^4-x^6*y^5+x^9*y-9*x^6*y^4-x^8*y+13*x^6*y^3+3*x^5*y^4-3*x ^6*y^2-8*x^5*y^3-3*x^4*y^4-x^6*y+6*x^5*y^2+4*x^4*y^3-2*x^5*y-4*x^4*y^2-8*x^3*y^ 3+3*x^4*y+6*x^3*y^2-x^4-x^3*y-4*x^2*y^2+x^3+3*x^2*y-2*x*y+x-1)/(x^16*y^7-4*x^16 *y^6+6*x^16*y^5+x^15*y^6-4*x^16*y^4-3*x^15*y^5+x^16*y^3+3*x^15*y^4+x^13*y^6-x^ 15*y^3-3*x^13*y^5+3*x^13*y^4+x^12*y^5-x^13*y^3-2*x^12*y^4+x^12*y^3-x^11*y^4+2*x ^11*y^3-x^10*y^4-x^11*y^2+2*x^10*y^3-x^9*y^4-x^10*y^2-x^9*y^3+x^8*y^4+3*x^9*y^2 -2*x^8*y^3-x^9*y+x^8*y-x^6*y^3-x^6*y^2+x^6*y-x^4*y+x^4+x^3*y-x^3-x^2*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3113983677 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 131" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 11 19 10 19 9 ) | ) C(m, n) x y | = - (x y - 8 x y + 25 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 10 19 8 18 9 17 10 19 7 18 8 + x y - 40 x y - 6 x y - 2 x y + 35 x y + 14 x y 17 9 16 10 19 6 18 7 17 8 16 9 + 12 x y - x y - 16 x y - 16 x y - 29 x y + x y 19 5 18 6 17 7 16 8 15 9 18 5 + 3 x y + 9 x y + 36 x y + 7 x y + 2 x y - 2 x y 17 6 16 7 15 8 14 9 17 5 16 6 - 24 x y - 20 x y - 8 x y + x y + 8 x y + 24 x y 15 7 14 8 17 4 16 5 15 6 14 7 + 13 x y + 2 x y - x y - 16 x y - 12 x y - 8 x y 13 8 16 4 15 5 14 6 13 7 12 8 16 3 + 3 x y + 6 x y + 8 x y + 3 x y - x y + x y - x y 15 4 14 5 13 6 12 7 15 3 14 4 - 4 x y + 5 x y - 14 x y + 2 x y + x y - 3 x y 13 5 12 6 11 7 13 4 12 5 10 7 13 3 + 20 x y - 5 x y - x y - 9 x y - 2 x y + 4 x y + x y 12 4 11 5 10 6 12 3 11 4 10 5 + 6 x y + 7 x y - 9 x y - 2 x y - 9 x y + 3 x y 9 6 11 3 10 4 9 5 8 6 11 2 10 3 + 5 x y + 2 x y + 9 x y - 11 x y - x y + x y - 10 x y 9 4 8 5 10 2 9 3 8 4 7 5 9 2 + 5 x y - 5 x y + 3 x y + 5 x y + 3 x y - 7 x y - 5 x y 8 3 7 4 6 5 9 6 4 8 6 3 5 4 + 3 x y + 6 x y - x y + x y - 9 x y - x y + 13 x y + 3 x y 6 2 5 3 4 4 6 5 2 4 3 5 - 3 x y - 8 x y - 3 x y - x y + 6 x y + 4 x y - 2 x y 4 2 3 3 4 3 2 4 3 2 2 3 2 - 4 x y - 8 x y + 3 x y + 6 x y - x - x y - 4 x y + x + 3 x y / 16 7 16 6 16 5 15 6 16 4 - 2 x y + x - 1) / (x y - 4 x y + 6 x y + x y - 4 x y / 15 5 16 3 15 4 13 6 15 3 13 5 13 4 - 3 x y + x y + 3 x y + x y - x y - 3 x y + 3 x y 12 5 13 3 12 4 12 3 11 4 11 3 10 4 + x y - x y - 2 x y + x y - x y + 2 x y - x y 11 2 10 3 9 4 10 2 9 3 8 4 9 2 8 3 - x y + 2 x y - x y - x y - x y + x y + 3 x y - 2 x y 9 8 6 3 6 2 6 4 4 3 3 2 - x y + x y - x y - x y + x y - x y + x + x y - x - x y - x + 1) and in Maple notation -(x^19*y^11-8*x^19*y^10+25*x^19*y^9+x^18*y^10-40*x^19*y^8-6*x^18*y^9-2*x^17*y^ 10+35*x^19*y^7+14*x^18*y^8+12*x^17*y^9-x^16*y^10-16*x^19*y^6-16*x^18*y^7-29*x^ 17*y^8+x^16*y^9+3*x^19*y^5+9*x^18*y^6+36*x^17*y^7+7*x^16*y^8+2*x^15*y^9-2*x^18* y^5-24*x^17*y^6-20*x^16*y^7-8*x^15*y^8+x^14*y^9+8*x^17*y^5+24*x^16*y^6+13*x^15* y^7+2*x^14*y^8-x^17*y^4-16*x^16*y^5-12*x^15*y^6-8*x^14*y^7+3*x^13*y^8+6*x^16*y^ 4+8*x^15*y^5+3*x^14*y^6-x^13*y^7+x^12*y^8-x^16*y^3-4*x^15*y^4+5*x^14*y^5-14*x^ 13*y^6+2*x^12*y^7+x^15*y^3-3*x^14*y^4+20*x^13*y^5-5*x^12*y^6-x^11*y^7-9*x^13*y^ 4-2*x^12*y^5+4*x^10*y^7+x^13*y^3+6*x^12*y^4+7*x^11*y^5-9*x^10*y^6-2*x^12*y^3-9* x^11*y^4+3*x^10*y^5+5*x^9*y^6+2*x^11*y^3+9*x^10*y^4-11*x^9*y^5-x^8*y^6+x^11*y^2 -10*x^10*y^3+5*x^9*y^4-5*x^8*y^5+3*x^10*y^2+5*x^9*y^3+3*x^8*y^4-7*x^7*y^5-5*x^9 *y^2+3*x^8*y^3+6*x^7*y^4-x^6*y^5+x^9*y-9*x^6*y^4-x^8*y+13*x^6*y^3+3*x^5*y^4-3*x ^6*y^2-8*x^5*y^3-3*x^4*y^4-x^6*y+6*x^5*y^2+4*x^4*y^3-2*x^5*y-4*x^4*y^2-8*x^3*y^ 3+3*x^4*y+6*x^3*y^2-x^4-x^3*y-4*x^2*y^2+x^3+3*x^2*y-2*x*y+x-1)/(x^16*y^7-4*x^16 *y^6+6*x^16*y^5+x^15*y^6-4*x^16*y^4-3*x^15*y^5+x^16*y^3+3*x^15*y^4+x^13*y^6-x^ 15*y^3-3*x^13*y^5+3*x^13*y^4+x^12*y^5-x^13*y^3-2*x^12*y^4+x^12*y^3-x^11*y^4+2*x ^11*y^3-x^10*y^4-x^11*y^2+2*x^10*y^3-x^9*y^4-x^10*y^2-x^9*y^3+x^8*y^4+3*x^9*y^2 -2*x^8*y^3-x^9*y+x^8*y-x^6*y^3-x^6*y^2+x^6*y-x^4*y+x^4+x^3*y-x^3-x^2*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3113983677 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 132" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 11 17 10 17 9 15 11 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 10 17 7 16 8 15 9 14 10 17 6 + 2 x y - 5 x y + x y + 3 x y - 2 x y + 4 x y 16 7 15 8 14 9 13 10 17 5 16 6 - 4 x y - 11 x y + 8 x y + 2 x y - x y + 6 x y 15 7 14 8 13 9 16 5 14 7 13 8 + 9 x y - 12 x y - 11 x y - 4 x y + 10 x y + 16 x y 12 9 16 4 15 5 14 6 12 8 11 9 15 4 + 3 x y + x y - 3 x y - 10 x y - 9 x y + x y + x y 14 5 13 6 12 7 11 8 14 4 13 5 + 11 x y - 15 x y + 2 x y + 6 x y - 5 x y + 11 x y 12 6 11 7 10 8 14 3 13 4 12 5 + 13 x y - 13 x y + 2 x y - x y - 8 x y - 8 x y 11 6 10 7 9 8 14 2 13 3 12 4 11 5 - 5 x y - 4 x y - 2 x y + x y + 8 x y - x y + 18 x y 10 6 9 7 13 2 12 3 11 4 10 5 9 6 + 3 x y + 5 x y - 3 x y - 5 x y - 3 x y - 2 x y + 3 x y 8 7 12 2 11 3 9 5 8 6 12 11 2 - 3 x y + 7 x y - 5 x y - 13 x y + 4 x y - 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 9 3 + 4 x y + 5 x y + 7 x y - 8 x y + 3 x y - 3 x y + 4 x y 8 4 7 5 11 10 9 2 8 3 7 4 6 5 - 3 x y - x y - x - x y - x y - 13 x y + 17 x y - 3 x y 10 9 8 2 7 3 6 4 5 5 8 7 2 + x - x y + 7 x y - 5 x y - x y + 2 x y + x y - 8 x y 6 3 5 4 7 6 2 5 3 4 4 6 + 9 x y - 4 x y + 4 x y - 3 x y - x y + 3 x y - 4 x y 5 2 4 3 6 4 2 3 3 5 4 3 2 + 5 x y - 5 x y + 2 x + x y + 8 x y - 2 x + x y - 6 x y 2 2 2 / 10 6 10 5 10 4 + 4 x y - 3 x y + 2 x y - x + 1) / ((x - 1) (x y - x y - x y / 9 4 8 5 10 2 9 3 10 9 2 8 3 10 8 2 + x y - x y + x y - 2 x y + x y + x y + x y - x + x y 7 3 6 4 8 7 2 6 3 6 2 5 2 4 3 4 2 - x y - x y - x y + x y - x y + 2 x y - x y + x y + x y 5 3 2 + 2 x + x y + x y - 1)) and in Maple notation (x^17*y^11-4*x^17*y^10+5*x^17*y^9-x^15*y^11+2*x^15*y^10-5*x^17*y^7+x^16*y^8+3*x ^15*y^9-2*x^14*y^10+4*x^17*y^6-4*x^16*y^7-11*x^15*y^8+8*x^14*y^9+2*x^13*y^10-x^ 17*y^5+6*x^16*y^6+9*x^15*y^7-12*x^14*y^8-11*x^13*y^9-4*x^16*y^5+10*x^14*y^7+16* x^13*y^8+3*x^12*y^9+x^16*y^4-3*x^15*y^5-10*x^14*y^6-9*x^12*y^8+x^11*y^9+x^15*y^ 4+11*x^14*y^5-15*x^13*y^6+2*x^12*y^7+6*x^11*y^8-5*x^14*y^4+11*x^13*y^5+13*x^12* y^6-13*x^11*y^7+2*x^10*y^8-x^14*y^3-8*x^13*y^4-8*x^12*y^5-5*x^11*y^6-4*x^10*y^7 -2*x^9*y^8+x^14*y^2+8*x^13*y^3-x^12*y^4+18*x^11*y^5+3*x^10*y^6+5*x^9*y^7-3*x^13 *y^2-5*x^12*y^3-3*x^11*y^4-2*x^10*y^5+3*x^9*y^6-3*x^8*y^7+7*x^12*y^2-5*x^11*y^3 -13*x^9*y^5+4*x^8*y^6-2*x^12*y-x^11*y^2+4*x^10*y^3+5*x^9*y^4+7*x^8*y^5-8*x^7*y^ 6+3*x^11*y-3*x^10*y^2+4*x^9*y^3-3*x^8*y^4-x^7*y^5-x^11-x^10*y-x^9*y^2-13*x^8*y^ 3+17*x^7*y^4-3*x^6*y^5+x^10-x^9*y+7*x^8*y^2-5*x^7*y^3-x^6*y^4+2*x^5*y^5+x^8*y-8 *x^7*y^2+9*x^6*y^3-4*x^5*y^4+4*x^7*y-3*x^6*y^2-x^5*y^3+3*x^4*y^4-4*x^6*y+5*x^5* y^2-5*x^4*y^3+2*x^6+x^4*y^2+8*x^3*y^3-2*x^5+x^4*y-6*x^3*y^2+4*x^2*y^2-3*x^2*y+2 *x*y-x+1)/(x-1)/(x^10*y^6-x^10*y^5-x^10*y^4+x^9*y^4-x^8*y^5+x^10*y^2-2*x^9*y^3+ x^10*y+x^9*y^2+x^8*y^3-x^10+x^8*y^2-x^7*y^3-x^6*y^4-x^8*y+x^7*y^2-x^6*y^3+2*x^6 *y^2-x^5*y^2+x^4*y^3+x^4*y^2+2*x^5+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3086757890 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 133" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 2, 1, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 11 17 10 17 9 15 11 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 10 17 7 16 8 15 9 14 10 17 6 + 2 x y - 5 x y + x y + 3 x y - 2 x y + 4 x y 16 7 15 8 14 9 13 10 17 5 16 6 - 4 x y - 11 x y + 8 x y + 2 x y - x y + 6 x y 15 7 14 8 13 9 16 5 14 7 13 8 + 9 x y - 12 x y - 11 x y - 4 x y + 10 x y + 16 x y 12 9 16 4 15 5 14 6 12 8 11 9 15 4 + 3 x y + x y - 3 x y - 10 x y - 9 x y + x y + x y 14 5 13 6 12 7 11 8 14 4 13 5 + 11 x y - 15 x y + 2 x y + 6 x y - 5 x y + 11 x y 12 6 11 7 10 8 14 3 13 4 12 5 + 13 x y - 13 x y + 2 x y - x y - 8 x y - 8 x y 11 6 10 7 9 8 14 2 13 3 12 4 11 5 - 5 x y - 4 x y - 2 x y + x y + 8 x y - x y + 18 x y 10 6 9 7 13 2 12 3 11 4 10 5 9 6 + 3 x y + 5 x y - 3 x y - 5 x y - 3 x y - 2 x y + 3 x y 8 7 12 2 11 3 9 5 8 6 12 11 2 - 3 x y + 7 x y - 5 x y - 13 x y + 4 x y - 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 9 3 + 4 x y + 5 x y + 7 x y - 8 x y + 3 x y - 3 x y + 4 x y 8 4 7 5 11 10 9 2 8 3 7 4 6 5 - 3 x y - x y - x - x y - x y - 13 x y + 17 x y - 3 x y 10 9 8 2 7 3 6 4 5 5 8 7 2 + x - x y + 7 x y - 5 x y - x y + 2 x y + x y - 8 x y 6 3 5 4 7 6 2 5 3 4 4 6 + 9 x y - 4 x y + 4 x y - 3 x y - x y + 3 x y - 4 x y 5 2 4 3 6 4 2 3 3 5 4 3 2 + 5 x y - 5 x y + 2 x + x y + 8 x y - 2 x + x y - 6 x y 2 2 2 / 10 6 10 5 10 4 + 4 x y - 3 x y + 2 x y - x + 1) / ((x - 1) (x y - x y - x y / 9 4 8 5 10 2 9 3 10 9 2 8 3 10 8 2 + x y - x y + x y - 2 x y + x y + x y + x y - x + x y 7 3 6 4 8 7 2 6 3 6 2 5 2 4 3 4 2 - x y - x y - x y + x y - x y + 2 x y - x y + x y + x y 5 3 2 + 2 x + x y + x y - 1)) and in Maple notation (x^17*y^11-4*x^17*y^10+5*x^17*y^9-x^15*y^11+2*x^15*y^10-5*x^17*y^7+x^16*y^8+3*x ^15*y^9-2*x^14*y^10+4*x^17*y^6-4*x^16*y^7-11*x^15*y^8+8*x^14*y^9+2*x^13*y^10-x^ 17*y^5+6*x^16*y^6+9*x^15*y^7-12*x^14*y^8-11*x^13*y^9-4*x^16*y^5+10*x^14*y^7+16* x^13*y^8+3*x^12*y^9+x^16*y^4-3*x^15*y^5-10*x^14*y^6-9*x^12*y^8+x^11*y^9+x^15*y^ 4+11*x^14*y^5-15*x^13*y^6+2*x^12*y^7+6*x^11*y^8-5*x^14*y^4+11*x^13*y^5+13*x^12* y^6-13*x^11*y^7+2*x^10*y^8-x^14*y^3-8*x^13*y^4-8*x^12*y^5-5*x^11*y^6-4*x^10*y^7 -2*x^9*y^8+x^14*y^2+8*x^13*y^3-x^12*y^4+18*x^11*y^5+3*x^10*y^6+5*x^9*y^7-3*x^13 *y^2-5*x^12*y^3-3*x^11*y^4-2*x^10*y^5+3*x^9*y^6-3*x^8*y^7+7*x^12*y^2-5*x^11*y^3 -13*x^9*y^5+4*x^8*y^6-2*x^12*y-x^11*y^2+4*x^10*y^3+5*x^9*y^4+7*x^8*y^5-8*x^7*y^ 6+3*x^11*y-3*x^10*y^2+4*x^9*y^3-3*x^8*y^4-x^7*y^5-x^11-x^10*y-x^9*y^2-13*x^8*y^ 3+17*x^7*y^4-3*x^6*y^5+x^10-x^9*y+7*x^8*y^2-5*x^7*y^3-x^6*y^4+2*x^5*y^5+x^8*y-8 *x^7*y^2+9*x^6*y^3-4*x^5*y^4+4*x^7*y-3*x^6*y^2-x^5*y^3+3*x^4*y^4-4*x^6*y+5*x^5* y^2-5*x^4*y^3+2*x^6+x^4*y^2+8*x^3*y^3-2*x^5+x^4*y-6*x^3*y^2+4*x^2*y^2-3*x^2*y+2 *x*y-x+1)/(x-1)/(x^10*y^6-x^10*y^5-x^10*y^4+x^9*y^4-x^8*y^5+x^10*y^2-2*x^9*y^3+ x^10*y+x^9*y^2+x^8*y^3-x^10+x^8*y^2-x^7*y^3-x^6*y^4-x^8*y+x^7*y^2-x^6*y^3+2*x^6 *y^2-x^5*y^2+x^4*y^3+x^4*y^2+2*x^5+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3086757890 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 134" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 7 18 8 16 10 19 6 ) | ) C(m, n) x y | = - (x y + 2 x y - x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 7 17 8 16 9 15 10 19 5 18 6 - 10 x y - 2 x y + 2 x y - x y + 10 x y + 20 x y 17 7 16 8 14 10 19 4 18 5 17 6 + 9 x y + x y + 2 x y - 10 x y - 20 x y - 16 x y 16 7 15 8 14 9 19 3 18 4 17 5 - 6 x y + 7 x y - 5 x y + 5 x y + 10 x y + 14 x y 16 6 15 7 14 8 13 9 19 2 18 3 17 4 + 9 x y - 10 x y + x y + 5 x y - x y - 2 x y - 6 x y 16 5 15 6 14 7 13 8 12 9 17 3 - 10 x y + 4 x y + 7 x y - 5 x y - 2 x y + x y 16 4 15 5 14 6 13 7 12 8 16 3 + 7 x y - 2 x y - 6 x y - 11 x y + 4 x y - 2 x y 15 4 14 5 13 6 12 7 11 8 15 3 + 6 x y - 3 x y + 12 x y + 10 x y - 3 x y - 7 x y 14 4 13 5 12 6 11 7 10 8 15 2 + 9 x y + 4 x y - 23 x y + 3 x y - 3 x y + 4 x y 14 3 13 4 12 5 11 6 10 7 15 14 2 - 7 x y - x y + 4 x y + 9 x y + 4 x y - x y + 2 x y 13 3 12 4 11 5 9 7 13 2 12 3 - 8 x y + 12 x y - 10 x y - 8 x y + 4 x y - 2 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 - 7 x y - x y + x y + 3 x y - 5 x y + 9 x y - 3 x y 9 5 8 6 12 11 2 10 3 9 4 8 5 + 14 x y - 3 x y + 2 x y + x y + 3 x y - 7 x y - 12 x y 7 6 11 10 2 8 4 7 5 11 10 + 7 x y - 3 x y - 2 x y + 10 x y + 2 x y + x + 3 x y 9 2 8 3 7 4 6 5 10 9 8 2 7 3 - x y + 12 x y - 15 x y + 4 x y - x + x y - 8 x y + 5 x y 6 4 8 7 2 6 3 5 4 7 6 2 + 3 x y - x y + 7 x y - 10 x y + 7 x y - 4 x y + 2 x y 5 3 4 4 6 5 2 4 3 6 5 4 2 - 4 x y - 3 x y + 4 x y - 3 x y + 5 x y - 2 x - x y - x y 3 3 5 4 3 2 2 2 2 / - 8 x y + 2 x - x y + 6 x y - 4 x y + 3 x y - 2 x y + x - 1) / / 10 6 10 5 10 4 9 4 8 5 10 2 9 3 ((x - 1) (x y - x y - x y + x y - x y + x y - 2 x y 10 9 2 8 3 10 8 2 7 3 6 4 8 7 2 + x y + x y + x y - x + x y - x y - x y - x y + x y 6 3 6 2 5 2 4 3 4 2 5 3 2 - x y + 2 x y - x y + x y + x y + 2 x + x y + x y - 1)) and in Maple notation -(x^19*y^7+2*x^18*y^8-x^16*y^10-5*x^19*y^6-10*x^18*y^7-2*x^17*y^8+2*x^16*y^9-x^ 15*y^10+10*x^19*y^5+20*x^18*y^6+9*x^17*y^7+x^16*y^8+2*x^14*y^10-10*x^19*y^4-20* x^18*y^5-16*x^17*y^6-6*x^16*y^7+7*x^15*y^8-5*x^14*y^9+5*x^19*y^3+10*x^18*y^4+14 *x^17*y^5+9*x^16*y^6-10*x^15*y^7+x^14*y^8+5*x^13*y^9-x^19*y^2-2*x^18*y^3-6*x^17 *y^4-10*x^16*y^5+4*x^15*y^6+7*x^14*y^7-5*x^13*y^8-2*x^12*y^9+x^17*y^3+7*x^16*y^ 4-2*x^15*y^5-6*x^14*y^6-11*x^13*y^7+4*x^12*y^8-2*x^16*y^3+6*x^15*y^4-3*x^14*y^5 +12*x^13*y^6+10*x^12*y^7-3*x^11*y^8-7*x^15*y^3+9*x^14*y^4+4*x^13*y^5-23*x^12*y^ 6+3*x^11*y^7-3*x^10*y^8+4*x^15*y^2-7*x^14*y^3-x^13*y^4+4*x^12*y^5+9*x^11*y^6+4* x^10*y^7-x^15*y+2*x^14*y^2-8*x^13*y^3+12*x^12*y^4-10*x^11*y^5-8*x^9*y^7+4*x^13* y^2-2*x^12*y^3-7*x^11*y^4-x^10*y^5+x^9*y^6+3*x^8*y^7-5*x^12*y^2+9*x^11*y^3-3*x^ 10*y^4+14*x^9*y^5-3*x^8*y^6+2*x^12*y+x^11*y^2+3*x^10*y^3-7*x^9*y^4-12*x^8*y^5+7 *x^7*y^6-3*x^11*y-2*x^10*y^2+10*x^8*y^4+2*x^7*y^5+x^11+3*x^10*y-x^9*y^2+12*x^8* y^3-15*x^7*y^4+4*x^6*y^5-x^10+x^9*y-8*x^8*y^2+5*x^7*y^3+3*x^6*y^4-x^8*y+7*x^7*y ^2-10*x^6*y^3+7*x^5*y^4-4*x^7*y+2*x^6*y^2-4*x^5*y^3-3*x^4*y^4+4*x^6*y-3*x^5*y^2 +5*x^4*y^3-2*x^6-x^5*y-x^4*y^2-8*x^3*y^3+2*x^5-x^4*y+6*x^3*y^2-4*x^2*y^2+3*x^2* y-2*x*y+x-1)/(x-1)/(x^10*y^6-x^10*y^5-x^10*y^4+x^9*y^4-x^8*y^5+x^10*y^2-2*x^9*y ^3+x^10*y+x^9*y^2+x^8*y^3-x^10+x^8*y^2-x^7*y^3-x^6*y^4-x^8*y+x^7*y^2-x^6*y^3+2* x^6*y^2-x^5*y^2+x^4*y^3+x^4*y^2+2*x^5+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3080534171 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 135" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 7 18 8 16 10 19 6 ) | ) C(m, n) x y | = - (x y + 2 x y - x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 7 17 8 16 9 15 10 19 5 18 6 - 10 x y - 2 x y + 2 x y - x y + 10 x y + 20 x y 17 7 16 8 14 10 19 4 18 5 17 6 + 9 x y + x y + 2 x y - 10 x y - 20 x y - 16 x y 16 7 15 8 14 9 19 3 18 4 17 5 - 6 x y + 7 x y - 5 x y + 5 x y + 10 x y + 14 x y 16 6 15 7 14 8 13 9 19 2 18 3 17 4 + 9 x y - 10 x y + x y + 5 x y - x y - 2 x y - 6 x y 16 5 15 6 14 7 13 8 12 9 17 3 - 10 x y + 4 x y + 7 x y - 5 x y - 2 x y + x y 16 4 15 5 14 6 13 7 12 8 16 3 + 7 x y - 2 x y - 6 x y - 11 x y + 4 x y - 2 x y 15 4 14 5 13 6 12 7 11 8 15 3 + 6 x y - 3 x y + 12 x y + 10 x y - 3 x y - 7 x y 14 4 13 5 12 6 11 7 10 8 15 2 + 9 x y + 4 x y - 23 x y + 3 x y - 3 x y + 4 x y 14 3 13 4 12 5 11 6 10 7 15 14 2 - 7 x y - x y + 4 x y + 9 x y + 4 x y - x y + 2 x y 13 3 12 4 11 5 9 7 13 2 12 3 - 8 x y + 12 x y - 10 x y - 8 x y + 4 x y - 2 x y 11 4 10 5 9 6 8 7 12 2 11 3 10 4 - 7 x y - x y + x y + 3 x y - 5 x y + 9 x y - 3 x y 9 5 8 6 12 11 2 10 3 9 4 8 5 + 14 x y - 3 x y + 2 x y + x y + 3 x y - 7 x y - 12 x y 7 6 11 10 2 8 4 7 5 11 10 + 7 x y - 3 x y - 2 x y + 10 x y + 2 x y + x + 3 x y 9 2 8 3 7 4 6 5 10 9 8 2 7 3 - x y + 12 x y - 15 x y + 4 x y - x + x y - 8 x y + 5 x y 6 4 8 7 2 6 3 5 4 7 6 2 + 3 x y - x y + 7 x y - 10 x y + 7 x y - 4 x y + 2 x y 5 3 4 4 6 5 2 4 3 6 5 4 2 - 4 x y - 3 x y + 4 x y - 3 x y + 5 x y - 2 x - x y - x y 3 3 5 4 3 2 2 2 2 / - 8 x y + 2 x - x y + 6 x y - 4 x y + 3 x y - 2 x y + x - 1) / / 10 6 10 5 10 4 9 4 8 5 10 2 9 3 ((x - 1) (x y - x y - x y + x y - x y + x y - 2 x y 10 9 2 8 3 10 8 2 7 3 6 4 8 7 2 + x y + x y + x y - x + x y - x y - x y - x y + x y 6 3 6 2 5 2 4 3 4 2 5 3 2 - x y + 2 x y - x y + x y + x y + 2 x + x y + x y - 1)) and in Maple notation -(x^19*y^7+2*x^18*y^8-x^16*y^10-5*x^19*y^6-10*x^18*y^7-2*x^17*y^8+2*x^16*y^9-x^ 15*y^10+10*x^19*y^5+20*x^18*y^6+9*x^17*y^7+x^16*y^8+2*x^14*y^10-10*x^19*y^4-20* x^18*y^5-16*x^17*y^6-6*x^16*y^7+7*x^15*y^8-5*x^14*y^9+5*x^19*y^3+10*x^18*y^4+14 *x^17*y^5+9*x^16*y^6-10*x^15*y^7+x^14*y^8+5*x^13*y^9-x^19*y^2-2*x^18*y^3-6*x^17 *y^4-10*x^16*y^5+4*x^15*y^6+7*x^14*y^7-5*x^13*y^8-2*x^12*y^9+x^17*y^3+7*x^16*y^ 4-2*x^15*y^5-6*x^14*y^6-11*x^13*y^7+4*x^12*y^8-2*x^16*y^3+6*x^15*y^4-3*x^14*y^5 +12*x^13*y^6+10*x^12*y^7-3*x^11*y^8-7*x^15*y^3+9*x^14*y^4+4*x^13*y^5-23*x^12*y^ 6+3*x^11*y^7-3*x^10*y^8+4*x^15*y^2-7*x^14*y^3-x^13*y^4+4*x^12*y^5+9*x^11*y^6+4* x^10*y^7-x^15*y+2*x^14*y^2-8*x^13*y^3+12*x^12*y^4-10*x^11*y^5-8*x^9*y^7+4*x^13* y^2-2*x^12*y^3-7*x^11*y^4-x^10*y^5+x^9*y^6+3*x^8*y^7-5*x^12*y^2+9*x^11*y^3-3*x^ 10*y^4+14*x^9*y^5-3*x^8*y^6+2*x^12*y+x^11*y^2+3*x^10*y^3-7*x^9*y^4-12*x^8*y^5+7 *x^7*y^6-3*x^11*y-2*x^10*y^2+10*x^8*y^4+2*x^7*y^5+x^11+3*x^10*y-x^9*y^2+12*x^8* y^3-15*x^7*y^4+4*x^6*y^5-x^10+x^9*y-8*x^8*y^2+5*x^7*y^3+3*x^6*y^4-x^8*y+7*x^7*y ^2-10*x^6*y^3+7*x^5*y^4-4*x^7*y+2*x^6*y^2-4*x^5*y^3-3*x^4*y^4+4*x^6*y-3*x^5*y^2 +5*x^4*y^3-2*x^6-x^5*y-x^4*y^2-8*x^3*y^3+2*x^5-x^4*y+6*x^3*y^2-4*x^2*y^2+3*x^2* y-2*x*y+x-1)/(x-1)/(x^10*y^6-x^10*y^5-x^10*y^4+x^9*y^4-x^8*y^5+x^10*y^2-2*x^9*y ^3+x^10*y+x^9*y^2+x^8*y^3-x^10+x^8*y^2-x^7*y^3-x^6*y^4-x^8*y+x^7*y^2-x^6*y^3+2* x^6*y^2-x^5*y^2+x^4*y^3+x^4*y^2+2*x^5+x^3*y+x^2*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3080534171 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 136" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 9 13 8 13 7 12 8 ) | ) C(m, n) x y | = - (x y - 3 x y + 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 6 12 7 13 5 12 6 11 7 13 4 12 5 + 2 x y - 2 x y - 3 x y - x y - x y + x y + 5 x y 11 6 10 7 12 4 11 5 10 6 9 7 12 3 + 3 x y + x y - 4 x y - 3 x y + 3 x y - x y + x y 11 4 10 5 9 6 10 4 9 5 8 6 10 3 + x y - 12 x y - 4 x y + 7 x y + 19 x y - 3 x y + 5 x y 9 4 10 2 9 3 8 4 7 5 9 2 7 4 - 14 x y - 4 x y - 10 x y + 8 x y - 3 x y + 12 x y + 3 x y 6 5 9 8 2 7 3 6 4 5 5 8 8 - 2 x y - 2 x y - 10 x y + 3 x y + x y - x y + 6 x y - x 7 6 2 5 3 4 4 7 6 5 2 - 6 x y + 2 x y - 5 x y - 3 x y + 3 x + 2 x y + 4 x y 4 3 6 4 2 3 3 5 4 3 2 3 + 14 x y - 3 x - 12 x y - 8 x y + x + 2 x y + 12 x y - 6 x y 2 2 3 2 2 / 8 4 7 4 - 4 x y + x + 6 x y - 3 x - 2 x y + 3 x - 1) / (x y - x y / 8 2 7 2 8 6 2 7 6 5 3 2 - 2 x y + 4 x y + x - 2 x y - 3 x + 3 x - x - x + 3 x - 3 x + 1) and in Maple notation -(x^13*y^9-3*x^13*y^8+2*x^13*y^7+x^12*y^8+2*x^13*y^6-2*x^12*y^7-3*x^13*y^5-x^12 *y^6-x^11*y^7+x^13*y^4+5*x^12*y^5+3*x^11*y^6+x^10*y^7-4*x^12*y^4-3*x^11*y^5+3*x ^10*y^6-x^9*y^7+x^12*y^3+x^11*y^4-12*x^10*y^5-4*x^9*y^6+7*x^10*y^4+19*x^9*y^5-3 *x^8*y^6+5*x^10*y^3-14*x^9*y^4-4*x^10*y^2-10*x^9*y^3+8*x^8*y^4-3*x^7*y^5+12*x^9 *y^2+3*x^7*y^4-2*x^6*y^5-2*x^9*y-10*x^8*y^2+3*x^7*y^3+x^6*y^4-x^5*y^5+6*x^8*y-x ^8-6*x^7*y+2*x^6*y^2-5*x^5*y^3-3*x^4*y^4+3*x^7+2*x^6*y+4*x^5*y^2+14*x^4*y^3-3*x ^6-12*x^4*y^2-8*x^3*y^3+x^5+2*x^4*y+12*x^3*y^2-6*x^3*y-4*x^2*y^2+x^3+6*x^2*y-3* x^2-2*x*y+3*x-1)/(x^8*y^4-x^7*y^4-2*x^8*y^2+4*x^7*y^2+x^8-2*x^6*y^2-3*x^7+3*x^6 -x^5-x^3+3*x^2-3*x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3012555362 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 137" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 9 13 8 13 7 12 8 ) | ) C(m, n) x y | = - (x y - 3 x y + 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 6 12 7 13 5 12 6 11 7 13 4 12 5 + 2 x y - 2 x y - 3 x y - x y - x y + x y + 5 x y 11 6 10 7 12 4 11 5 10 6 9 7 12 3 + 3 x y + x y - 4 x y - 3 x y + 3 x y - x y + x y 11 4 10 5 9 6 10 4 9 5 8 6 10 3 + x y - 12 x y - 4 x y + 7 x y + 19 x y - 3 x y + 5 x y 9 4 10 2 9 3 8 4 7 5 9 2 7 4 - 14 x y - 4 x y - 10 x y + 8 x y - 3 x y + 12 x y + 3 x y 6 5 9 8 2 7 3 6 4 5 5 8 8 - 2 x y - 2 x y - 10 x y + 3 x y + x y - x y + 6 x y - x 7 6 2 5 3 4 4 7 6 5 2 - 6 x y + 2 x y - 5 x y - 3 x y + 3 x + 2 x y + 4 x y 4 3 6 4 2 3 3 5 4 3 2 3 + 14 x y - 3 x - 12 x y - 8 x y + x + 2 x y + 12 x y - 6 x y 2 2 3 2 2 / - 4 x y + x + 6 x y - 3 x - 2 x y + 3 x - 1) / ((x - 1) / 7 4 7 2 6 2 7 6 5 2 (x y - 2 x y + 2 x y + x - 2 x + x - x + 2 x - 1)) and in Maple notation -(x^13*y^9-3*x^13*y^8+2*x^13*y^7+x^12*y^8+2*x^13*y^6-2*x^12*y^7-3*x^13*y^5-x^12 *y^6-x^11*y^7+x^13*y^4+5*x^12*y^5+3*x^11*y^6+x^10*y^7-4*x^12*y^4-3*x^11*y^5+3*x ^10*y^6-x^9*y^7+x^12*y^3+x^11*y^4-12*x^10*y^5-4*x^9*y^6+7*x^10*y^4+19*x^9*y^5-3 *x^8*y^6+5*x^10*y^3-14*x^9*y^4-4*x^10*y^2-10*x^9*y^3+8*x^8*y^4-3*x^7*y^5+12*x^9 *y^2+3*x^7*y^4-2*x^6*y^5-2*x^9*y-10*x^8*y^2+3*x^7*y^3+x^6*y^4-x^5*y^5+6*x^8*y-x ^8-6*x^7*y+2*x^6*y^2-5*x^5*y^3-3*x^4*y^4+3*x^7+2*x^6*y+4*x^5*y^2+14*x^4*y^3-3*x ^6-12*x^4*y^2-8*x^3*y^3+x^5+2*x^4*y+12*x^3*y^2-6*x^3*y-4*x^2*y^2+x^3+6*x^2*y-3* x^2-2*x*y+3*x-1)/(x-1)/(x^7*y^4-2*x^7*y^2+2*x^6*y^2+x^7-2*x^6+x^5-x^2+2*x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.3012555362 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 138" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 14 7 13 8 14 6 ) | ) C(m, n) x y | = (2 x y - 8 x y - 2 x y + 12 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 14 5 13 6 12 7 14 4 + 10 x y + x y - 8 x y - 20 x y - 6 x y + 2 x y 13 5 12 6 11 7 13 4 12 5 11 6 + 20 x y + 15 x y + 2 x y - 10 x y - 20 x y - 2 x y 13 3 12 4 11 5 10 6 12 3 11 4 + 2 x y + 15 x y - 4 x y - 7 x y - 6 x y + 4 x y 10 5 9 6 12 2 11 3 10 4 9 5 + 24 x y + 4 x y + x y + 2 x y - 30 x y - 16 x y 11 2 10 3 9 4 8 5 7 6 10 2 - 2 x y + 16 x y + 26 x y + 2 x y - 2 x y - 3 x y 9 3 8 4 7 5 6 6 9 2 7 4 6 5 - 22 x y - x y + 10 x y - x y + 10 x y - 20 x y - 6 x y 9 8 2 7 3 6 4 5 5 8 7 2 - 2 x y - 5 x y + 18 x y + 20 x y - 2 x y + 4 x y - 6 x y 6 3 6 2 5 3 4 4 6 5 2 4 3 - 20 x y + 10 x y + 8 x y - 6 x y - 4 x y - 8 x y + 10 x y 6 5 4 2 3 3 5 4 3 2 4 2 2 + x + 4 x y - x y - 8 x y - 2 x - 4 x y + 8 x y + x - 4 x y 2 2 / 12 4 12 3 11 4 + 4 x y - x - 2 x y + 2 x - 1) / (x y - 2 x y - 2 x y / 12 2 11 3 10 4 11 2 10 3 10 2 9 2 + x y + 4 x y + x y - 2 x y - 2 x y + x y + 2 x y 9 8 2 8 7 2 7 6 5 5 - 2 x y - 4 x y + 4 x y + 2 x y - 2 x y + x + 2 x y - 2 x 4 4 3 2 - 4 x y + x + 2 x y - x + 2 x - 1) and in Maple notation (2*x^14*y^8-8*x^14*y^7-2*x^13*y^8+12*x^14*y^6+10*x^13*y^7+x^12*y^8-8*x^14*y^5-\ 20*x^13*y^6-6*x^12*y^7+2*x^14*y^4+20*x^13*y^5+15*x^12*y^6+2*x^11*y^7-10*x^13*y^ 4-20*x^12*y^5-2*x^11*y^6+2*x^13*y^3+15*x^12*y^4-4*x^11*y^5-7*x^10*y^6-6*x^12*y^ 3+4*x^11*y^4+24*x^10*y^5+4*x^9*y^6+x^12*y^2+2*x^11*y^3-30*x^10*y^4-16*x^9*y^5-2 *x^11*y^2+16*x^10*y^3+26*x^9*y^4+2*x^8*y^5-2*x^7*y^6-3*x^10*y^2-22*x^9*y^3-x^8* y^4+10*x^7*y^5-x^6*y^6+10*x^9*y^2-20*x^7*y^4-6*x^6*y^5-2*x^9*y-5*x^8*y^2+18*x^7 *y^3+20*x^6*y^4-2*x^5*y^5+4*x^8*y-6*x^7*y^2-20*x^6*y^3+10*x^6*y^2+8*x^5*y^3-6*x ^4*y^4-4*x^6*y-8*x^5*y^2+10*x^4*y^3+x^6+4*x^5*y-x^4*y^2-8*x^3*y^3-2*x^5-4*x^4*y +8*x^3*y^2+x^4-4*x^2*y^2+4*x^2*y-x^2-2*x*y+2*x-1)/(x^12*y^4-2*x^12*y^3-2*x^11*y ^4+x^12*y^2+4*x^11*y^3+x^10*y^4-2*x^11*y^2-2*x^10*y^3+x^10*y^2+2*x^9*y^2-2*x^9* y-4*x^8*y^2+4*x^8*y+2*x^7*y^2-2*x^7*y+x^6+2*x^5*y-2*x^5-4*x^4*y+x^4+2*x^3*y-x^2 +2*x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2889272911 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 139" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 8 13 7 13 6 12 7 ) | ) C(m, n) x y | = (x y - 6 x y + 12 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 5 12 6 11 7 13 4 12 5 11 6 - 10 x y - 11 x y + 2 x y + 3 x y + 15 x y - 5 x y 10 7 12 4 11 5 10 6 12 3 11 4 10 5 + x y - 9 x y + 2 x y + x y + 2 x y + 4 x y - 13 x y 9 6 11 3 10 4 9 5 11 2 10 3 9 4 + x y - 4 x y + 18 x y + x y + x y - 6 x y - 9 x y 8 5 10 2 9 3 7 5 9 2 8 3 7 4 6 5 + x y - x y + 11 x y - x y - 4 x y - 5 x y + 4 x y + x y 8 2 7 3 6 4 8 7 2 6 3 5 4 + 6 x y - 7 x y - 8 x y - 2 x y + 2 x y + 9 x y + 5 x y 7 6 2 5 3 4 4 6 5 2 4 3 + 2 x y - 4 x y - 6 x y + 3 x y + 2 x y + 4 x y + 2 x y 5 4 2 3 3 5 4 3 2 4 3 - 2 x y - 4 x y + 8 x y + x + 2 x y - 4 x y - x - 2 x y 2 2 2 / 11 4 11 3 10 4 + 4 x y - 2 x y + 2 x y - x + 1) / (x y - 2 x y - x y / 11 2 10 3 10 2 8 2 8 7 2 7 5 + x y + 2 x y - x y + 2 x y - 2 x y - 2 x y + 2 x y + x 4 4 3 + 2 x y - x - 2 x y - x + 1) and in Maple notation (x^13*y^8-6*x^13*y^7+12*x^13*y^6+3*x^12*y^7-10*x^13*y^5-11*x^12*y^6+2*x^11*y^7+ 3*x^13*y^4+15*x^12*y^5-5*x^11*y^6+x^10*y^7-9*x^12*y^4+2*x^11*y^5+x^10*y^6+2*x^ 12*y^3+4*x^11*y^4-13*x^10*y^5+x^9*y^6-4*x^11*y^3+18*x^10*y^4+x^9*y^5+x^11*y^2-6 *x^10*y^3-9*x^9*y^4+x^8*y^5-x^10*y^2+11*x^9*y^3-x^7*y^5-4*x^9*y^2-5*x^8*y^3+4*x ^7*y^4+x^6*y^5+6*x^8*y^2-7*x^7*y^3-8*x^6*y^4-2*x^8*y+2*x^7*y^2+9*x^6*y^3+5*x^5* y^4+2*x^7*y-4*x^6*y^2-6*x^5*y^3+3*x^4*y^4+2*x^6*y+4*x^5*y^2+2*x^4*y^3-2*x^5*y-4 *x^4*y^2+8*x^3*y^3+x^5+2*x^4*y-4*x^3*y^2-x^4-2*x^3*y+4*x^2*y^2-2*x^2*y+2*x*y-x+ 1)/(x^11*y^4-2*x^11*y^3-x^10*y^4+x^11*y^2+2*x^10*y^3-x^10*y^2+2*x^8*y^2-2*x^8*y -2*x^7*y^2+2*x^7*y+x^5+2*x^4*y-x^4-2*x^3*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2875859755 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 140" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2], [1, 2, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 8 13 7 13 6 12 7 ) | ) C(m, n) x y | = (x y - 6 x y + 12 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 5 12 6 11 7 13 4 12 5 11 6 - 10 x y - 11 x y + 2 x y + 3 x y + 15 x y - 5 x y 10 7 12 4 11 5 10 6 12 3 11 4 10 5 + x y - 9 x y + 2 x y + x y + 2 x y + 4 x y - 13 x y 9 6 11 3 10 4 9 5 11 2 10 3 9 4 + x y - 4 x y + 18 x y + x y + x y - 6 x y - 9 x y 8 5 10 2 9 3 7 5 9 2 8 3 7 4 6 5 + x y - x y + 11 x y - x y - 4 x y - 5 x y + 4 x y + x y 8 2 7 3 6 4 8 7 2 6 3 5 4 + 6 x y - 7 x y - 8 x y - 2 x y + 2 x y + 9 x y + 5 x y 7 6 2 5 3 4 4 6 5 2 4 3 + 2 x y - 4 x y - 6 x y + 3 x y + 2 x y + 4 x y + 2 x y 5 4 2 3 3 5 4 3 2 4 3 - 2 x y - 4 x y + 8 x y + x + 2 x y - 4 x y - x - 2 x y 2 2 2 / 11 4 11 3 10 4 + 4 x y - 2 x y + 2 x y - x + 1) / (x y - 2 x y - x y / 11 2 10 3 10 2 8 2 8 7 2 7 5 + x y + 2 x y - x y + 2 x y - 2 x y - 2 x y + 2 x y + x 4 4 3 + 2 x y - x - 2 x y - x + 1) and in Maple notation (x^13*y^8-6*x^13*y^7+12*x^13*y^6+3*x^12*y^7-10*x^13*y^5-11*x^12*y^6+2*x^11*y^7+ 3*x^13*y^4+15*x^12*y^5-5*x^11*y^6+x^10*y^7-9*x^12*y^4+2*x^11*y^5+x^10*y^6+2*x^ 12*y^3+4*x^11*y^4-13*x^10*y^5+x^9*y^6-4*x^11*y^3+18*x^10*y^4+x^9*y^5+x^11*y^2-6 *x^10*y^3-9*x^9*y^4+x^8*y^5-x^10*y^2+11*x^9*y^3-x^7*y^5-4*x^9*y^2-5*x^8*y^3+4*x ^7*y^4+x^6*y^5+6*x^8*y^2-7*x^7*y^3-8*x^6*y^4-2*x^8*y+2*x^7*y^2+9*x^6*y^3+5*x^5* y^4+2*x^7*y-4*x^6*y^2-6*x^5*y^3+3*x^4*y^4+2*x^6*y+4*x^5*y^2+2*x^4*y^3-2*x^5*y-4 *x^4*y^2+8*x^3*y^3+x^5+2*x^4*y-4*x^3*y^2-x^4-2*x^3*y+4*x^2*y^2-2*x^2*y+2*x*y-x+ 1)/(x^11*y^4-2*x^11*y^3-x^10*y^4+x^11*y^2+2*x^10*y^3-x^10*y^2+2*x^8*y^2-2*x^8*y -2*x^7*y^2+2*x^7*y+x^5+2*x^4*y-x^4-2*x^3*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2875859755 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 141" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 14 7 13 8 14 6 ) | ) C(m, n) x y | = - (2 x y - 4 x y - 2 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 12 8 14 5 13 6 12 7 11 8 14 4 + 2 x y - x y + 8 x y + 8 x y + 4 x y + 2 x y - 8 x y 13 5 12 6 11 7 10 8 14 3 13 4 - 16 x y - 7 x y - 6 x y + x y + 4 x y + 10 x y 12 5 11 6 14 2 13 3 12 4 11 5 10 6 + 8 x y + 4 x y - x y - 2 x y - 7 x y + 6 x y - 7 x y 9 7 12 3 11 4 9 6 12 2 11 3 10 4 + 2 x y + 4 x y - 6 x y - 2 x y - x y - 4 x y + 20 x y 9 5 8 6 11 2 10 3 9 4 8 5 10 2 + 4 x y + 3 x y + 4 x y - 18 x y - 16 x y - 8 x y + 4 x y 9 3 8 4 9 2 8 3 7 4 9 8 2 + 20 x y + 8 x y - 10 x y - 2 x y + 8 x y + 2 x y + 4 x y 7 3 6 4 8 7 2 6 3 5 4 6 2 - 12 x y - 15 x y - 4 x y + 4 x y + 20 x y + 4 x y - 10 x y 5 3 4 4 6 5 2 4 3 6 5 4 2 - 10 x y + 3 x y + 4 x y + 8 x y - 8 x y - x - 4 x y + 2 x y 3 3 5 4 3 2 4 2 2 2 2 + 8 x y + 2 x + 4 x y - 8 x y - x + 4 x y - 4 x y + x + 2 x y / 2 - 2 x + 1) / ((x - 2 x + 1) / 10 4 10 3 10 2 7 2 7 4 3 (x y - 2 x y + x y + 2 x y - 2 x y + x + 2 x y - 1)) and in Maple notation -(2*x^14*y^8-4*x^14*y^7-2*x^13*y^8-x^14*y^6+2*x^13*y^7-x^12*y^8+8*x^14*y^5+8*x^ 13*y^6+4*x^12*y^7+2*x^11*y^8-8*x^14*y^4-16*x^13*y^5-7*x^12*y^6-6*x^11*y^7+x^10* y^8+4*x^14*y^3+10*x^13*y^4+8*x^12*y^5+4*x^11*y^6-x^14*y^2-2*x^13*y^3-7*x^12*y^4 +6*x^11*y^5-7*x^10*y^6+2*x^9*y^7+4*x^12*y^3-6*x^11*y^4-2*x^9*y^6-x^12*y^2-4*x^ 11*y^3+20*x^10*y^4+4*x^9*y^5+3*x^8*y^6+4*x^11*y^2-18*x^10*y^3-16*x^9*y^4-8*x^8* y^5+4*x^10*y^2+20*x^9*y^3+8*x^8*y^4-10*x^9*y^2-2*x^8*y^3+8*x^7*y^4+2*x^9*y+4*x^ 8*y^2-12*x^7*y^3-15*x^6*y^4-4*x^8*y+4*x^7*y^2+20*x^6*y^3+4*x^5*y^4-10*x^6*y^2-\ 10*x^5*y^3+3*x^4*y^4+4*x^6*y+8*x^5*y^2-8*x^4*y^3-x^6-4*x^5*y+2*x^4*y^2+8*x^3*y^ 3+2*x^5+4*x^4*y-8*x^3*y^2-x^4+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+1)/(x^2-2*x+1)/(x ^10*y^4-2*x^10*y^3+x^10*y^2+2*x^7*y^2-2*x^7*y+x^4+2*x^3*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2870920079 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 142" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 8 17 9 16 10 18 7 ) | ) C(m, n) x y | = (x y + 2 x y + x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 16 9 18 6 17 7 16 8 14 10 - 8 x y - 4 x y + 6 x y + 12 x y + 7 x y - x y 18 5 17 6 16 7 15 8 14 9 18 4 - 4 x y - 8 x y - 10 x y - 5 x y + 2 x y + x y 17 5 16 6 15 7 14 8 13 9 16 5 + 2 x y + 13 x y + 15 x y - x y - 2 x y - 10 x y 15 6 14 7 13 8 12 9 16 4 15 5 - 15 x y - 3 x y + 2 x y - 2 x y + 3 x y + 5 x y 14 6 13 7 12 8 14 5 13 6 12 7 + 11 x y + 12 x y + 6 x y - 13 x y - 25 x y - 8 x y 11 8 14 4 13 5 12 6 11 7 13 4 - 4 x y + 5 x y + 16 x y + 14 x y + 8 x y - 3 x y 12 5 11 6 10 7 12 4 11 5 10 6 - 21 x y - 7 x y - 2 x y + 14 x y + 8 x y + x y 9 7 8 8 12 3 11 4 10 5 9 6 11 3 + 3 x y + x y - 3 x y - 7 x y + 7 x y - 4 x y + 2 x y 10 4 9 5 8 6 7 7 10 3 9 4 8 5 - 5 x y + 5 x y + 6 x y + 2 x y - 3 x y + x y - 6 x y 7 6 10 2 9 3 8 4 7 5 6 6 9 2 + 3 x y + 2 x y - 7 x y - 3 x y - 3 x y + 2 x y + x y 8 3 7 4 6 5 9 8 2 7 3 6 4 5 5 + x y - x y + 2 x y + x y - x y - 6 x y - 12 x y + 4 x y 8 7 2 6 3 5 4 7 6 2 5 3 4 4 + x y + 2 x y + 2 x y - 5 x y + x y + 2 x y - 11 x y - x y 5 2 4 3 5 4 2 3 3 4 2 2 3 + 6 x y - 4 x y + x y - 4 x y - 6 x y + 4 x y - 3 x y + x / 10 4 10 3 10 2 9 3 8 4 9 2 - 2 x y - 1) / (x y - 2 x y + x y + x y - x y - 2 x y / 9 8 2 7 3 6 3 7 6 2 5 2 5 + x y + x y - x y - 2 x y + x y + 3 x y - x y + 2 x y 4 2 2 3 + 2 x y + x y + x - 1) and in Maple notation (x^18*y^8+2*x^17*y^9+x^16*y^10-4*x^18*y^7-8*x^17*y^8-4*x^16*y^9+6*x^18*y^6+12*x ^17*y^7+7*x^16*y^8-x^14*y^10-4*x^18*y^5-8*x^17*y^6-10*x^16*y^7-5*x^15*y^8+2*x^ 14*y^9+x^18*y^4+2*x^17*y^5+13*x^16*y^6+15*x^15*y^7-x^14*y^8-2*x^13*y^9-10*x^16* y^5-15*x^15*y^6-3*x^14*y^7+2*x^13*y^8-2*x^12*y^9+3*x^16*y^4+5*x^15*y^5+11*x^14* y^6+12*x^13*y^7+6*x^12*y^8-13*x^14*y^5-25*x^13*y^6-8*x^12*y^7-4*x^11*y^8+5*x^14 *y^4+16*x^13*y^5+14*x^12*y^6+8*x^11*y^7-3*x^13*y^4-21*x^12*y^5-7*x^11*y^6-2*x^ 10*y^7+14*x^12*y^4+8*x^11*y^5+x^10*y^6+3*x^9*y^7+x^8*y^8-3*x^12*y^3-7*x^11*y^4+ 7*x^10*y^5-4*x^9*y^6+2*x^11*y^3-5*x^10*y^4+5*x^9*y^5+6*x^8*y^6+2*x^7*y^7-3*x^10 *y^3+x^9*y^4-6*x^8*y^5+3*x^7*y^6+2*x^10*y^2-7*x^9*y^3-3*x^8*y^4-3*x^7*y^5+2*x^6 *y^6+x^9*y^2+x^8*y^3-x^7*y^4+2*x^6*y^5+x^9*y-x^8*y^2-6*x^7*y^3-12*x^6*y^4+4*x^5 *y^5+x^8*y+2*x^7*y^2+2*x^6*y^3-5*x^5*y^4+x^7*y+2*x^6*y^2-11*x^5*y^3-x^4*y^4+6*x ^5*y^2-4*x^4*y^3+x^5*y-4*x^4*y^2-6*x^3*y^3+4*x^4*y-3*x^2*y^2+x^3-2*x*y-1)/(x^10 *y^4-2*x^10*y^3+x^10*y^2+x^9*y^3-x^8*y^4-2*x^9*y^2+x^9*y+x^8*y^2-x^7*y^3-2*x^6* y^3+x^7*y+3*x^6*y^2-x^5*y^2+2*x^5*y+2*x^4*y+x^2*y^2+x^3-1) As the length of the word goes to infinity, the average number of good neigh\ 14 13 bors of a random word of length n tends to n times, 2 (2 a + 2 a 12 11 10 9 8 7 6 5 4 + 2 a - 8 a - 20 a - 38 a - 49 a - 46 a - 33 a - 10 a + 3 a 3 2 / 12 11 10 9 8 + 10 a + 10 a + 4 a + 1) / (a (6 a + 17 a + 42 a + 69 a + 95 a / 7 6 5 4 3 2 + 111 a + 106 a + 92 a + 65 a + 40 a + 20 a + 7 a + 2)) 6 5 4 3 2 where a is the root of the polynomial, x + x + 2 x + x + x - 1, and in decimals this is, 0.2835283350 BTW the ratio for words with, 500, letters is, 0.2868639547 ------------------------------------------------ "Theorem Number 143" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 15 9 16 7 15 8 ) | ) C(m, n) x y | = (x y + x y - 4 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 6 15 7 14 8 16 5 15 6 14 7 + 6 x y + 3 x y + 3 x y - 4 x y - x y - 10 x y 13 8 16 4 14 6 13 7 12 8 14 5 13 6 - 2 x y + x y + 12 x y + 9 x y - x y - 6 x y - 16 x y 12 7 11 8 14 4 13 5 12 6 11 7 13 4 - x y - 2 x y + x y + 14 x y + 7 x y + 2 x y - 6 x y 12 5 11 6 10 7 13 3 12 4 11 5 10 6 - 6 x y - x y - 4 x y + x y - x y + 5 x y + 4 x y 12 3 11 4 10 5 9 6 12 2 11 3 10 4 + 3 x y - 3 x y - x y - 7 x y - x y - 3 x y + 6 x y 9 5 8 6 11 2 10 3 9 4 8 5 7 6 + 12 x y + 2 x y + 2 x y - 7 x y - 12 x y - 9 x y + 2 x y 10 2 9 3 8 4 7 5 6 6 9 2 8 3 + 2 x y + 12 x y + 11 x y + 3 x y + 2 x y - 5 x y - 7 x y 7 4 6 5 8 2 6 4 5 5 8 7 2 6 3 - 5 x y + 2 x y + 5 x y - x y + 3 x y - 2 x y - x y + x y 5 4 7 6 2 5 3 4 4 7 6 5 2 + 2 x y + 3 x y - 4 x y - 7 x y + 5 x y - x + x y + 4 x y 4 3 6 5 4 2 3 3 5 4 3 2 4 - 2 x y + x - 2 x y - 2 x y + 8 x y + x + 2 x y - 4 x y - x 3 2 2 3 2 2 / 11 5 - 2 x y + 4 x y + x - 2 x y - x + 2 x y - x + 1) / (x y / 11 4 11 3 10 4 11 2 10 3 10 2 7 3 - 3 x y + 3 x y + x y - x y - 2 x y + x y - x y 7 2 6 3 7 6 2 7 6 5 2 6 5 4 3 + x y - x y + x y + x y - x - x y - 2 x y + x + x - x + x 2 - x - x + 1) and in Maple notation (x^16*y^8+x^15*y^9-4*x^16*y^7-3*x^15*y^8+6*x^16*y^6+3*x^15*y^7+3*x^14*y^8-4*x^ 16*y^5-x^15*y^6-10*x^14*y^7-2*x^13*y^8+x^16*y^4+12*x^14*y^6+9*x^13*y^7-x^12*y^8 -6*x^14*y^5-16*x^13*y^6-x^12*y^7-2*x^11*y^8+x^14*y^4+14*x^13*y^5+7*x^12*y^6+2*x ^11*y^7-6*x^13*y^4-6*x^12*y^5-x^11*y^6-4*x^10*y^7+x^13*y^3-x^12*y^4+5*x^11*y^5+ 4*x^10*y^6+3*x^12*y^3-3*x^11*y^4-x^10*y^5-7*x^9*y^6-x^12*y^2-3*x^11*y^3+6*x^10* y^4+12*x^9*y^5+2*x^8*y^6+2*x^11*y^2-7*x^10*y^3-12*x^9*y^4-9*x^8*y^5+2*x^7*y^6+2 *x^10*y^2+12*x^9*y^3+11*x^8*y^4+3*x^7*y^5+2*x^6*y^6-5*x^9*y^2-7*x^8*y^3-5*x^7*y ^4+2*x^6*y^5+5*x^8*y^2-x^6*y^4+3*x^5*y^5-2*x^8*y-x^7*y^2+x^6*y^3+2*x^5*y^4+3*x^ 7*y-4*x^6*y^2-7*x^5*y^3+5*x^4*y^4-x^7+x^6*y+4*x^5*y^2-2*x^4*y^3+x^6-2*x^5*y-2*x ^4*y^2+8*x^3*y^3+x^5+2*x^4*y-4*x^3*y^2-x^4-2*x^3*y+4*x^2*y^2+x^3-2*x^2*y-x^2+2* x*y-x+1)/(x^11*y^5-3*x^11*y^4+3*x^11*y^3+x^10*y^4-x^11*y^2-2*x^10*y^3+x^10*y^2- x^7*y^3+x^7*y^2-x^6*y^3+x^7*y+x^6*y^2-x^7-x^6*y-2*x^5*y^2+x^6+x^5-x^4+x^3-x^2-x +1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2797346646 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 144" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 2, 1], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 10 18 9 18 8 17 8 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 9 18 6 17 7 16 8 15 9 18 5 17 6 - x y - 5 x y + 8 x y + x y - 2 x y + 4 x y - 12 x y 16 7 15 8 18 4 17 5 16 6 15 7 14 8 + 6 x y + 6 x y - x y + 8 x y - 14 x y - 3 x y + x y 17 4 16 5 15 6 14 7 13 8 16 4 - 2 x y + 11 x y - 6 x y + x y + 2 x y - 3 x y 15 5 14 6 13 7 12 8 14 5 13 6 + 6 x y - 9 x y + 3 x y + x y + 11 x y - 20 x y 12 7 15 3 14 4 13 5 12 6 11 7 10 8 - 2 x y - x y - 4 x y + 21 x y - 6 x y - x y - x y 13 4 12 5 11 6 10 7 13 3 12 4 - 3 x y + 14 x y - 6 x y - x y - 4 x y - 6 x y 11 5 9 7 13 2 12 3 11 4 10 5 9 6 + 15 x y - 2 x y + x y - 2 x y - 5 x y + 11 x y + 2 x y 12 2 11 3 10 4 9 5 8 6 7 7 11 2 + x y - 5 x y - 11 x y + 9 x y + x y + x y + x y 9 4 8 5 7 6 11 10 2 9 3 8 4 - 10 x y + 6 x y + x y + x y + 2 x y - 4 x y - 11 x y 7 5 6 6 9 2 8 3 7 4 6 5 9 + 6 x y + 2 x y + 7 x y + 2 x y - 11 x y - 2 x y - 2 x y 8 2 6 4 5 5 7 2 6 3 5 4 6 2 + 2 x y - 7 x y - x y + 3 x y + 3 x y - 2 x y + 3 x y 5 3 4 4 5 2 4 3 5 4 2 3 3 4 + x y - 6 x y + 3 x y + 3 x y - x y + 3 x y + 2 x y - x y 3 2 2 / 10 3 10 2 9 3 10 9 2 - x y + x y + 1) / ((x y - 2 x y + 2 x y + x y - 4 x y / 8 3 9 8 2 8 7 2 7 6 2 7 6 + x y + 2 x y - 2 x y + x y - 2 x y + x y - x y + x + x y 5 2 5 4 2 4 3 - 2 x y + 2 x y - x y + 2 x y + x y + x y - 1) (x y - 1)) and in Maple notation (x^18*y^10-4*x^18*y^9+5*x^18*y^8-2*x^17*y^8-x^16*y^9-5*x^18*y^6+8*x^17*y^7+x^16 *y^8-2*x^15*y^9+4*x^18*y^5-12*x^17*y^6+6*x^16*y^7+6*x^15*y^8-x^18*y^4+8*x^17*y^ 5-14*x^16*y^6-3*x^15*y^7+x^14*y^8-2*x^17*y^4+11*x^16*y^5-6*x^15*y^6+x^14*y^7+2* x^13*y^8-3*x^16*y^4+6*x^15*y^5-9*x^14*y^6+3*x^13*y^7+x^12*y^8+11*x^14*y^5-20*x^ 13*y^6-2*x^12*y^7-x^15*y^3-4*x^14*y^4+21*x^13*y^5-6*x^12*y^6-x^11*y^7-x^10*y^8-\ 3*x^13*y^4+14*x^12*y^5-6*x^11*y^6-x^10*y^7-4*x^13*y^3-6*x^12*y^4+15*x^11*y^5-2* x^9*y^7+x^13*y^2-2*x^12*y^3-5*x^11*y^4+11*x^10*y^5+2*x^9*y^6+x^12*y^2-5*x^11*y^ 3-11*x^10*y^4+9*x^9*y^5+x^8*y^6+x^7*y^7+x^11*y^2-10*x^9*y^4+6*x^8*y^5+x^7*y^6+x ^11*y+2*x^10*y^2-4*x^9*y^3-11*x^8*y^4+6*x^7*y^5+2*x^6*y^6+7*x^9*y^2+2*x^8*y^3-\ 11*x^7*y^4-2*x^6*y^5-2*x^9*y+2*x^8*y^2-7*x^6*y^4-x^5*y^5+3*x^7*y^2+3*x^6*y^3-2* x^5*y^4+3*x^6*y^2+x^5*y^3-6*x^4*y^4+3*x^5*y^2+3*x^4*y^3-x^5*y+3*x^4*y^2+2*x^3*y ^3-x^4*y-x^3*y+x^2*y^2+1)/(x^10*y^3-2*x^10*y^2+2*x^9*y^3+x^10*y-4*x^9*y^2+x^8*y ^3+2*x^9*y-2*x^8*y^2+x^8*y-2*x^7*y^2+x^7*y-x^6*y^2+x^7+x^6*y-2*x^5*y^2+2*x^5*y- x^4*y^2+2*x^4*y+x^3*y+x*y-1)/(x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2727503097 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 145" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [2, 1, 1, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 10 18 9 18 8 17 8 ) | ) C(m, n) x y | = (x y - 4 x y + 5 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 9 18 6 17 7 16 8 15 9 18 5 17 6 - x y - 5 x y + 8 x y + x y - 2 x y + 4 x y - 12 x y 16 7 15 8 18 4 17 5 16 6 15 7 14 8 + 6 x y + 6 x y - x y + 8 x y - 14 x y - 3 x y + x y 17 4 16 5 15 6 14 7 13 8 16 4 - 2 x y + 11 x y - 6 x y + x y + 2 x y - 3 x y 15 5 14 6 13 7 12 8 14 5 13 6 + 6 x y - 9 x y + 3 x y + x y + 11 x y - 20 x y 12 7 15 3 14 4 13 5 12 6 11 7 10 8 - 2 x y - x y - 4 x y + 21 x y - 6 x y - x y - x y 13 4 12 5 11 6 10 7 13 3 12 4 - 3 x y + 14 x y - 6 x y - x y - 4 x y - 6 x y 11 5 9 7 13 2 12 3 11 4 10 5 9 6 + 15 x y - 2 x y + x y - 2 x y - 5 x y + 11 x y + 2 x y 12 2 11 3 10 4 9 5 8 6 7 7 11 2 + x y - 5 x y - 11 x y + 9 x y + x y + x y + x y 9 4 8 5 7 6 11 10 2 9 3 8 4 - 10 x y + 6 x y + x y + x y + 2 x y - 4 x y - 11 x y 7 5 6 6 9 2 8 3 7 4 6 5 9 + 6 x y + 2 x y + 7 x y + 2 x y - 11 x y - 2 x y - 2 x y 8 2 6 4 5 5 7 2 6 3 5 4 6 2 + 2 x y - 7 x y - x y + 3 x y + 3 x y - 2 x y + 3 x y 5 3 4 4 5 2 4 3 5 4 2 3 3 4 + x y - 6 x y + 3 x y + 3 x y - x y + 3 x y + 2 x y - x y 3 2 2 / 10 3 10 2 9 3 10 9 2 - x y + x y + 1) / ((x y - 2 x y + 2 x y + x y - 4 x y / 8 3 9 8 2 8 7 2 7 6 2 7 6 + x y + 2 x y - 2 x y + x y - 2 x y + x y - x y + x + x y 5 2 5 4 2 4 3 - 2 x y + 2 x y - x y + 2 x y + x y + x y - 1) (x y - 1)) and in Maple notation (x^18*y^10-4*x^18*y^9+5*x^18*y^8-2*x^17*y^8-x^16*y^9-5*x^18*y^6+8*x^17*y^7+x^16 *y^8-2*x^15*y^9+4*x^18*y^5-12*x^17*y^6+6*x^16*y^7+6*x^15*y^8-x^18*y^4+8*x^17*y^ 5-14*x^16*y^6-3*x^15*y^7+x^14*y^8-2*x^17*y^4+11*x^16*y^5-6*x^15*y^6+x^14*y^7+2* x^13*y^8-3*x^16*y^4+6*x^15*y^5-9*x^14*y^6+3*x^13*y^7+x^12*y^8+11*x^14*y^5-20*x^ 13*y^6-2*x^12*y^7-x^15*y^3-4*x^14*y^4+21*x^13*y^5-6*x^12*y^6-x^11*y^7-x^10*y^8-\ 3*x^13*y^4+14*x^12*y^5-6*x^11*y^6-x^10*y^7-4*x^13*y^3-6*x^12*y^4+15*x^11*y^5-2* x^9*y^7+x^13*y^2-2*x^12*y^3-5*x^11*y^4+11*x^10*y^5+2*x^9*y^6+x^12*y^2-5*x^11*y^ 3-11*x^10*y^4+9*x^9*y^5+x^8*y^6+x^7*y^7+x^11*y^2-10*x^9*y^4+6*x^8*y^5+x^7*y^6+x ^11*y+2*x^10*y^2-4*x^9*y^3-11*x^8*y^4+6*x^7*y^5+2*x^6*y^6+7*x^9*y^2+2*x^8*y^3-\ 11*x^7*y^4-2*x^6*y^5-2*x^9*y+2*x^8*y^2-7*x^6*y^4-x^5*y^5+3*x^7*y^2+3*x^6*y^3-2* x^5*y^4+3*x^6*y^2+x^5*y^3-6*x^4*y^4+3*x^5*y^2+3*x^4*y^3-x^5*y+3*x^4*y^2+2*x^3*y ^3-x^4*y-x^3*y+x^2*y^2+1)/(x^10*y^3-2*x^10*y^2+2*x^9*y^3+x^10*y-4*x^9*y^2+x^8*y ^3+2*x^9*y-2*x^8*y^2+x^8*y-2*x^7*y^2+x^7*y-x^6*y^2+x^7+x^6*y-2*x^5*y^2+2*x^5*y- x^4*y^2+2*x^4*y+x^3*y+x*y-1)/(x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2727503097 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 146" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 17 8 17 7 ) | ) C(m, n) x y | = - (2 x y - 10 x y + 20 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 8 15 9 17 6 16 7 15 8 17 5 - 2 x y + 2 x y - 20 x y + 8 x y - 13 x y + 10 x y 16 6 15 7 14 8 17 4 16 5 15 6 - 12 x y + 31 x y - x y - 2 x y + 8 x y - 34 x y 14 7 16 4 15 5 14 6 13 7 12 8 15 4 - x y - 2 x y + 16 x y + 14 x y + 5 x y - 2 x y - x y 14 5 13 6 12 7 15 3 14 4 13 5 - 26 x y - 18 x y + 9 x y - x y + 19 x y + 23 x y 12 6 11 7 14 3 13 4 12 5 11 6 - 12 x y - 2 x y - 5 x y - 11 x y - x y + 5 x y 12 4 11 5 10 6 13 2 12 3 11 4 + 15 x y + x y + 7 x y + x y - 12 x y - 13 x y 10 5 9 6 12 2 11 3 10 4 9 5 - 17 x y + 2 x y + 3 x y + 13 x y + 13 x y - 3 x y 11 2 10 3 9 4 8 5 10 2 9 3 8 4 - 4 x y - 4 x y - 5 x y - 4 x y + 2 x y + 11 x y + 9 x y 7 5 10 9 2 8 3 7 4 6 5 9 8 2 - 6 x y - x y - 6 x y - 9 x y + 5 x y - x y + x y + 5 x y 7 3 8 7 2 6 3 5 4 7 6 2 4 4 + 7 x y - x y - 9 x y + 2 x y - 2 x y + 3 x y + x y + 3 x y 6 5 2 4 3 6 5 4 2 3 3 5 - 2 x y + 3 x y - 4 x y + x + x y + 2 x y + 8 x y - x 3 2 2 2 2 / 12 5 12 4 - 6 x y + 4 x y - 3 x y + 2 x y - x + 1) / (x y - 3 x y / 11 5 12 3 11 4 12 2 11 3 11 2 9 4 - x y + 3 x y + 3 x y - x y - 3 x y + x y - x y 10 2 9 3 8 4 10 9 2 8 3 9 8 2 8 - x y - x y + x y + x y + 3 x y - x y - x y - x y + x y 7 2 7 6 2 5 2 6 5 5 2 + 2 x y - x y + 2 x y - x y - x - x y + x + x y + x - 1) and in Maple notation -(2*x^17*y^9-10*x^17*y^8+20*x^17*y^7-2*x^16*y^8+2*x^15*y^9-20*x^17*y^6+8*x^16*y ^7-13*x^15*y^8+10*x^17*y^5-12*x^16*y^6+31*x^15*y^7-x^14*y^8-2*x^17*y^4+8*x^16*y ^5-34*x^15*y^6-x^14*y^7-2*x^16*y^4+16*x^15*y^5+14*x^14*y^6+5*x^13*y^7-2*x^12*y^ 8-x^15*y^4-26*x^14*y^5-18*x^13*y^6+9*x^12*y^7-x^15*y^3+19*x^14*y^4+23*x^13*y^5-\ 12*x^12*y^6-2*x^11*y^7-5*x^14*y^3-11*x^13*y^4-x^12*y^5+5*x^11*y^6+15*x^12*y^4+x ^11*y^5+7*x^10*y^6+x^13*y^2-12*x^12*y^3-13*x^11*y^4-17*x^10*y^5+2*x^9*y^6+3*x^ 12*y^2+13*x^11*y^3+13*x^10*y^4-3*x^9*y^5-4*x^11*y^2-4*x^10*y^3-5*x^9*y^4-4*x^8* y^5+2*x^10*y^2+11*x^9*y^3+9*x^8*y^4-6*x^7*y^5-x^10*y-6*x^9*y^2-9*x^8*y^3+5*x^7* y^4-x^6*y^5+x^9*y+5*x^8*y^2+7*x^7*y^3-x^8*y-9*x^7*y^2+2*x^6*y^3-2*x^5*y^4+3*x^7 *y+x^6*y^2+3*x^4*y^4-2*x^6*y+3*x^5*y^2-4*x^4*y^3+x^6+x^5*y+2*x^4*y^2+8*x^3*y^3- x^5-6*x^3*y^2+4*x^2*y^2-3*x^2*y+2*x*y-x+1)/(x^12*y^5-3*x^12*y^4-x^11*y^5+3*x^12 *y^3+3*x^11*y^4-x^12*y^2-3*x^11*y^3+x^11*y^2-x^9*y^4-x^10*y^2-x^9*y^3+x^8*y^4+x ^10*y+3*x^9*y^2-x^8*y^3-x^9*y-x^8*y^2+x^8*y+2*x^7*y^2-x^7*y+2*x^6*y^2-x^5*y^2-x ^6-x^5*y+x^5+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 11 10 9 8 7 6 5 4 3 2 (a - 2 a + 2 a - 2 a - 3 a + 7 a - 7 a + 9 a - 10 a + 2 a + 1) 6 5 4 / 5 3 2 2 (7 a + 6 a - 5 a + 2 a + 1) / ((6 a - 4 a + 3 a - 2 a + 2) / 6 5 4 3 2 (a + a + a + 2 a + a + a + 1)) 7 6 5 2 where a is the root of the polynomial, x + x - x + x + x - 1, and in decimals this is, 0.2676610850 BTW the ratio for words with, 500, letters is, 0.2706383211 ------------------------------------------------ "Theorem Number 147" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 17 8 17 7 ) | ) C(m, n) x y | = - (2 x y - 10 x y + 20 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 8 15 9 17 6 16 7 15 8 17 5 - 2 x y + 2 x y - 20 x y + 8 x y - 13 x y + 10 x y 16 6 15 7 14 8 17 4 16 5 15 6 - 12 x y + 31 x y - x y - 2 x y + 8 x y - 34 x y 14 7 16 4 15 5 14 6 13 7 12 8 15 4 - x y - 2 x y + 16 x y + 14 x y + 5 x y - 2 x y - x y 14 5 13 6 12 7 15 3 14 4 13 5 - 26 x y - 18 x y + 9 x y - x y + 19 x y + 23 x y 12 6 11 7 14 3 13 4 12 5 11 6 - 12 x y - 2 x y - 5 x y - 11 x y - x y + 5 x y 12 4 11 5 10 6 13 2 12 3 11 4 + 15 x y + x y + 7 x y + x y - 12 x y - 13 x y 10 5 9 6 12 2 11 3 10 4 9 5 - 17 x y + 2 x y + 3 x y + 13 x y + 13 x y - 3 x y 11 2 10 3 9 4 8 5 10 2 9 3 8 4 - 4 x y - 4 x y - 5 x y - 4 x y + 2 x y + 11 x y + 9 x y 7 5 10 9 2 8 3 7 4 6 5 9 8 2 - 6 x y - x y - 6 x y - 9 x y + 5 x y - x y + x y + 5 x y 7 3 8 7 2 6 3 5 4 7 6 2 4 4 + 7 x y - x y - 9 x y + 2 x y - 2 x y + 3 x y + x y + 3 x y 6 5 2 4 3 6 5 4 2 3 3 5 - 2 x y + 3 x y - 4 x y + x + x y + 2 x y + 8 x y - x 3 2 2 2 2 / 12 5 12 4 - 6 x y + 4 x y - 3 x y + 2 x y - x + 1) / (x y - 3 x y / 11 5 12 3 11 4 12 2 11 3 11 2 9 4 - x y + 3 x y + 3 x y - x y - 3 x y + x y - x y 10 2 9 3 8 4 10 9 2 8 3 9 8 2 8 - x y - x y + x y + x y + 3 x y - x y - x y - x y + x y 7 2 7 6 2 5 2 6 5 5 2 + 2 x y - x y + 2 x y - x y - x - x y + x + x y + x - 1) and in Maple notation -(2*x^17*y^9-10*x^17*y^8+20*x^17*y^7-2*x^16*y^8+2*x^15*y^9-20*x^17*y^6+8*x^16*y ^7-13*x^15*y^8+10*x^17*y^5-12*x^16*y^6+31*x^15*y^7-x^14*y^8-2*x^17*y^4+8*x^16*y ^5-34*x^15*y^6-x^14*y^7-2*x^16*y^4+16*x^15*y^5+14*x^14*y^6+5*x^13*y^7-2*x^12*y^ 8-x^15*y^4-26*x^14*y^5-18*x^13*y^6+9*x^12*y^7-x^15*y^3+19*x^14*y^4+23*x^13*y^5-\ 12*x^12*y^6-2*x^11*y^7-5*x^14*y^3-11*x^13*y^4-x^12*y^5+5*x^11*y^6+15*x^12*y^4+x ^11*y^5+7*x^10*y^6+x^13*y^2-12*x^12*y^3-13*x^11*y^4-17*x^10*y^5+2*x^9*y^6+3*x^ 12*y^2+13*x^11*y^3+13*x^10*y^4-3*x^9*y^5-4*x^11*y^2-4*x^10*y^3-5*x^9*y^4-4*x^8* y^5+2*x^10*y^2+11*x^9*y^3+9*x^8*y^4-6*x^7*y^5-x^10*y-6*x^9*y^2-9*x^8*y^3+5*x^7* y^4-x^6*y^5+x^9*y+5*x^8*y^2+7*x^7*y^3-x^8*y-9*x^7*y^2+2*x^6*y^3-2*x^5*y^4+3*x^7 *y+x^6*y^2+3*x^4*y^4-2*x^6*y+3*x^5*y^2-4*x^4*y^3+x^6+x^5*y+2*x^4*y^2+8*x^3*y^3- x^5-6*x^3*y^2+4*x^2*y^2-3*x^2*y+2*x*y-x+1)/(x^12*y^5-3*x^12*y^4-x^11*y^5+3*x^12 *y^3+3*x^11*y^4-x^12*y^2-3*x^11*y^3+x^11*y^2-x^9*y^4-x^10*y^2-x^9*y^3+x^8*y^4+x ^10*y+3*x^9*y^2-x^8*y^3-x^9*y-x^8*y^2+x^8*y+2*x^7*y^2-x^7*y+2*x^6*y^2-x^5*y^2-x ^6-x^5*y+x^5+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 11 10 9 8 7 6 5 4 3 2 (a - 2 a + 2 a - 2 a - 3 a + 7 a - 7 a + 9 a - 10 a + 2 a + 1) 6 5 4 / 5 3 2 2 (7 a + 6 a - 5 a + 2 a + 1) / ((6 a - 4 a + 3 a - 2 a + 2) / 6 5 4 3 2 (a + a + a + 2 a + a + a + 1)) 7 6 5 2 where a is the root of the polynomial, x + x - x + x + x - 1, and in decimals this is, 0.2676610850 BTW the ratio for words with, 500, letters is, 0.2706383211 ------------------------------------------------ "Theorem Number 148" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [2, 1, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 10 18 9 18 8 17 9 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 10 18 7 17 8 16 9 18 6 17 7 + x y - 4 x y - 22 x y + 2 x y + x y + 28 x y 16 8 15 9 17 6 16 7 15 8 17 5 - 2 x y + 8 x y - 12 x y - 26 x y - 14 x y - 2 x y 16 6 15 7 14 8 17 4 16 5 15 6 + 51 x y + 2 x y + 16 x y + 2 x y - 28 x y - 10 x y 14 7 16 4 15 5 14 6 13 7 12 8 - 28 x y - 2 x y + 36 x y - 2 x y + 26 x y - x y 16 3 15 4 14 5 13 6 12 7 15 3 + 4 x y - 26 x y + 14 x y - 46 x y - 2 x y + 2 x y 14 4 13 5 12 6 11 7 15 2 14 3 + 11 x y - 4 x y + 34 x y - 2 x y + 2 x y - 12 x y 13 4 12 5 11 6 14 2 13 3 12 4 + 38 x y - 46 x y - 4 x y + x y - 10 x y - 7 x y 11 5 10 6 13 2 12 3 11 4 9 6 + 42 x y - 8 x y - 4 x y + 32 x y - 50 x y - 2 x y 12 2 11 3 10 4 9 5 8 6 11 2 - 10 x y - 2 x y + 42 x y - 6 x y - x y + 20 x y 10 3 9 4 8 5 11 10 2 9 3 8 4 - 42 x y + 2 x y - 4 x y - 4 x y + x y + 28 x y - 3 x y 7 5 10 9 2 8 3 7 4 10 8 2 - 2 x y + 8 x y - 24 x y + 4 x y - 4 x y - x + 11 x y 7 3 6 4 9 8 7 2 6 3 5 4 7 - 2 x y - 2 x y + 2 x - 8 x y + 4 x y - 6 x y + 2 x y + 4 x y 6 2 5 3 4 4 7 5 2 4 3 6 5 + 5 x y + 4 x y - 3 x y - 2 x - 8 x y + 8 x y + x + 2 x y 4 2 3 3 4 3 2 4 2 2 3 2 - 2 x y - 8 x y - 4 x y + 8 x y + x - 4 x y - 2 x + 4 x y / 12 6 12 5 11 5 12 3 11 4 - 2 x y + 2 x - 1) / (x y - 2 x y + 2 x y + 2 x y - 4 x y / 12 2 10 4 11 2 10 3 11 10 2 9 3 - x y + 4 x y + 4 x y - 6 x y - 2 x y - x y + 4 x y 10 9 2 10 8 2 9 8 7 7 6 + 4 x y - 6 x y - x + 4 x y + 2 x - 4 x y + 2 x y - 2 x + x 4 3 + x - 2 x + 2 x - 1) and in Maple notation (x^18*y^10-4*x^18*y^9+6*x^18*y^8+6*x^17*y^9+x^16*y^10-4*x^18*y^7-22*x^17*y^8+2* x^16*y^9+x^18*y^6+28*x^17*y^7-2*x^16*y^8+8*x^15*y^9-12*x^17*y^6-26*x^16*y^7-14* x^15*y^8-2*x^17*y^5+51*x^16*y^6+2*x^15*y^7+16*x^14*y^8+2*x^17*y^4-28*x^16*y^5-\ 10*x^15*y^6-28*x^14*y^7-2*x^16*y^4+36*x^15*y^5-2*x^14*y^6+26*x^13*y^7-x^12*y^8+ 4*x^16*y^3-26*x^15*y^4+14*x^14*y^5-46*x^13*y^6-2*x^12*y^7+2*x^15*y^3+11*x^14*y^ 4-4*x^13*y^5+34*x^12*y^6-2*x^11*y^7+2*x^15*y^2-12*x^14*y^3+38*x^13*y^4-46*x^12* y^5-4*x^11*y^6+x^14*y^2-10*x^13*y^3-7*x^12*y^4+42*x^11*y^5-8*x^10*y^6-4*x^13*y^ 2+32*x^12*y^3-50*x^11*y^4-2*x^9*y^6-10*x^12*y^2-2*x^11*y^3+42*x^10*y^4-6*x^9*y^ 5-x^8*y^6+20*x^11*y^2-42*x^10*y^3+2*x^9*y^4-4*x^8*y^5-4*x^11*y+x^10*y^2+28*x^9* y^3-3*x^8*y^4-2*x^7*y^5+8*x^10*y-24*x^9*y^2+4*x^8*y^3-4*x^7*y^4-x^10+11*x^8*y^2 -2*x^7*y^3-2*x^6*y^4+2*x^9-8*x^8*y+4*x^7*y^2-6*x^6*y^3+2*x^5*y^4+4*x^7*y+5*x^6* y^2+4*x^5*y^3-3*x^4*y^4-2*x^7-8*x^5*y^2+8*x^4*y^3+x^6+2*x^5*y-2*x^4*y^2-8*x^3*y ^3-4*x^4*y+8*x^3*y^2+x^4-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^12*y^6-2*x^12* y^5+2*x^11*y^5+2*x^12*y^3-4*x^11*y^4-x^12*y^2+4*x^10*y^4+4*x^11*y^2-6*x^10*y^3-\ 2*x^11*y-x^10*y^2+4*x^9*y^3+4*x^10*y-6*x^9*y^2-x^10+4*x^8*y^2+2*x^9-4*x^8*y+2*x ^7*y-2*x^7+x^6+x^4-2*x^3+2*x-1) As the length of the word goes to infinity, the average number of good neigh\ 16 15 14 bors of a random word of length n tends to n times, (a + 4 a + 7 a 12 11 10 9 8 7 6 5 4 - 11 a - 14 a + 12 a + 16 a + 2 a - 18 a - 14 a + 20 a + 4 a 3 / 5 3 2 8 7 6 3 - 8 a + 1) / ((3 a + 2 a - 3 a + 1) (a + 2 a + 2 a + 2 a + 1)) / 6 4 3 where a is the root of the polynomial, x + x - 2 x + 2 x - 1, and in decimals this is, 0.2672326404 BTW the ratio for words with, 500, letters is, 0.2696261525 ------------------------------------------------ "Theorem Number 149" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = (x y + x y - 4 x y - 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 8 14 5 13 6 12 7 11 8 14 4 13 5 + x y + 6 x y + 12 x y - x y + x y - 4 x y - 10 x y 12 6 14 3 13 4 12 5 11 6 10 7 - 5 x y + x y + 3 x y + 11 x y - 3 x y + 2 x y 12 4 11 5 10 6 12 3 11 4 10 5 9 6 - 8 x y - x y - x y + 2 x y + 7 x y - 6 x y + x y 11 3 10 4 9 5 8 6 11 2 9 4 8 5 - 5 x y + 6 x y - 2 x y - 2 x y + x y + x y + 8 x y 7 6 10 2 8 4 7 5 6 6 8 3 7 4 6 5 - x y - x y - 4 x y - 3 x y - x y - 6 x y + 5 x y - x y 8 2 7 3 6 4 5 5 7 2 6 3 5 4 + 4 x y + 5 x y - 3 x y - 2 x y - 8 x y + 5 x y - x y 7 6 2 4 4 6 5 2 4 3 6 5 + 2 x y + 2 x y - 3 x y - 4 x y + 2 x y + 6 x y + x + 2 x y 4 2 3 3 5 3 2 4 3 2 2 2 2 - 4 x y - 8 x y - 2 x + 8 x y + x - 2 x y - 4 x y + 4 x y - x / 11 4 11 3 10 4 11 2 10 3 - 2 x y + 2 x - 1) / (x y - 2 x y - x y + x y + 2 x y / 10 2 6 2 5 2 6 5 4 2 - x y - 2 x y + 2 x y + x - 2 x + x - x + 2 x - 1) and in Maple notation (x^14*y^7+x^13*y^8-4*x^14*y^6-6*x^13*y^7+x^12*y^8+6*x^14*y^5+12*x^13*y^6-x^12*y ^7+x^11*y^8-4*x^14*y^4-10*x^13*y^5-5*x^12*y^6+x^14*y^3+3*x^13*y^4+11*x^12*y^5-3 *x^11*y^6+2*x^10*y^7-8*x^12*y^4-x^11*y^5-x^10*y^6+2*x^12*y^3+7*x^11*y^4-6*x^10* y^5+x^9*y^6-5*x^11*y^3+6*x^10*y^4-2*x^9*y^5-2*x^8*y^6+x^11*y^2+x^9*y^4+8*x^8*y^ 5-x^7*y^6-x^10*y^2-4*x^8*y^4-3*x^7*y^5-x^6*y^6-6*x^8*y^3+5*x^7*y^4-x^6*y^5+4*x^ 8*y^2+5*x^7*y^3-3*x^6*y^4-2*x^5*y^5-8*x^7*y^2+5*x^6*y^3-x^5*y^4+2*x^7*y+2*x^6*y ^2-3*x^4*y^4-4*x^6*y+2*x^5*y^2+6*x^4*y^3+x^6+2*x^5*y-4*x^4*y^2-8*x^3*y^3-2*x^5+ 8*x^3*y^2+x^4-2*x^3*y-4*x^2*y^2+4*x^2*y-x^2-2*x*y+2*x-1)/(x^11*y^4-2*x^11*y^3-x ^10*y^4+x^11*y^2+2*x^10*y^3-x^10*y^2-2*x^6*y^2+2*x^5*y^2+x^6-2*x^5+x^4-x^2+2*x-\ 1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2538723300 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 150" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = (x y + x y - 4 x y - 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 8 14 5 13 6 12 7 11 8 14 4 13 5 + x y + 6 x y + 12 x y - x y + x y - 4 x y - 10 x y 12 6 14 3 13 4 12 5 11 6 10 7 - 5 x y + x y + 3 x y + 11 x y - 3 x y + 2 x y 12 4 11 5 10 6 12 3 11 4 10 5 9 6 - 8 x y - x y - x y + 2 x y + 7 x y - 6 x y + x y 11 3 10 4 9 5 8 6 11 2 9 4 8 5 - 5 x y + 6 x y - 2 x y - 2 x y + x y + x y + 8 x y 7 6 10 2 8 4 7 5 6 6 8 3 7 4 6 5 - x y - x y - 4 x y - 3 x y - x y - 6 x y + 5 x y - x y 8 2 7 3 6 4 5 5 7 2 6 3 5 4 + 4 x y + 5 x y - 3 x y - 2 x y - 8 x y + 5 x y - x y 7 6 2 4 4 6 5 2 4 3 6 5 + 2 x y + 2 x y - 3 x y - 4 x y + 2 x y + 6 x y + x + 2 x y 4 2 3 3 5 3 2 4 3 2 2 2 2 - 4 x y - 8 x y - 2 x + 8 x y + x - 2 x y - 4 x y + 4 x y - x / 11 4 11 3 10 4 11 2 10 3 - 2 x y + 2 x - 1) / (x y - 2 x y - x y + x y + 2 x y / 10 2 6 2 5 2 6 5 4 2 - x y - 2 x y + 2 x y + x - 2 x + x - x + 2 x - 1) and in Maple notation (x^14*y^7+x^13*y^8-4*x^14*y^6-6*x^13*y^7+x^12*y^8+6*x^14*y^5+12*x^13*y^6-x^12*y ^7+x^11*y^8-4*x^14*y^4-10*x^13*y^5-5*x^12*y^6+x^14*y^3+3*x^13*y^4+11*x^12*y^5-3 *x^11*y^6+2*x^10*y^7-8*x^12*y^4-x^11*y^5-x^10*y^6+2*x^12*y^3+7*x^11*y^4-6*x^10* y^5+x^9*y^6-5*x^11*y^3+6*x^10*y^4-2*x^9*y^5-2*x^8*y^6+x^11*y^2+x^9*y^4+8*x^8*y^ 5-x^7*y^6-x^10*y^2-4*x^8*y^4-3*x^7*y^5-x^6*y^6-6*x^8*y^3+5*x^7*y^4-x^6*y^5+4*x^ 8*y^2+5*x^7*y^3-3*x^6*y^4-2*x^5*y^5-8*x^7*y^2+5*x^6*y^3-x^5*y^4+2*x^7*y+2*x^6*y ^2-3*x^4*y^4-4*x^6*y+2*x^5*y^2+6*x^4*y^3+x^6+2*x^5*y-4*x^4*y^2-8*x^3*y^3-2*x^5+ 8*x^3*y^2+x^4-2*x^3*y-4*x^2*y^2+4*x^2*y-x^2-2*x*y+2*x-1)/(x^11*y^4-2*x^11*y^3-x ^10*y^4+x^11*y^2+2*x^10*y^3-x^10*y^2-2*x^6*y^2+2*x^5*y^2+x^6-2*x^5+x^4-x^2+2*x-\ 1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2538723300 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 151" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 6 11 7 10 8 12 5 ) | ) C(m, n) x y | = (x y + 2 x y + x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 6 10 7 12 4 11 5 10 6 12 3 10 5 - 5 x y - 2 x y + 3 x y + 3 x y + x y - x y - 3 x y 9 6 11 3 10 4 9 5 8 6 7 7 11 2 - 2 x y + x y + 4 x y + 2 x y - 3 x y - x y - x y 10 3 9 4 8 5 10 2 8 4 7 5 6 6 9 2 + x y + x y + 3 x y - 2 x y + 3 x y + 3 x y - x y - x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 8 - 3 x y - x y + 2 x y - x y - 5 x y - x y - x y + x y 7 2 5 4 5 3 4 4 6 5 2 4 3 6 + 4 x y + x y + 2 x y - 2 x y + 2 x y + 2 x y + 3 x y - x 5 4 2 3 3 4 3 2 2 / - 2 x y + 2 x y + 4 x y - x - x y + 2 x y + x y + 1) / ( / 6 5 2 6 4 2 5 4 4 3 x y + x y - x + x y - x - x y - x - x y - x y + 1) and in Maple notation (x^12*y^6+2*x^11*y^7+x^10*y^8-3*x^12*y^5-5*x^11*y^6-2*x^10*y^7+3*x^12*y^4+3*x^ 11*y^5+x^10*y^6-x^12*y^3-3*x^10*y^5-2*x^9*y^6+x^11*y^3+4*x^10*y^4+2*x^9*y^5-3*x ^8*y^6-x^7*y^7-x^11*y^2+x^10*y^3+x^9*y^4+3*x^8*y^5-2*x^10*y^2+3*x^8*y^4+3*x^7*y ^5-x^6*y^6-x^9*y^2-3*x^8*y^3-x^7*y^4+2*x^6*y^5-x^8*y^2-5*x^7*y^3-x^6*y^4-x^5*y^ 5+x^8*y+4*x^7*y^2+x^5*y^4+2*x^5*y^3-2*x^4*y^4+2*x^6*y+2*x^5*y^2+3*x^4*y^3-x^6-2 *x^5*y+2*x^4*y^2+4*x^3*y^3-x^4-x^3*y+2*x^2*y^2+x*y+1)/(x^6*y+x^5*y^2-x^6+x^4*y^ 2-x^5-x^4*y-x^4-x^3*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2509972359 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 152" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 8 13 7 13 6 12 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 5 12 6 13 4 12 5 11 6 10 7 13 3 - 10 x y - 8 x y + 5 x y + 12 x y + x y + x y - x y 12 4 11 5 10 6 9 7 12 3 11 4 10 5 - 8 x y - 4 x y - x y + x y + 2 x y + 6 x y - 2 x y 9 6 11 3 10 4 8 6 11 2 10 3 9 4 - x y - 4 x y + x y + 3 x y + x y + 4 x y + x y 8 5 7 6 10 2 9 3 8 4 7 5 6 6 - 8 x y - 2 x y - 4 x y - 3 x y + 7 x y + 5 x y - 2 x y 10 9 2 7 4 6 5 9 8 2 7 3 6 4 + x y + 3 x y - 7 x y - 3 x y - x y - 3 x y + 8 x y + 8 x y 5 5 8 7 2 6 3 5 4 7 6 2 - 5 x y + x y - 3 x y - 8 x y - 2 x y - 2 x y + 6 x y 5 3 4 4 7 6 5 2 4 3 6 5 + 10 x y - 5 x y + x - x y - 5 x y + 2 x y - x + 2 x y 4 2 3 3 5 4 3 2 4 3 2 2 3 + 4 x y - 8 x y - x - 3 x y + 4 x y + x + 3 x y - 4 x y - x 2 2 / 10 4 10 3 10 2 9 3 + 2 x y + x - 2 x y + x - 1) / (x y - 3 x y + 3 x y + x y / 10 9 2 9 7 3 7 2 7 6 2 7 6 5 - x y - 2 x y + x y - x y + x y + x y - x y - x + x - x y 5 4 4 3 3 2 + x + x y - x - x y + x - x - x + 1) and in Maple notation -(x^13*y^8-5*x^13*y^7+10*x^13*y^6+2*x^12*y^7-10*x^13*y^5-8*x^12*y^6+5*x^13*y^4+ 12*x^12*y^5+x^11*y^6+x^10*y^7-x^13*y^3-8*x^12*y^4-4*x^11*y^5-x^10*y^6+x^9*y^7+2 *x^12*y^3+6*x^11*y^4-2*x^10*y^5-x^9*y^6-4*x^11*y^3+x^10*y^4+3*x^8*y^6+x^11*y^2+ 4*x^10*y^3+x^9*y^4-8*x^8*y^5-2*x^7*y^6-4*x^10*y^2-3*x^9*y^3+7*x^8*y^4+5*x^7*y^5 -2*x^6*y^6+x^10*y+3*x^9*y^2-7*x^7*y^4-3*x^6*y^5-x^9*y-3*x^8*y^2+8*x^7*y^3+8*x^6 *y^4-5*x^5*y^5+x^8*y-3*x^7*y^2-8*x^6*y^3-2*x^5*y^4-2*x^7*y+6*x^6*y^2+10*x^5*y^3 -5*x^4*y^4+x^7-x^6*y-5*x^5*y^2+2*x^4*y^3-x^6+2*x^5*y+4*x^4*y^2-8*x^3*y^3-x^5-3* x^4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2*y^2-x^3+2*x^2*y+x^2-2*x*y+x-1)/(x^10*y^4-3*x^ 10*y^3+3*x^10*y^2+x^9*y^3-x^10*y-2*x^9*y^2+x^9*y-x^7*y^3+x^7*y^2+x^7*y-x^6*y^2- x^7+x^6-x^5*y+x^5+x^4*y-x^4-x^3*y+x^3-x^2-x+1) As the length of the word goes to infinity, the average number of good neigh\ 11 10 9 bors of a random word of length n tends to n times, 2 (a + 2 a + 2 a 8 7 6 5 4 3 2 / 2 + 3 a - 2 a - 9 a - 4 a - 5 a - 5 a + 3 a + 2 a + 1) / ((a + 1) / 2 2 (a - a + 1) (a + 1) (2 a + 1)) 2 where a is the root of the polynomial, x + x - 1, and in decimals this is, 0.2111456219 BTW the ratio for words with, 500, letters is, 0.2156468295 ------------------------------------------------ "Theorem Number 153" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 8 13 7 13 6 12 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 5 12 6 13 4 12 5 11 6 10 7 13 3 - 10 x y - 8 x y + 5 x y + 12 x y + x y + x y - x y 12 4 11 5 10 6 9 7 12 3 11 4 10 5 - 8 x y - 4 x y - x y + x y + 2 x y + 6 x y - 2 x y 9 6 11 3 10 4 8 6 11 2 10 3 9 4 8 5 - x y - 4 x y + x y + x y + x y + 4 x y + x y - 4 x y 7 6 10 2 9 3 8 4 7 5 6 6 10 9 2 - x y - 4 x y - 3 x y + 4 x y + 3 x y - x y + x y + 3 x y 8 3 7 4 6 5 9 8 2 7 3 6 4 + 3 x y - 5 x y - 4 x y - x y - 6 x y + 5 x y + 7 x y 5 5 8 6 3 5 4 7 6 2 5 3 - 3 x y + 2 x y - 8 x y - 4 x y - 3 x y + 8 x y + 11 x y 4 4 7 6 5 2 4 3 6 5 4 2 - 5 x y + x - 2 x y - 7 x y + 2 x y - x + 3 x y + 4 x y 3 3 5 4 3 2 4 3 2 2 3 2 - 8 x y - x - 3 x y + 4 x y + x + 3 x y - 4 x y - x + 2 x y 2 / 10 4 10 3 10 2 9 3 10 + x - 2 x y + x - 1) / (x y - 3 x y + 3 x y + x y - x y / 9 2 9 7 3 7 2 7 6 2 7 6 5 5 - 2 x y + x y - x y + x y + x y - x y - x + x - x y + x 4 4 3 3 2 + x y - x - x y + x - x - x + 1) and in Maple notation -(x^13*y^8-5*x^13*y^7+10*x^13*y^6+2*x^12*y^7-10*x^13*y^5-8*x^12*y^6+5*x^13*y^4+ 12*x^12*y^5+x^11*y^6+x^10*y^7-x^13*y^3-8*x^12*y^4-4*x^11*y^5-x^10*y^6+x^9*y^7+2 *x^12*y^3+6*x^11*y^4-2*x^10*y^5-x^9*y^6-4*x^11*y^3+x^10*y^4+x^8*y^6+x^11*y^2+4* x^10*y^3+x^9*y^4-4*x^8*y^5-x^7*y^6-4*x^10*y^2-3*x^9*y^3+4*x^8*y^4+3*x^7*y^5-x^6 *y^6+x^10*y+3*x^9*y^2+3*x^8*y^3-5*x^7*y^4-4*x^6*y^5-x^9*y-6*x^8*y^2+5*x^7*y^3+7 *x^6*y^4-3*x^5*y^5+2*x^8*y-8*x^6*y^3-4*x^5*y^4-3*x^7*y+8*x^6*y^2+11*x^5*y^3-5*x ^4*y^4+x^7-2*x^6*y-7*x^5*y^2+2*x^4*y^3-x^6+3*x^5*y+4*x^4*y^2-8*x^3*y^3-x^5-3*x^ 4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2*y^2-x^3+2*x^2*y+x^2-2*x*y+x-1)/(x^10*y^4-3*x^10 *y^3+3*x^10*y^2+x^9*y^3-x^10*y-2*x^9*y^2+x^9*y-x^7*y^3+x^7*y^2+x^7*y-x^6*y^2-x^ 7+x^6-x^5*y+x^5+x^4*y-x^4-x^3*y+x^3-x^2-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 10 9 8 7 6 5 4 3 2 2 (a + a + a + 2 a - 4 a - 4 a + a - 7 a + 2 a + a + 1) ----------------------------------------------------------------- 5 3 2 (a + a + a + 1) (2 a + 1) 2 where a is the root of the polynomial, x + x - 1, and in decimals this is, 0.2111456216 BTW the ratio for words with, 500, letters is, 0.2155165948 ------------------------------------------------ "Theorem Number 154" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = (x y - 4 x y - x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 16 6 15 7 14 8 16 5 15 6 + 7 x y - 4 x y - 17 x y - 3 x y + 2 x y + 20 x y 14 7 13 8 16 4 15 5 14 6 13 7 + 17 x y - x y - 4 x y - 14 x y - 37 x y - 5 x y 12 8 16 3 15 4 14 5 13 6 12 7 + x y + 6 x y + 9 x y + 40 x y + 30 x y - 4 x y 16 2 15 3 14 4 13 5 12 6 11 7 - 4 x y - 7 x y - 23 x y - 51 x y - 6 x y + 3 x y 16 15 2 14 3 13 4 12 5 11 6 10 7 + x y + 4 x y + 7 x y + 37 x y + 31 x y - 10 x y - x y 15 14 2 13 3 12 4 11 5 10 6 9 7 - x y - x y - 10 x y - 36 x y + 2 x y + 10 x y + x y 13 2 12 3 11 4 10 5 9 6 13 + x y + 17 x y + 21 x y - 18 x y - 6 x y - 2 x y 12 2 11 3 9 5 8 6 13 12 11 2 - 4 x y - 25 x y + 15 x y + 3 x y + x + 2 x y + 10 x y 10 3 9 4 8 5 7 6 12 10 2 8 4 + 20 x y - 13 x y - 11 x y + x y - x - 11 x y + 21 x y 7 5 6 6 11 10 9 2 8 3 7 4 6 5 + 3 x y - x y - x - x y + 3 x y - 20 x y - 23 x y + 5 x y 10 9 8 2 7 3 6 4 5 5 9 8 + x + x y + 5 x y + 31 x y + 9 x y - 4 x y - x + 2 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 - 12 x y - 25 x y + 7 x y - 3 x y + 15 x y + 4 x y - 6 x y 7 6 5 2 4 3 6 5 4 2 3 3 5 + 3 x - x y - 10 x y + 13 x y - x + 6 x y - x y - 8 x y - 2 x 4 3 2 4 3 2 2 3 2 - 7 x y + 8 x y + 2 x + x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x / 10 4 10 3 10 2 9 3 10 9 2 9 - 1) / ((x y - 3 x y + 3 x y + x y - x y - 2 x y + x y / 7 3 7 2 7 6 2 7 6 5 5 4 4 3 - x y + x y + x y - x y - x + x - x y + x + x y - x - x y 3 2 + x - x - x + 1) (x - 1)) and in Maple notation (x^16*y^9-4*x^16*y^8-x^15*y^9+6*x^16*y^7+7*x^15*y^8-4*x^16*y^6-17*x^15*y^7-3*x^ 14*y^8+2*x^16*y^5+20*x^15*y^6+17*x^14*y^7-x^13*y^8-4*x^16*y^4-14*x^15*y^5-37*x^ 14*y^6-5*x^13*y^7+x^12*y^8+6*x^16*y^3+9*x^15*y^4+40*x^14*y^5+30*x^13*y^6-4*x^12 *y^7-4*x^16*y^2-7*x^15*y^3-23*x^14*y^4-51*x^13*y^5-6*x^12*y^6+3*x^11*y^7+x^16*y +4*x^15*y^2+7*x^14*y^3+37*x^13*y^4+31*x^12*y^5-10*x^11*y^6-x^10*y^7-x^15*y-x^14 *y^2-10*x^13*y^3-36*x^12*y^4+2*x^11*y^5+10*x^10*y^6+x^9*y^7+x^13*y^2+17*x^12*y^ 3+21*x^11*y^4-18*x^10*y^5-6*x^9*y^6-2*x^13*y-4*x^12*y^2-25*x^11*y^3+15*x^9*y^5+ 3*x^8*y^6+x^13+2*x^12*y+10*x^11*y^2+20*x^10*y^3-13*x^9*y^4-11*x^8*y^5+x^7*y^6-x ^12-11*x^10*y^2+21*x^8*y^4+3*x^7*y^5-x^6*y^6-x^11-x^10*y+3*x^9*y^2-20*x^8*y^3-\ 23*x^7*y^4+5*x^6*y^5+x^10+x^9*y+5*x^8*y^2+31*x^7*y^3+9*x^6*y^4-4*x^5*y^5-x^9+2* x^8*y-12*x^7*y^2-25*x^6*y^3+7*x^5*y^4-3*x^7*y+15*x^6*y^2+4*x^5*y^3-6*x^4*y^4+3* x^7-x^6*y-10*x^5*y^2+13*x^4*y^3-x^6+6*x^5*y-x^4*y^2-8*x^3*y^3-2*x^5-7*x^4*y+8*x ^3*y^2+2*x^4+x^3*y-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^10*y^4-3*x^10*y^3+3* x^10*y^2+x^9*y^3-x^10*y-2*x^9*y^2+x^9*y-x^7*y^3+x^7*y^2+x^7*y-x^6*y^2-x^7+x^6-x ^5*y+x^5+x^4*y-x^4-x^3*y+x^3-x^2-x+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 7 6 5 4 3 2 2 2 (a + 2 a + 3 a + 7 a + 5 a - 3 a - 2 a - 1) (3 a - 2) (a + a + 1) / 2 6 5 4 3 (a - 1) / ((2 a + 1) (a + a + a - a - 1)) / 2 where a is the root of the polynomial, (x + x - 1) (x - 1), and in decimals this is, 0.2111456213 BTW the ratio for words with, 500, letters is, 0.2148569238 ------------------------------------------------ "Theorem Number 155" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 9 16 8 15 9 16 7 ) | ) C(m, n) x y | = (x y - 4 x y - x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 8 16 6 15 7 14 8 16 5 15 6 + 7 x y - 4 x y - 17 x y - 3 x y + 2 x y + 20 x y 14 7 13 8 16 4 15 5 14 6 13 7 + 17 x y - x y - 4 x y - 14 x y - 37 x y - 5 x y 12 8 16 3 15 4 14 5 13 6 12 7 + x y + 6 x y + 9 x y + 40 x y + 30 x y - 4 x y 16 2 15 3 14 4 13 5 12 6 11 7 - 4 x y - 7 x y - 23 x y - 51 x y - 6 x y + 3 x y 16 15 2 14 3 13 4 12 5 11 6 10 7 + x y + 4 x y + 7 x y + 37 x y + 31 x y - 10 x y - x y 15 14 2 13 3 12 4 11 5 10 6 9 7 - x y - x y - 10 x y - 36 x y + 2 x y + 10 x y + x y 13 2 12 3 11 4 10 5 9 6 13 + x y + 17 x y + 21 x y - 18 x y - 6 x y - 2 x y 12 2 11 3 9 5 8 6 13 12 11 2 - 4 x y - 25 x y + 15 x y + 3 x y + x + 2 x y + 10 x y 10 3 9 4 8 5 7 6 12 10 2 8 4 + 20 x y - 13 x y - 11 x y + x y - x - 11 x y + 21 x y 7 5 6 6 11 10 9 2 8 3 7 4 6 5 + 3 x y - x y - x - x y + 3 x y - 20 x y - 23 x y + 5 x y 10 9 8 2 7 3 6 4 5 5 9 8 + x + x y + 5 x y + 31 x y + 9 x y - 4 x y - x + 2 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 - 12 x y - 25 x y + 7 x y - 3 x y + 15 x y + 4 x y - 6 x y 7 6 5 2 4 3 6 5 4 2 3 3 5 + 3 x - x y - 10 x y + 13 x y - x + 6 x y - x y - 8 x y - 2 x 4 3 2 4 3 2 2 3 2 - 7 x y + 8 x y + 2 x + x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x / 10 4 10 3 10 2 9 3 10 9 2 9 - 1) / ((x y - 3 x y + 3 x y + x y - x y - 2 x y + x y / 7 3 7 2 7 6 2 7 6 5 5 4 4 3 - x y + x y + x y - x y - x + x - x y + x + x y - x - x y 3 2 + x - x - x + 1) (x - 1)) and in Maple notation (x^16*y^9-4*x^16*y^8-x^15*y^9+6*x^16*y^7+7*x^15*y^8-4*x^16*y^6-17*x^15*y^7-3*x^ 14*y^8+2*x^16*y^5+20*x^15*y^6+17*x^14*y^7-x^13*y^8-4*x^16*y^4-14*x^15*y^5-37*x^ 14*y^6-5*x^13*y^7+x^12*y^8+6*x^16*y^3+9*x^15*y^4+40*x^14*y^5+30*x^13*y^6-4*x^12 *y^7-4*x^16*y^2-7*x^15*y^3-23*x^14*y^4-51*x^13*y^5-6*x^12*y^6+3*x^11*y^7+x^16*y +4*x^15*y^2+7*x^14*y^3+37*x^13*y^4+31*x^12*y^5-10*x^11*y^6-x^10*y^7-x^15*y-x^14 *y^2-10*x^13*y^3-36*x^12*y^4+2*x^11*y^5+10*x^10*y^6+x^9*y^7+x^13*y^2+17*x^12*y^ 3+21*x^11*y^4-18*x^10*y^5-6*x^9*y^6-2*x^13*y-4*x^12*y^2-25*x^11*y^3+15*x^9*y^5+ 3*x^8*y^6+x^13+2*x^12*y+10*x^11*y^2+20*x^10*y^3-13*x^9*y^4-11*x^8*y^5+x^7*y^6-x ^12-11*x^10*y^2+21*x^8*y^4+3*x^7*y^5-x^6*y^6-x^11-x^10*y+3*x^9*y^2-20*x^8*y^3-\ 23*x^7*y^4+5*x^6*y^5+x^10+x^9*y+5*x^8*y^2+31*x^7*y^3+9*x^6*y^4-4*x^5*y^5-x^9+2* x^8*y-12*x^7*y^2-25*x^6*y^3+7*x^5*y^4-3*x^7*y+15*x^6*y^2+4*x^5*y^3-6*x^4*y^4+3* x^7-x^6*y-10*x^5*y^2+13*x^4*y^3-x^6+6*x^5*y-x^4*y^2-8*x^3*y^3-2*x^5-7*x^4*y+8*x ^3*y^2+2*x^4+x^3*y-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^10*y^4-3*x^10*y^3+3* x^10*y^2+x^9*y^3-x^10*y-2*x^9*y^2+x^9*y-x^7*y^3+x^7*y^2+x^7*y-x^6*y^2-x^7+x^6-x ^5*y+x^5+x^4*y-x^4-x^3*y+x^3-x^2-x+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 7 6 5 4 3 2 2 2 (a + 2 a + 3 a + 7 a + 5 a - 3 a - 2 a - 1) (3 a - 2) (a + a + 1) / 2 6 5 4 3 (a - 1) / ((2 a + 1) (a + a + a - a - 1)) / 2 where a is the root of the polynomial, (x + x - 1) (x - 1), and in decimals this is, 0.2111456213 BTW the ratio for words with, 500, letters is, 0.2148569238 ------------------------------------------------ "Theorem Number 156" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 6 12 7 13 5 12 6 ) | ) C(m, n) x y | = - (x y + x y - 3 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 7 13 4 12 5 11 6 10 7 13 3 12 4 - x y + 3 x y + 4 x y + 6 x y - x y - x y - 4 x y 11 5 10 6 9 7 12 3 11 4 10 5 9 6 - 11 x y - x y - x y + 3 x y + 8 x y + 10 x y - 3 x y 8 7 12 2 11 3 10 4 9 5 8 6 10 3 - x y - x y - 2 x y - 13 x y + 6 x y - x y + 5 x y 8 5 7 6 9 3 8 4 7 5 6 6 9 2 8 3 - 3 x y - 2 x y - 3 x y + 11 x y + x y + x y + x y - 8 x y 7 4 6 5 8 2 7 3 6 4 5 5 7 2 - 6 x y - 2 x y + 2 x y + 12 x y + 8 x y + 3 x y - 6 x y 6 3 5 4 7 6 2 5 3 4 4 6 - 7 x y - 3 x y + x y + 4 x y + 4 x y + 5 x y - 3 x y 5 2 4 3 6 5 4 2 3 3 5 4 - 3 x y - 6 x y + x + 2 x y + 4 x y + 8 x y - x - x y 3 2 3 2 2 2 / - 6 x y + x y + 4 x y - 3 x y + 2 x y - x + 1) / ( / 6 6 5 5 4 3 2 x y - x - x y + x + x y - x y + x y + x - 1) and in Maple notation -(x^13*y^6+x^12*y^7-3*x^13*y^5-3*x^12*y^6-x^11*y^7+3*x^13*y^4+4*x^12*y^5+6*x^11 *y^6-x^10*y^7-x^13*y^3-4*x^12*y^4-11*x^11*y^5-x^10*y^6-x^9*y^7+3*x^12*y^3+8*x^ 11*y^4+10*x^10*y^5-3*x^9*y^6-x^8*y^7-x^12*y^2-2*x^11*y^3-13*x^10*y^4+6*x^9*y^5- x^8*y^6+5*x^10*y^3-3*x^8*y^5-2*x^7*y^6-3*x^9*y^3+11*x^8*y^4+x^7*y^5+x^6*y^6+x^9 *y^2-8*x^8*y^3-6*x^7*y^4-2*x^6*y^5+2*x^8*y^2+12*x^7*y^3+8*x^6*y^4+3*x^5*y^5-6*x ^7*y^2-7*x^6*y^3-3*x^5*y^4+x^7*y+4*x^6*y^2+4*x^5*y^3+5*x^4*y^4-3*x^6*y-3*x^5*y^ 2-6*x^4*y^3+x^6+2*x^5*y+4*x^4*y^2+8*x^3*y^3-x^5-x^4*y-6*x^3*y^2+x^3*y+4*x^2*y^2 -3*x^2*y+2*x*y-x+1)/(x^6*y-x^6-x^5*y+x^5+x^4*y-x^3*y+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ 12 11 10 bors of a random word of length n tends to n times, 2 (a + 2 a + 2 a 9 8 7 6 4 3 2 / - 5 a + 5 a - 4 a - 15 a - 3 a - 3 a + 3 a + 2 a + 1) / ((a + 1) / 3 2 3 2 (4 a - 3 a + 2 a + 1) (2 a + 1) (a + 1)) 4 3 2 where a is the root of the polynomial, x - x + x + x - 1, and in decimals this is, 0.2068668122 BTW the ratio for words with, 500, letters is, 0.2112659309 ------------------------------------------------ "Theorem Number 157" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [2, 1, 1, 2], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 15 6 15 5 14 6 15 4 ) | ) C(m, n) x y | = (x y - 3 x y - x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 5 13 6 12 7 15 3 14 4 13 5 12 6 + 3 x y - 2 x y - x y - x y - 3 x y + 5 x y + 4 x y 11 7 14 3 13 4 12 5 11 6 13 3 12 4 - x y + x y - 4 x y - 7 x y + 2 x y + x y + 8 x y 11 5 12 3 11 4 10 5 9 6 12 2 11 3 + 5 x y - 6 x y - 14 x y - x y + x y + 2 x y + 10 x y 10 4 9 5 11 2 9 4 8 5 7 6 8 4 + x y - 6 x y - 2 x y + 5 x y + 4 x y + x y - 5 x y 9 2 8 3 7 4 6 5 9 8 2 7 3 6 4 + x y + 4 x y + 9 x y + x y - x y - 2 x y - 14 x y - 8 x y 5 5 7 2 6 3 5 4 6 2 5 3 4 4 - x y + 5 x y + 15 x y + 3 x y - 9 x y - 6 x y - 3 x y 6 5 2 4 3 5 4 2 3 3 4 3 2 + x y + 7 x y + 4 x y - 3 x y - 4 x y - 8 x y + 3 x y + 6 x y 4 3 2 2 3 2 / - x - x y - 4 x y + x + 3 x y - 2 x y + x - 1) / ( / 8 2 8 6 2 4 4 3 3 2 x y - x y - x y + x y - x - x y + x + x y + x - 1) and in Maple notation (x^15*y^6-3*x^15*y^5-x^14*y^6+3*x^15*y^4+3*x^14*y^5-2*x^13*y^6-x^12*y^7-x^15*y^ 3-3*x^14*y^4+5*x^13*y^5+4*x^12*y^6-x^11*y^7+x^14*y^3-4*x^13*y^4-7*x^12*y^5+2*x^ 11*y^6+x^13*y^3+8*x^12*y^4+5*x^11*y^5-6*x^12*y^3-14*x^11*y^4-x^10*y^5+x^9*y^6+2 *x^12*y^2+10*x^11*y^3+x^10*y^4-6*x^9*y^5-2*x^11*y^2+5*x^9*y^4+4*x^8*y^5+x^7*y^6 -5*x^8*y^4+x^9*y^2+4*x^8*y^3+9*x^7*y^4+x^6*y^5-x^9*y-2*x^8*y^2-14*x^7*y^3-8*x^6 *y^4-x^5*y^5+5*x^7*y^2+15*x^6*y^3+3*x^5*y^4-9*x^6*y^2-6*x^5*y^3-3*x^4*y^4+x^6*y +7*x^5*y^2+4*x^4*y^3-3*x^5*y-4*x^4*y^2-8*x^3*y^3+3*x^4*y+6*x^3*y^2-x^4-x^3*y-4* x^2*y^2+x^3+3*x^2*y-2*x*y+x-1)/(x^8*y^2-x^8*y-x^6*y^2+x^4*y-x^4-x^3*y+x^3+x^2*y +x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.2034466117 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 158" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 10 19 9 19 8 ) | ) C(m, n) x y | = - (x y - 6 x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 9 19 7 18 8 17 9 19 6 18 7 + 3 x y - 20 x y - 16 x y + x y + 15 x y + 35 x y 17 8 16 9 19 5 18 6 17 7 16 8 19 4 - 2 x y + x y - 6 x y - 40 x y - 5 x y - 2 x y + x y 18 5 17 6 16 7 15 8 18 4 17 5 + 25 x y + 20 x y - 2 x y + 3 x y - 8 x y - 25 x y 16 6 15 7 14 8 18 3 17 4 16 5 + 6 x y - 12 x y + 2 x y + x y + 14 x y + x y 15 6 14 7 17 3 16 4 15 5 14 6 + 21 x y - 3 x y - 3 x y - 10 x y - 25 x y - 15 x y 13 7 16 3 15 4 14 5 13 6 12 7 + 5 x y + 8 x y + 24 x y + 46 x y - 13 x y - x y 11 8 16 2 15 3 14 4 13 5 12 6 - x y - 2 x y - 15 x y - 49 x y + 6 x y + 12 x y 11 7 15 2 14 3 13 4 12 5 11 6 + 3 x y + 4 x y + 24 x y + 12 x y - 34 x y - x y 10 7 14 2 13 3 12 4 11 5 10 6 - 2 x y - 6 x y - 19 x y + 34 x y + x y + 2 x y 9 7 14 13 2 12 3 11 4 10 5 9 6 - 2 x y + x y + 14 x y - 6 x y - 7 x y + 11 x y + 3 x y 13 12 2 11 3 10 4 9 5 8 6 13 - 6 x y - 10 x y + 7 x y - 16 x y + x y - 4 x y + x 12 11 2 10 3 9 4 8 5 7 6 12 + 7 x y - 3 x y + x y - 13 x y + 3 x y - 2 x y - 2 x 11 10 2 9 3 8 4 10 9 2 8 3 + x y + 5 x y + 22 x y + 19 x y - 2 x y - 11 x y - 28 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 9 + x y - 4 x y + x - x y + 16 x y + x y + 2 x y - x y + x 8 7 2 6 3 5 4 7 5 3 4 4 7 - 5 x y - 7 x y + x y - x y + 9 x y + 6 x y - 3 x y - 3 x 6 5 2 4 3 6 5 4 2 3 3 4 - 3 x y - 2 x y + 8 x y + 3 x - x y - 2 x y - 8 x y - 2 x y 3 2 4 2 2 3 2 / + 8 x y - x - 4 x y - x + 4 x y - 2 x y + 2 x - 1) / ((x - 1) ( / 12 3 12 2 11 3 12 11 2 10 3 12 11 x y - 3 x y - x y + 3 x y + 3 x y + x y - x - 3 x y 10 2 9 3 11 10 9 2 10 9 8 2 7 2 8 - x y + x y + x - x y - 2 x y + x + x y + x y + x y - x 7 6 6 5 5 4 2 - x - 2 x y + 2 x + 2 x y - x - x + x + x - 1)) and in Maple notation -(x^19*y^10-6*x^19*y^9+15*x^19*y^8+3*x^18*y^9-20*x^19*y^7-16*x^18*y^8+x^17*y^9+ 15*x^19*y^6+35*x^18*y^7-2*x^17*y^8+x^16*y^9-6*x^19*y^5-40*x^18*y^6-5*x^17*y^7-2 *x^16*y^8+x^19*y^4+25*x^18*y^5+20*x^17*y^6-2*x^16*y^7+3*x^15*y^8-8*x^18*y^4-25* x^17*y^5+6*x^16*y^6-12*x^15*y^7+2*x^14*y^8+x^18*y^3+14*x^17*y^4+x^16*y^5+21*x^ 15*y^6-3*x^14*y^7-3*x^17*y^3-10*x^16*y^4-25*x^15*y^5-15*x^14*y^6+5*x^13*y^7+8*x ^16*y^3+24*x^15*y^4+46*x^14*y^5-13*x^13*y^6-x^12*y^7-x^11*y^8-2*x^16*y^2-15*x^ 15*y^3-49*x^14*y^4+6*x^13*y^5+12*x^12*y^6+3*x^11*y^7+4*x^15*y^2+24*x^14*y^3+12* x^13*y^4-34*x^12*y^5-x^11*y^6-2*x^10*y^7-6*x^14*y^2-19*x^13*y^3+34*x^12*y^4+x^ 11*y^5+2*x^10*y^6-2*x^9*y^7+x^14*y+14*x^13*y^2-6*x^12*y^3-7*x^11*y^4+11*x^10*y^ 5+3*x^9*y^6-6*x^13*y-10*x^12*y^2+7*x^11*y^3-16*x^10*y^4+x^9*y^5-4*x^8*y^6+x^13+ 7*x^12*y-3*x^11*y^2+x^10*y^3-13*x^9*y^4+3*x^8*y^5-2*x^7*y^6-2*x^12+x^11*y+5*x^ 10*y^2+22*x^9*y^3+19*x^8*y^4-2*x^10*y-11*x^9*y^2-28*x^8*y^3+x^7*y^4-4*x^6*y^5+x ^10-x^9*y+16*x^8*y^2+x^7*y^3+2*x^6*y^4-x^5*y^5+x^9-5*x^8*y-7*x^7*y^2+x^6*y^3-x^ 5*y^4+9*x^7*y+6*x^5*y^3-3*x^4*y^4-3*x^7-3*x^6*y-2*x^5*y^2+8*x^4*y^3+3*x^6-x^5*y -2*x^4*y^2-8*x^3*y^3-2*x^4*y+8*x^3*y^2-x^4-4*x^2*y^2-x^3+4*x^2*y-2*x*y+2*x-1)/( x-1)/(x^12*y^3-3*x^12*y^2-x^11*y^3+3*x^12*y+3*x^11*y^2+x^10*y^3-x^12-3*x^11*y-x ^10*y^2+x^9*y^3+x^11-x^10*y-2*x^9*y^2+x^10+x^9*y+x^8*y^2+x^7*y^2-x^8-x^7-2*x^6* y+2*x^6+2*x^5*y-x^5-x^4+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1778342345 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 159" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 10 19 9 19 8 ) | ) C(m, n) x y | = - (x y - 6 x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 18 9 19 7 18 8 17 9 19 6 18 7 + 3 x y - 20 x y - 16 x y + x y + 15 x y + 35 x y 17 8 16 9 19 5 18 6 17 7 16 8 19 4 - 2 x y + x y - 6 x y - 40 x y - 5 x y - 2 x y + x y 18 5 17 6 16 7 15 8 18 4 17 5 + 25 x y + 20 x y - 2 x y + 3 x y - 8 x y - 25 x y 16 6 15 7 14 8 18 3 17 4 16 5 + 6 x y - 12 x y + 2 x y + x y + 14 x y + x y 15 6 14 7 17 3 16 4 15 5 14 6 + 21 x y - 3 x y - 3 x y - 10 x y - 25 x y - 15 x y 13 7 16 3 15 4 14 5 13 6 12 7 + 5 x y + 8 x y + 24 x y + 46 x y - 13 x y - x y 11 8 16 2 15 3 14 4 13 5 12 6 - x y - 2 x y - 15 x y - 49 x y + 6 x y + 12 x y 11 7 15 2 14 3 13 4 12 5 11 6 + 3 x y + 4 x y + 24 x y + 12 x y - 34 x y - x y 10 7 14 2 13 3 12 4 11 5 10 6 - 2 x y - 6 x y - 19 x y + 34 x y + x y + 2 x y 9 7 14 13 2 12 3 11 4 10 5 9 6 - 2 x y + x y + 14 x y - 6 x y - 7 x y + 11 x y + 3 x y 13 12 2 11 3 10 4 9 5 8 6 13 - 6 x y - 10 x y + 7 x y - 16 x y + x y - 4 x y + x 12 11 2 10 3 9 4 8 5 7 6 12 + 7 x y - 3 x y + x y - 13 x y + 3 x y - 2 x y - 2 x 11 10 2 9 3 8 4 10 9 2 8 3 + x y + 5 x y + 22 x y + 19 x y - 2 x y - 11 x y - 28 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 9 + x y - 4 x y + x - x y + 16 x y + x y + 2 x y - x y + x 8 7 2 6 3 5 4 7 5 3 4 4 7 - 5 x y - 7 x y + x y - x y + 9 x y + 6 x y - 3 x y - 3 x 6 5 2 4 3 6 5 4 2 3 3 4 - 3 x y - 2 x y + 8 x y + 3 x - x y - 2 x y - 8 x y - 2 x y 3 2 4 2 2 3 2 / + 8 x y - x - 4 x y - x + 4 x y - 2 x y + 2 x - 1) / ((x - 1) ( / 12 3 12 2 11 3 12 11 2 10 3 12 11 x y - 3 x y - x y + 3 x y + 3 x y + x y - x - 3 x y 10 2 9 3 11 10 9 2 10 9 8 2 7 2 8 - x y + x y + x - x y - 2 x y + x + x y + x y + x y - x 7 6 6 5 5 4 2 - x - 2 x y + 2 x + 2 x y - x - x + x + x - 1)) and in Maple notation -(x^19*y^10-6*x^19*y^9+15*x^19*y^8+3*x^18*y^9-20*x^19*y^7-16*x^18*y^8+x^17*y^9+ 15*x^19*y^6+35*x^18*y^7-2*x^17*y^8+x^16*y^9-6*x^19*y^5-40*x^18*y^6-5*x^17*y^7-2 *x^16*y^8+x^19*y^4+25*x^18*y^5+20*x^17*y^6-2*x^16*y^7+3*x^15*y^8-8*x^18*y^4-25* x^17*y^5+6*x^16*y^6-12*x^15*y^7+2*x^14*y^8+x^18*y^3+14*x^17*y^4+x^16*y^5+21*x^ 15*y^6-3*x^14*y^7-3*x^17*y^3-10*x^16*y^4-25*x^15*y^5-15*x^14*y^6+5*x^13*y^7+8*x ^16*y^3+24*x^15*y^4+46*x^14*y^5-13*x^13*y^6-x^12*y^7-x^11*y^8-2*x^16*y^2-15*x^ 15*y^3-49*x^14*y^4+6*x^13*y^5+12*x^12*y^6+3*x^11*y^7+4*x^15*y^2+24*x^14*y^3+12* x^13*y^4-34*x^12*y^5-x^11*y^6-2*x^10*y^7-6*x^14*y^2-19*x^13*y^3+34*x^12*y^4+x^ 11*y^5+2*x^10*y^6-2*x^9*y^7+x^14*y+14*x^13*y^2-6*x^12*y^3-7*x^11*y^4+11*x^10*y^ 5+3*x^9*y^6-6*x^13*y-10*x^12*y^2+7*x^11*y^3-16*x^10*y^4+x^9*y^5-4*x^8*y^6+x^13+ 7*x^12*y-3*x^11*y^2+x^10*y^3-13*x^9*y^4+3*x^8*y^5-2*x^7*y^6-2*x^12+x^11*y+5*x^ 10*y^2+22*x^9*y^3+19*x^8*y^4-2*x^10*y-11*x^9*y^2-28*x^8*y^3+x^7*y^4-4*x^6*y^5+x ^10-x^9*y+16*x^8*y^2+x^7*y^3+2*x^6*y^4-x^5*y^5+x^9-5*x^8*y-7*x^7*y^2+x^6*y^3-x^ 5*y^4+9*x^7*y+6*x^5*y^3-3*x^4*y^4-3*x^7-3*x^6*y-2*x^5*y^2+8*x^4*y^3+3*x^6-x^5*y -2*x^4*y^2-8*x^3*y^3-2*x^4*y+8*x^3*y^2-x^4-4*x^2*y^2-x^3+4*x^2*y-2*x*y+2*x-1)/( x-1)/(x^12*y^3-3*x^12*y^2-x^11*y^3+3*x^12*y+3*x^11*y^2+x^10*y^3-x^12-3*x^11*y-x ^10*y^2+x^9*y^3+x^11-x^10*y-2*x^9*y^2+x^10+x^9*y+x^8*y^2+x^7*y^2-x^8-x^7-2*x^6* y+2*x^6+2*x^5*y-x^5-x^4+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1778342345 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 160" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 18 5 17 6 16 7 + 7 x y - 10 x y - 20 x y + 5 x y + 30 x y + 3 x y 18 4 17 5 16 6 15 7 17 4 16 5 - x y - 25 x y - 14 x y + x y + 11 x y + 26 x y 14 7 17 3 16 4 15 5 14 6 13 7 + 5 x y - 2 x y - 24 x y - 10 x y - 19 x y - x y 16 3 15 4 14 5 13 6 12 7 16 2 + 11 x y + 20 x y + 29 x y + 5 x y - x y - 2 x y 15 3 14 4 13 5 12 6 15 2 14 3 - 15 x y - 25 x y - 7 x y + 13 x y + 4 x y + 16 x y 13 4 12 5 11 6 10 7 14 2 13 3 + 5 x y - 34 x y - 4 x y - x y - 8 x y - 9 x y 12 4 11 5 14 13 2 12 3 11 4 + 34 x y + 16 x y + 2 x y + 13 x y - 11 x y - 20 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 + 4 x y - 2 x y - 7 x y - 5 x y + 8 x y + 2 x y + 6 x y 8 6 13 12 10 3 9 4 8 5 12 10 2 - x y + x + 6 x y - 9 x y - 17 x y - 6 x y - 2 x + 3 x y 9 3 8 4 9 2 8 3 7 4 6 5 10 + 27 x y + 16 x y - 16 x y - 16 x y + 6 x y - 2 x y + x 9 8 2 7 3 6 4 9 8 7 2 6 3 + x y + 13 x y - 10 x y - 4 x y + x - 6 x y - x y + 6 x y 7 5 3 4 4 7 6 5 2 4 3 6 + 8 x y + 4 x y - 3 x y - 3 x - 4 x y - 3 x y + 8 x y + 3 x 4 2 3 3 4 3 2 4 2 2 3 2 - 2 x y - 8 x y - 2 x y + 8 x y - x - 4 x y - x + 4 x y / 13 3 13 2 12 3 13 12 2 - 2 x y + 2 x - 1) / (x y - 3 x y - 2 x y + 3 x y + 6 x y / 11 3 13 12 11 2 12 11 10 2 9 3 + 2 x y - x - 6 x y - 4 x y + 2 x + 2 x y - x y - x y 10 9 2 10 9 9 7 2 7 7 6 + 2 x y + 3 x y - x - x y - x - x y - 2 x y + 3 x + 4 x y 6 5 4 3 - 3 x - 2 x y + x + x - 2 x + 1) and in Maple notation -(x^18*y^9-5*x^18*y^8-x^17*y^9+10*x^18*y^7+7*x^17*y^8-10*x^18*y^6-20*x^17*y^7+5 *x^18*y^5+30*x^17*y^6+3*x^16*y^7-x^18*y^4-25*x^17*y^5-14*x^16*y^6+x^15*y^7+11*x ^17*y^4+26*x^16*y^5+5*x^14*y^7-2*x^17*y^3-24*x^16*y^4-10*x^15*y^5-19*x^14*y^6-x ^13*y^7+11*x^16*y^3+20*x^15*y^4+29*x^14*y^5+5*x^13*y^6-x^12*y^7-2*x^16*y^2-15*x ^15*y^3-25*x^14*y^4-7*x^13*y^5+13*x^12*y^6+4*x^15*y^2+16*x^14*y^3+5*x^13*y^4-34 *x^12*y^5-4*x^11*y^6-x^10*y^7-8*x^14*y^2-9*x^13*y^3+34*x^12*y^4+16*x^11*y^5+2*x ^14*y+13*x^13*y^2-11*x^12*y^3-20*x^11*y^4+4*x^10*y^5-2*x^9*y^6-7*x^13*y-5*x^12* y^2+8*x^11*y^3+2*x^10*y^4+6*x^9*y^5-x^8*y^6+x^13+6*x^12*y-9*x^10*y^3-17*x^9*y^4 -6*x^8*y^5-2*x^12+3*x^10*y^2+27*x^9*y^3+16*x^8*y^4-16*x^9*y^2-16*x^8*y^3+6*x^7* y^4-2*x^6*y^5+x^10+x^9*y+13*x^8*y^2-10*x^7*y^3-4*x^6*y^4+x^9-6*x^8*y-x^7*y^2+6* x^6*y^3+8*x^7*y+4*x^5*y^3-3*x^4*y^4-3*x^7-4*x^6*y-3*x^5*y^2+8*x^4*y^3+3*x^6-2*x ^4*y^2-8*x^3*y^3-2*x^4*y+8*x^3*y^2-x^4-4*x^2*y^2-x^3+4*x^2*y-2*x*y+2*x-1)/(x^13 *y^3-3*x^13*y^2-2*x^12*y^3+3*x^13*y+6*x^12*y^2+2*x^11*y^3-x^13-6*x^12*y-4*x^11* y^2+2*x^12+2*x^11*y-x^10*y^2-x^9*y^3+2*x^10*y+3*x^9*y^2-x^10-x^9*y-x^9-x^7*y^2-\ 2*x^7*y+3*x^7+4*x^6*y-3*x^6-2*x^5*y+x^4+x^3-2*x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1773241167 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 161" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y - x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 18 5 17 6 16 7 + 7 x y - 10 x y - 20 x y + 5 x y + 30 x y + 3 x y 18 4 17 5 16 6 15 7 17 4 16 5 - x y - 25 x y - 14 x y + x y + 11 x y + 26 x y 14 7 17 3 16 4 15 5 14 6 13 7 + 5 x y - 2 x y - 24 x y - 10 x y - 19 x y - x y 16 3 15 4 14 5 13 6 12 7 16 2 + 11 x y + 20 x y + 29 x y + 5 x y - x y - 2 x y 15 3 14 4 13 5 12 6 15 2 14 3 - 15 x y - 25 x y - 7 x y + 13 x y + 4 x y + 16 x y 13 4 12 5 11 6 10 7 14 2 13 3 + 5 x y - 34 x y - 4 x y - x y - 8 x y - 9 x y 12 4 11 5 14 13 2 12 3 11 4 + 34 x y + 16 x y + 2 x y + 13 x y - 11 x y - 20 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 + 4 x y - 2 x y - 7 x y - 5 x y + 8 x y + 2 x y + 6 x y 8 6 13 12 10 3 9 4 8 5 12 10 2 - x y + x + 6 x y - 9 x y - 17 x y - 6 x y - 2 x + 3 x y 9 3 8 4 9 2 8 3 7 4 6 5 10 + 27 x y + 16 x y - 16 x y - 16 x y + 6 x y - 2 x y + x 9 8 2 7 3 6 4 9 8 7 2 6 3 + x y + 13 x y - 10 x y - 4 x y + x - 6 x y - x y + 6 x y 7 5 3 4 4 7 6 5 2 4 3 6 + 8 x y + 4 x y - 3 x y - 3 x - 4 x y - 3 x y + 8 x y + 3 x 4 2 3 3 4 3 2 4 2 2 3 2 - 2 x y - 8 x y - 2 x y + 8 x y - x - 4 x y - x + 4 x y / 12 3 12 2 11 3 12 - 2 x y + 2 x - 1) / ((x - 1) (x y - 3 x y - x y + 3 x y / 11 2 10 3 12 11 10 2 9 3 11 10 + 3 x y + x y - x - 3 x y - x y + x y + x - x y 9 2 10 9 8 2 7 2 8 7 6 6 5 - 2 x y + x + x y + x y + x y - x - x - 2 x y + 2 x + 2 x y 5 4 2 - x - x + x + x - 1)) and in Maple notation -(x^18*y^9-5*x^18*y^8-x^17*y^9+10*x^18*y^7+7*x^17*y^8-10*x^18*y^6-20*x^17*y^7+5 *x^18*y^5+30*x^17*y^6+3*x^16*y^7-x^18*y^4-25*x^17*y^5-14*x^16*y^6+x^15*y^7+11*x ^17*y^4+26*x^16*y^5+5*x^14*y^7-2*x^17*y^3-24*x^16*y^4-10*x^15*y^5-19*x^14*y^6-x ^13*y^7+11*x^16*y^3+20*x^15*y^4+29*x^14*y^5+5*x^13*y^6-x^12*y^7-2*x^16*y^2-15*x ^15*y^3-25*x^14*y^4-7*x^13*y^5+13*x^12*y^6+4*x^15*y^2+16*x^14*y^3+5*x^13*y^4-34 *x^12*y^5-4*x^11*y^6-x^10*y^7-8*x^14*y^2-9*x^13*y^3+34*x^12*y^4+16*x^11*y^5+2*x ^14*y+13*x^13*y^2-11*x^12*y^3-20*x^11*y^4+4*x^10*y^5-2*x^9*y^6-7*x^13*y-5*x^12* y^2+8*x^11*y^3+2*x^10*y^4+6*x^9*y^5-x^8*y^6+x^13+6*x^12*y-9*x^10*y^3-17*x^9*y^4 -6*x^8*y^5-2*x^12+3*x^10*y^2+27*x^9*y^3+16*x^8*y^4-16*x^9*y^2-16*x^8*y^3+6*x^7* y^4-2*x^6*y^5+x^10+x^9*y+13*x^8*y^2-10*x^7*y^3-4*x^6*y^4+x^9-6*x^8*y-x^7*y^2+6* x^6*y^3+8*x^7*y+4*x^5*y^3-3*x^4*y^4-3*x^7-4*x^6*y-3*x^5*y^2+8*x^4*y^3+3*x^6-2*x ^4*y^2-8*x^3*y^3-2*x^4*y+8*x^3*y^2-x^4-4*x^2*y^2-x^3+4*x^2*y-2*x*y+2*x-1)/(x-1) /(x^12*y^3-3*x^12*y^2-x^11*y^3+3*x^12*y+3*x^11*y^2+x^10*y^3-x^12-3*x^11*y-x^10* y^2+x^9*y^3+x^11-x^10*y-2*x^9*y^2+x^10+x^9*y+x^8*y^2+x^7*y^2-x^8-x^7-2*x^6*y+2* x^6+2*x^5*y-x^5-x^4+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1773241167 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 162" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = (x y + 2 x y - 4 x y - 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 5 13 6 12 7 14 4 13 5 12 6 + 6 x y + 9 x y + 3 x y - 4 x y - 6 x y - 9 x y 11 7 14 3 13 4 12 5 11 6 13 3 12 4 + 2 x y + x y + 4 x y + 9 x y - x y - 3 x y - 3 x y 11 5 10 6 13 2 11 4 10 5 9 6 11 3 - 7 x y + 2 x y + x y + 6 x y - 3 x y - x y + 4 x y 9 5 8 6 11 2 9 4 8 5 11 10 2 + 5 x y - x y - 5 x y - 3 x y + 2 x y + x y + 2 x y 9 3 8 4 7 5 6 6 10 9 2 8 3 - 6 x y - x y - 3 x y - 2 x y - x y + 6 x y + 3 x y 7 4 9 8 2 7 3 6 4 5 5 8 7 2 + 3 x y - x y - 6 x y + 3 x y + 5 x y - 4 x y + 3 x y - x y 6 3 5 4 7 6 2 5 3 4 4 7 6 - 5 x y - x y - 3 x y + 4 x y + 9 x y - 5 x y + x - x y 5 2 4 3 6 5 4 2 3 3 5 4 - 5 x y + 2 x y - x + 2 x y + 4 x y - 8 x y - x - 3 x y 3 2 4 3 2 2 3 2 2 / + 4 x y + x + 3 x y - 4 x y - x + 2 x y + x - 2 x y + x - 1) / / 11 3 11 2 10 3 11 10 2 10 9 2 9 (x y - 2 x y - x y + x y + 2 x y - x y + x y - x y 8 2 8 7 2 7 7 6 5 5 4 4 - 2 x y + 2 x y + x y - 2 x y + x - x + x y - x - x y + x 3 3 2 + x y - x + x + x - 1) and in Maple notation (x^14*y^7+2*x^13*y^8-4*x^14*y^6-7*x^13*y^7+6*x^14*y^5+9*x^13*y^6+3*x^12*y^7-4*x ^14*y^4-6*x^13*y^5-9*x^12*y^6+2*x^11*y^7+x^14*y^3+4*x^13*y^4+9*x^12*y^5-x^11*y^ 6-3*x^13*y^3-3*x^12*y^4-7*x^11*y^5+2*x^10*y^6+x^13*y^2+6*x^11*y^4-3*x^10*y^5-x^ 9*y^6+4*x^11*y^3+5*x^9*y^5-x^8*y^6-5*x^11*y^2-3*x^9*y^4+2*x^8*y^5+x^11*y+2*x^10 *y^2-6*x^9*y^3-x^8*y^4-3*x^7*y^5-2*x^6*y^6-x^10*y+6*x^9*y^2+3*x^8*y^3+3*x^7*y^4 -x^9*y-6*x^8*y^2+3*x^7*y^3+5*x^6*y^4-4*x^5*y^5+3*x^8*y-x^7*y^2-5*x^6*y^3-x^5*y^ 4-3*x^7*y+4*x^6*y^2+9*x^5*y^3-5*x^4*y^4+x^7-x^6*y-5*x^5*y^2+2*x^4*y^3-x^6+2*x^5 *y+4*x^4*y^2-8*x^3*y^3-x^5-3*x^4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2*y^2-x^3+2*x^2*y+ x^2-2*x*y+x-1)/(x^11*y^3-2*x^11*y^2-x^10*y^3+x^11*y+2*x^10*y^2-x^10*y+x^9*y^2-x ^9*y-2*x^8*y^2+2*x^8*y+x^7*y^2-2*x^7*y+x^7-x^6+x^5*y-x^5-x^4*y+x^4+x^3*y-x^3+x^ 2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1686150553 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 163" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 13 8 14 6 13 7 ) | ) C(m, n) x y | = (x y + 2 x y - 4 x y - 7 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 5 13 6 12 7 14 4 13 5 12 6 + 6 x y + 9 x y + 3 x y - 4 x y - 6 x y - 9 x y 11 7 14 3 13 4 12 5 11 6 13 3 12 4 + 2 x y + x y + 4 x y + 9 x y - x y - 3 x y - 3 x y 11 5 10 6 13 2 11 4 10 5 9 6 11 3 - 7 x y + 2 x y + x y + 6 x y - 3 x y - x y + 4 x y 9 5 8 6 11 2 9 4 8 5 11 10 2 + 5 x y - x y - 5 x y - 3 x y + 2 x y + x y + 2 x y 9 3 8 4 7 5 6 6 10 9 2 8 3 - 6 x y - x y - 3 x y - 2 x y - x y + 6 x y + 3 x y 7 4 9 8 2 7 3 6 4 5 5 8 7 2 + 3 x y - x y - 6 x y + 3 x y + 5 x y - 4 x y + 3 x y - x y 6 3 5 4 7 6 2 5 3 4 4 7 6 - 5 x y - x y - 3 x y + 4 x y + 9 x y - 5 x y + x - x y 5 2 4 3 6 5 4 2 3 3 5 4 - 5 x y + 2 x y - x + 2 x y + 4 x y - 8 x y - x - 3 x y 3 2 4 3 2 2 3 2 2 / + 4 x y + x + 3 x y - 4 x y - x + 2 x y + x - 2 x y + x - 1) / / 10 3 10 2 10 8 2 8 7 2 7 6 ((x - 1) (x y - 2 x y + x y + x y - x y - x y + x y - x y 6 5 4 3 2 + x - x y - x - x y - x + 1)) and in Maple notation (x^14*y^7+2*x^13*y^8-4*x^14*y^6-7*x^13*y^7+6*x^14*y^5+9*x^13*y^6+3*x^12*y^7-4*x ^14*y^4-6*x^13*y^5-9*x^12*y^6+2*x^11*y^7+x^14*y^3+4*x^13*y^4+9*x^12*y^5-x^11*y^ 6-3*x^13*y^3-3*x^12*y^4-7*x^11*y^5+2*x^10*y^6+x^13*y^2+6*x^11*y^4-3*x^10*y^5-x^ 9*y^6+4*x^11*y^3+5*x^9*y^5-x^8*y^6-5*x^11*y^2-3*x^9*y^4+2*x^8*y^5+x^11*y+2*x^10 *y^2-6*x^9*y^3-x^8*y^4-3*x^7*y^5-2*x^6*y^6-x^10*y+6*x^9*y^2+3*x^8*y^3+3*x^7*y^4 -x^9*y-6*x^8*y^2+3*x^7*y^3+5*x^6*y^4-4*x^5*y^5+3*x^8*y-x^7*y^2-5*x^6*y^3-x^5*y^ 4-3*x^7*y+4*x^6*y^2+9*x^5*y^3-5*x^4*y^4+x^7-x^6*y-5*x^5*y^2+2*x^4*y^3-x^6+2*x^5 *y+4*x^4*y^2-8*x^3*y^3-x^5-3*x^4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2*y^2-x^3+2*x^2*y+ x^2-2*x*y+x-1)/(x-1)/(x^10*y^3-2*x^10*y^2+x^10*y+x^8*y^2-x^8*y-x^7*y^2+x^7*y-x^ 6*y+x^6-x^5*y-x^4-x^3*y-x^2+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1686150553 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 164" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 16 6 15 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 5 15 6 14 7 13 8 16 4 15 5 14 6 - 10 x y - 5 x y + x y - x y + 5 x y + 10 x y - 4 x y 13 7 16 3 15 4 14 5 13 6 12 7 15 3 + 2 x y - x y - 10 x y + 6 x y + 4 x y - x y + 5 x y 14 4 13 5 12 6 11 7 15 2 14 3 13 4 - 4 x y - 16 x y + 2 x y + x y - x y + x y + 19 x y 12 5 11 6 13 3 12 4 11 5 10 6 + x y - 8 x y - 10 x y - 5 x y + 18 x y + 2 x y 13 2 12 3 11 4 10 5 9 6 12 2 11 3 + 2 x y + 4 x y - 15 x y - 10 x y + x y - x y + 2 x y 10 4 9 5 11 2 10 3 9 4 8 5 11 + 15 x y - 2 x y + 3 x y - 7 x y - 5 x y + x y - x y 10 2 9 3 8 4 7 5 6 6 10 9 2 8 3 - x y + 13 x y - x y + x y + 2 x y + x y - 8 x y - 5 x y 7 4 6 5 9 8 2 7 3 6 4 5 5 8 + 3 x y - x y + x y + 9 x y - 8 x y - 6 x y + 3 x y - 4 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 7 + x y + 9 x y + x y + 4 x y - 7 x y - 9 x y + 5 x y - x 6 5 2 4 3 6 5 4 2 3 3 5 + 2 x y + 7 x y - 2 x y + x - 3 x y - 4 x y + 8 x y + x 4 3 2 4 3 2 2 3 2 2 + 3 x y - 4 x y - x - 3 x y + 4 x y + x - 2 x y - x + 2 x y - x / 10 3 10 2 10 8 2 8 7 2 7 + 1) / ((x - 1) (x y - 2 x y + x y + x y - x y - x y + x y / 6 6 5 4 3 2 - x y + x - x y - x - x y - x + 1)) and in Maple notation -(x^16*y^8-5*x^16*y^7+10*x^16*y^6+x^15*y^7-10*x^16*y^5-5*x^15*y^6+x^14*y^7-x^13 *y^8+5*x^16*y^4+10*x^15*y^5-4*x^14*y^6+2*x^13*y^7-x^16*y^3-10*x^15*y^4+6*x^14*y ^5+4*x^13*y^6-x^12*y^7+5*x^15*y^3-4*x^14*y^4-16*x^13*y^5+2*x^12*y^6+x^11*y^7-x^ 15*y^2+x^14*y^3+19*x^13*y^4+x^12*y^5-8*x^11*y^6-10*x^13*y^3-5*x^12*y^4+18*x^11* y^5+2*x^10*y^6+2*x^13*y^2+4*x^12*y^3-15*x^11*y^4-10*x^10*y^5+x^9*y^6-x^12*y^2+2 *x^11*y^3+15*x^10*y^4-2*x^9*y^5+3*x^11*y^2-7*x^10*y^3-5*x^9*y^4+x^8*y^5-x^11*y- x^10*y^2+13*x^9*y^3-x^8*y^4+x^7*y^5+2*x^6*y^6+x^10*y-8*x^9*y^2-5*x^8*y^3+3*x^7* y^4-x^6*y^5+x^9*y+9*x^8*y^2-8*x^7*y^3-6*x^6*y^4+3*x^5*y^5-4*x^8*y+x^7*y^2+9*x^6 *y^3+x^5*y^4+4*x^7*y-7*x^6*y^2-9*x^5*y^3+5*x^4*y^4-x^7+2*x^6*y+7*x^5*y^2-2*x^4* y^3+x^6-3*x^5*y-4*x^4*y^2+8*x^3*y^3+x^5+3*x^4*y-4*x^3*y^2-x^4-3*x^3*y+4*x^2*y^2 +x^3-2*x^2*y-x^2+2*x*y-x+1)/(x-1)/(x^10*y^3-2*x^10*y^2+x^10*y+x^8*y^2-x^8*y-x^7 *y^2+x^7*y-x^6*y+x^6-x^5*y-x^4-x^3*y-x^2+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1683390423 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 165" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 1, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 8 16 7 16 6 15 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 5 15 6 14 7 13 8 16 4 15 5 14 6 - 10 x y - 5 x y + x y - x y + 5 x y + 10 x y - 4 x y 13 7 16 3 15 4 14 5 13 6 12 7 15 3 + 2 x y - x y - 10 x y + 6 x y + 4 x y - x y + 5 x y 14 4 13 5 12 6 11 7 15 2 14 3 13 4 - 4 x y - 16 x y + 2 x y + x y - x y + x y + 19 x y 12 5 11 6 13 3 12 4 11 5 10 6 + x y - 8 x y - 10 x y - 5 x y + 18 x y + 2 x y 13 2 12 3 11 4 10 5 9 6 12 2 11 3 + 2 x y + 4 x y - 15 x y - 10 x y + x y - x y + 2 x y 10 4 9 5 11 2 10 3 9 4 8 5 11 + 15 x y - 2 x y + 3 x y - 7 x y - 5 x y + x y - x y 10 2 9 3 8 4 7 5 6 6 10 9 2 8 3 - x y + 13 x y - x y + x y + 2 x y + x y - 8 x y - 5 x y 7 4 6 5 9 8 2 7 3 6 4 5 5 8 + 3 x y - x y + x y + 9 x y - 8 x y - 6 x y + 3 x y - 4 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 7 + x y + 9 x y + x y + 4 x y - 7 x y - 9 x y + 5 x y - x 6 5 2 4 3 6 5 4 2 3 3 5 + 2 x y + 7 x y - 2 x y + x - 3 x y - 4 x y + 8 x y + x 4 3 2 4 3 2 2 3 2 2 + 3 x y - 4 x y - x - 3 x y + 4 x y + x - 2 x y - x + 2 x y - x / 11 3 11 2 10 3 11 10 2 10 9 2 + 1) / (x y - 2 x y - x y + x y + 2 x y - x y + x y / 9 8 2 8 7 2 7 7 6 5 5 4 - x y - 2 x y + 2 x y + x y - 2 x y + x - x + x y - x - x y 4 3 3 2 + x + x y - x + x + x - 1) and in Maple notation -(x^16*y^8-5*x^16*y^7+10*x^16*y^6+x^15*y^7-10*x^16*y^5-5*x^15*y^6+x^14*y^7-x^13 *y^8+5*x^16*y^4+10*x^15*y^5-4*x^14*y^6+2*x^13*y^7-x^16*y^3-10*x^15*y^4+6*x^14*y ^5+4*x^13*y^6-x^12*y^7+5*x^15*y^3-4*x^14*y^4-16*x^13*y^5+2*x^12*y^6+x^11*y^7-x^ 15*y^2+x^14*y^3+19*x^13*y^4+x^12*y^5-8*x^11*y^6-10*x^13*y^3-5*x^12*y^4+18*x^11* y^5+2*x^10*y^6+2*x^13*y^2+4*x^12*y^3-15*x^11*y^4-10*x^10*y^5+x^9*y^6-x^12*y^2+2 *x^11*y^3+15*x^10*y^4-2*x^9*y^5+3*x^11*y^2-7*x^10*y^3-5*x^9*y^4+x^8*y^5-x^11*y- x^10*y^2+13*x^9*y^3-x^8*y^4+x^7*y^5+2*x^6*y^6+x^10*y-8*x^9*y^2-5*x^8*y^3+3*x^7* y^4-x^6*y^5+x^9*y+9*x^8*y^2-8*x^7*y^3-6*x^6*y^4+3*x^5*y^5-4*x^8*y+x^7*y^2+9*x^6 *y^3+x^5*y^4+4*x^7*y-7*x^6*y^2-9*x^5*y^3+5*x^4*y^4-x^7+2*x^6*y+7*x^5*y^2-2*x^4* y^3+x^6-3*x^5*y-4*x^4*y^2+8*x^3*y^3+x^5+3*x^4*y-4*x^3*y^2-x^4-3*x^3*y+4*x^2*y^2 +x^3-2*x^2*y-x^2+2*x*y-x+1)/(x^11*y^3-2*x^11*y^2-x^10*y^3+x^11*y+2*x^10*y^2-x^ 10*y+x^9*y^2-x^9*y-2*x^8*y^2+2*x^8*y+x^7*y^2-2*x^7*y+x^7-x^6+x^5*y-x^5-x^4*y+x^ 4+x^3*y-x^3+x^2+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1683390423 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 166" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 1, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 6 12 5 11 6 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 4 11 5 10 6 12 3 11 4 10 5 9 6 - 5 x y - 4 x y - 2 x y + x y + 6 x y + 5 x y - 2 x y 11 3 10 4 9 5 8 6 11 2 10 3 9 4 - 4 x y - 4 x y + 7 x y - 2 x y + x y + x y - 8 x y 8 5 7 6 9 3 8 4 7 5 8 3 7 4 - 2 x y - 2 x y + 3 x y + 14 x y + 2 x y - 15 x y - 6 x y 6 5 8 2 7 3 6 4 5 5 7 2 6 3 + 3 x y + 5 x y + 14 x y + x y + 4 x y - 11 x y - 8 x y 5 4 7 6 2 5 3 4 4 6 5 2 - 5 x y + 3 x y + 8 x y + 4 x y + 5 x y - 4 x y - 5 x y 4 3 6 5 4 2 3 3 5 4 3 2 4 - 6 x y + x + 4 x y + 4 x y + 8 x y - x - 3 x y - 6 x y + x 3 2 2 3 2 / 7 2 7 6 2 + x y + 4 x y - x - 3 x y + 2 x y - x + 1) / (x y - x y - x y / 6 6 5 5 4 4 3 3 2 + 2 x y - x - 2 x y + x + x y - x - x y + x + x y + x - 1) and in Maple notation -(2*x^12*y^7-7*x^12*y^6+9*x^12*y^5+x^11*y^6-5*x^12*y^4-4*x^11*y^5-2*x^10*y^6+x^ 12*y^3+6*x^11*y^4+5*x^10*y^5-2*x^9*y^6-4*x^11*y^3-4*x^10*y^4+7*x^9*y^5-2*x^8*y^ 6+x^11*y^2+x^10*y^3-8*x^9*y^4-2*x^8*y^5-2*x^7*y^6+3*x^9*y^3+14*x^8*y^4+2*x^7*y^ 5-15*x^8*y^3-6*x^7*y^4+3*x^6*y^5+5*x^8*y^2+14*x^7*y^3+x^6*y^4+4*x^5*y^5-11*x^7* y^2-8*x^6*y^3-5*x^5*y^4+3*x^7*y+8*x^6*y^2+4*x^5*y^3+5*x^4*y^4-4*x^6*y-5*x^5*y^2 -6*x^4*y^3+x^6+4*x^5*y+4*x^4*y^2+8*x^3*y^3-x^5-3*x^4*y-6*x^3*y^2+x^4+x^3*y+4*x^ 2*y^2-x^3-3*x^2*y+2*x*y-x+1)/(x^7*y^2-x^7*y-x^6*y^2+2*x^6*y-x^6-2*x^5*y+x^5+x^4 *y-x^4-x^3*y+x^3+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1631971797 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 167" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1], [1, 2, 2, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 12 7 12 6 11 7 12 5 ) | ) C(m, n) x y | = - (x y - 4 x y + x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 6 12 4 11 5 12 3 11 4 9 6 9 5 - 3 x y - 4 x y + 3 x y + x y - x y - 3 x y + 8 x y 8 6 9 4 8 5 7 6 9 3 8 4 7 5 - x y - 7 x y - 3 x y - x y + 2 x y + 13 x y + x y 8 3 7 4 6 5 8 2 7 3 6 4 5 5 - 13 x y - 2 x y - x y + 4 x y + 7 x y + 4 x y + 3 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 - 8 x y - 8 x y - 6 x y + 3 x y + 8 x y + 7 x y + 5 x y 6 5 2 4 3 6 5 4 2 3 3 5 - 4 x y - 6 x y - 6 x y + x + 4 x y + 4 x y + 8 x y - x 4 3 2 4 3 2 2 3 2 - 3 x y - 6 x y + x + x y + 4 x y - x - 3 x y + 2 x y - x + 1) / 7 2 7 6 2 6 6 5 5 4 4 3 / (x y - x y - x y + 2 x y - x - 2 x y + x + x y - x - x y / 3 2 + x + x y + x - 1) and in Maple notation -(x^12*y^7-4*x^12*y^6+x^11*y^7+6*x^12*y^5-3*x^11*y^6-4*x^12*y^4+3*x^11*y^5+x^12 *y^3-x^11*y^4-3*x^9*y^6+8*x^9*y^5-x^8*y^6-7*x^9*y^4-3*x^8*y^5-x^7*y^6+2*x^9*y^3 +13*x^8*y^4+x^7*y^5-13*x^8*y^3-2*x^7*y^4-x^6*y^5+4*x^8*y^2+7*x^7*y^3+4*x^6*y^4+ 3*x^5*y^5-8*x^7*y^2-8*x^6*y^3-6*x^5*y^4+3*x^7*y+8*x^6*y^2+7*x^5*y^3+5*x^4*y^4-4 *x^6*y-6*x^5*y^2-6*x^4*y^3+x^6+4*x^5*y+4*x^4*y^2+8*x^3*y^3-x^5-3*x^4*y-6*x^3*y^ 2+x^4+x^3*y+4*x^2*y^2-x^3-3*x^2*y+2*x*y-x+1)/(x^7*y^2-x^7*y-x^6*y^2+2*x^6*y-x^6 -2*x^5*y+x^5+x^4*y-x^4-x^3*y+x^3+x^2*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1628678852 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 168" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 10 17 9 17 8 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 9 17 7 16 8 15 9 17 6 16 7 + 2 x y - 10 x y - 9 x y + 2 x y + 5 x y + 16 x y 15 8 14 9 17 5 16 6 15 7 14 8 16 5 - 8 x y - x y - x y - 14 x y + 12 x y + 6 x y + 6 x y 15 6 14 7 13 8 16 4 15 5 14 6 - 8 x y - 13 x y - 2 x y - x y + 2 x y + 13 x y 13 7 12 8 14 5 13 6 12 7 11 8 14 4 + 7 x y - 2 x y - 6 x y - 8 x y + 2 x y - x y + x y 13 5 12 6 13 4 12 5 11 6 10 7 13 3 + 2 x y + 7 x y + 2 x y - 13 x y + 3 x y - x y - x y 12 4 10 6 9 7 12 3 11 4 10 5 9 6 + 7 x y - x y - 2 x y - x y - 4 x y + 7 x y + 6 x y 8 7 11 3 10 4 9 5 8 6 8 5 7 6 + x y + 2 x y - 6 x y - 6 x y - 5 x y + 8 x y + x y 10 2 9 3 8 4 7 5 6 6 9 2 8 3 + x y + 4 x y - 6 x y - 4 x y + 2 x y - 2 x y + x y 7 4 8 2 7 3 6 4 5 5 8 7 2 + 8 x y + 2 x y - 8 x y - 4 x y + 3 x y - x y + 2 x y 6 3 5 4 7 6 2 4 4 7 5 2 4 3 + 5 x y - 2 x y + 2 x y - 4 x y + 5 x y - x + x y - 10 x y 6 5 4 2 3 3 5 4 3 2 4 3 + x - 2 x y + 4 x y + 8 x y + x + 2 x y - 8 x y - x + x y 2 2 2 2 / + 4 x y - 4 x y + x + 2 x y - 2 x + 1) / ((x - 1) / 9 3 9 2 6 2 6 6 4 4 3 (x y - x y - x y - x y + x + x y - x - x y - x + 1)) and in Maple notation -(x^17*y^10-5*x^17*y^9+10*x^17*y^8+2*x^16*y^9-10*x^17*y^7-9*x^16*y^8+2*x^15*y^9 +5*x^17*y^6+16*x^16*y^7-8*x^15*y^8-x^14*y^9-x^17*y^5-14*x^16*y^6+12*x^15*y^7+6* x^14*y^8+6*x^16*y^5-8*x^15*y^6-13*x^14*y^7-2*x^13*y^8-x^16*y^4+2*x^15*y^5+13*x^ 14*y^6+7*x^13*y^7-2*x^12*y^8-6*x^14*y^5-8*x^13*y^6+2*x^12*y^7-x^11*y^8+x^14*y^4 +2*x^13*y^5+7*x^12*y^6+2*x^13*y^4-13*x^12*y^5+3*x^11*y^6-x^10*y^7-x^13*y^3+7*x^ 12*y^4-x^10*y^6-2*x^9*y^7-x^12*y^3-4*x^11*y^4+7*x^10*y^5+6*x^9*y^6+x^8*y^7+2*x^ 11*y^3-6*x^10*y^4-6*x^9*y^5-5*x^8*y^6+8*x^8*y^5+x^7*y^6+x^10*y^2+4*x^9*y^3-6*x^ 8*y^4-4*x^7*y^5+2*x^6*y^6-2*x^9*y^2+x^8*y^3+8*x^7*y^4+2*x^8*y^2-8*x^7*y^3-4*x^6 *y^4+3*x^5*y^5-x^8*y+2*x^7*y^2+5*x^6*y^3-2*x^5*y^4+2*x^7*y-4*x^6*y^2+5*x^4*y^4- x^7+x^5*y^2-10*x^4*y^3+x^6-2*x^5*y+4*x^4*y^2+8*x^3*y^3+x^5+2*x^4*y-8*x^3*y^2-x^ 4+x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+1)/(x-1)/(x^9*y^3-x^9*y^2-x^6*y^2-x^6*y +x^6+x^4*y-x^4-x^3*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1620504344 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 169" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 2], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 10 17 9 17 8 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 9 17 7 16 8 15 9 17 6 16 7 + 2 x y - 10 x y - 9 x y + 2 x y + 5 x y + 16 x y 15 8 14 9 17 5 16 6 15 7 14 8 16 5 - 8 x y - x y - x y - 14 x y + 12 x y + 6 x y + 6 x y 15 6 14 7 13 8 16 4 15 5 14 6 - 8 x y - 13 x y - 2 x y - x y + 2 x y + 13 x y 13 7 12 8 14 5 13 6 12 7 11 8 14 4 + 7 x y - 2 x y - 6 x y - 8 x y + 2 x y - x y + x y 13 5 12 6 13 4 12 5 11 6 10 7 13 3 + 2 x y + 7 x y + 2 x y - 13 x y + 3 x y - x y - x y 12 4 10 6 9 7 12 3 11 4 10 5 9 6 + 7 x y - x y - 2 x y - x y - 4 x y + 7 x y + 6 x y 8 7 11 3 10 4 9 5 8 6 8 5 7 6 + x y + 2 x y - 6 x y - 6 x y - 5 x y + 8 x y + x y 10 2 9 3 8 4 7 5 6 6 9 2 8 3 + x y + 4 x y - 6 x y - 4 x y + 2 x y - 2 x y + x y 7 4 8 2 7 3 6 4 5 5 8 7 2 + 8 x y + 2 x y - 8 x y - 4 x y + 3 x y - x y + 2 x y 6 3 5 4 7 6 2 4 4 7 5 2 4 3 + 5 x y - 2 x y + 2 x y - 4 x y + 5 x y - x + x y - 10 x y 6 5 4 2 3 3 5 4 3 2 4 3 + x - 2 x y + 4 x y + 8 x y + x + 2 x y - 8 x y - x + x y 2 2 2 2 / + 4 x y - 4 x y + x + 2 x y - 2 x + 1) / ((x - 1) / 9 3 9 2 6 2 6 6 4 4 3 (x y - x y - x y - x y + x + x y - x - x y - x + 1)) and in Maple notation -(x^17*y^10-5*x^17*y^9+10*x^17*y^8+2*x^16*y^9-10*x^17*y^7-9*x^16*y^8+2*x^15*y^9 +5*x^17*y^6+16*x^16*y^7-8*x^15*y^8-x^14*y^9-x^17*y^5-14*x^16*y^6+12*x^15*y^7+6* x^14*y^8+6*x^16*y^5-8*x^15*y^6-13*x^14*y^7-2*x^13*y^8-x^16*y^4+2*x^15*y^5+13*x^ 14*y^6+7*x^13*y^7-2*x^12*y^8-6*x^14*y^5-8*x^13*y^6+2*x^12*y^7-x^11*y^8+x^14*y^4 +2*x^13*y^5+7*x^12*y^6+2*x^13*y^4-13*x^12*y^5+3*x^11*y^6-x^10*y^7-x^13*y^3+7*x^ 12*y^4-x^10*y^6-2*x^9*y^7-x^12*y^3-4*x^11*y^4+7*x^10*y^5+6*x^9*y^6+x^8*y^7+2*x^ 11*y^3-6*x^10*y^4-6*x^9*y^5-5*x^8*y^6+8*x^8*y^5+x^7*y^6+x^10*y^2+4*x^9*y^3-6*x^ 8*y^4-4*x^7*y^5+2*x^6*y^6-2*x^9*y^2+x^8*y^3+8*x^7*y^4+2*x^8*y^2-8*x^7*y^3-4*x^6 *y^4+3*x^5*y^5-x^8*y+2*x^7*y^2+5*x^6*y^3-2*x^5*y^4+2*x^7*y-4*x^6*y^2+5*x^4*y^4- x^7+x^5*y^2-10*x^4*y^3+x^6-2*x^5*y+4*x^4*y^2+8*x^3*y^3+x^5+2*x^4*y-8*x^3*y^2-x^ 4+x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+1)/(x-1)/(x^9*y^3-x^9*y^2-x^6*y^2-x^6*y +x^6+x^4*y-x^4-x^3*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1620504344 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 170" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 16 8 18 5 17 6 - 5 x y - 10 x y + 10 x y + 3 x y + 5 x y - 10 x y 16 7 15 8 18 4 17 5 16 6 15 7 14 8 - 14 x y + x y - x y + 5 x y + 26 x y - 6 x y - x y 17 4 16 5 15 6 14 7 16 4 15 5 - x y - 24 x y + 14 x y + 7 x y + 11 x y - 16 x y 14 6 16 3 15 4 14 5 13 6 12 7 - 19 x y - 2 x y + 9 x y + 26 x y - 3 x y - 3 x y 15 3 14 4 13 5 12 6 11 7 14 3 - 2 x y - 19 x y + 10 x y + 13 x y - 2 x y + 7 x y 13 4 12 5 11 6 14 2 13 3 12 4 - 12 x y - 20 x y + 3 x y - x y + 6 x y + 14 x y 11 5 10 6 13 2 12 3 11 4 10 5 9 6 + x y - 6 x y - x y - 5 x y - x y + 12 x y - x y 12 2 11 3 10 4 9 5 8 6 11 2 10 3 + x y - 3 x y - 5 x y + 3 x y + 2 x y + 2 x y - 3 x y 9 4 8 5 10 2 9 3 8 4 7 5 9 2 - 8 x y - 7 x y + 2 x y + 10 x y + 8 x y + 2 x y - 4 x y 8 3 7 4 6 5 8 2 7 3 6 4 8 - 6 x y + 7 x y + 3 x y + 5 x y - 11 x y - 12 x y - 2 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 7 + x y + 15 x y + 5 x y + 3 x y - 9 x y - 6 x y + 3 x y - x 6 5 2 4 3 6 5 4 2 3 3 5 4 + x y + 4 x y - 8 x y + x - 3 x y + 4 x y + 8 x y + x + 2 x y 3 2 4 3 2 2 2 2 / - 8 x y - x + x y + 4 x y - 4 x y + x + 2 x y - 2 x + 1) / ( / 10 3 10 2 9 3 9 2 7 2 7 6 2 7 6 6 x y - x y - x y + x y - x y - x y + x y + x + x y - x 5 5 4 4 3 2 + x y - x - 2 x y + x + x y - x + 2 x - 1) and in Maple notation -(x^18*y^9-5*x^18*y^8+x^17*y^9+10*x^18*y^7-5*x^17*y^8-10*x^18*y^6+10*x^17*y^7+3 *x^16*y^8+5*x^18*y^5-10*x^17*y^6-14*x^16*y^7+x^15*y^8-x^18*y^4+5*x^17*y^5+26*x^ 16*y^6-6*x^15*y^7-x^14*y^8-x^17*y^4-24*x^16*y^5+14*x^15*y^6+7*x^14*y^7+11*x^16* y^4-16*x^15*y^5-19*x^14*y^6-2*x^16*y^3+9*x^15*y^4+26*x^14*y^5-3*x^13*y^6-3*x^12 *y^7-2*x^15*y^3-19*x^14*y^4+10*x^13*y^5+13*x^12*y^6-2*x^11*y^7+7*x^14*y^3-12*x^ 13*y^4-20*x^12*y^5+3*x^11*y^6-x^14*y^2+6*x^13*y^3+14*x^12*y^4+x^11*y^5-6*x^10*y ^6-x^13*y^2-5*x^12*y^3-x^11*y^4+12*x^10*y^5-x^9*y^6+x^12*y^2-3*x^11*y^3-5*x^10* y^4+3*x^9*y^5+2*x^8*y^6+2*x^11*y^2-3*x^10*y^3-8*x^9*y^4-7*x^8*y^5+2*x^10*y^2+10 *x^9*y^3+8*x^8*y^4+2*x^7*y^5-4*x^9*y^2-6*x^8*y^3+7*x^7*y^4+3*x^6*y^5+5*x^8*y^2-\ 11*x^7*y^3-12*x^6*y^4-2*x^8*y+x^7*y^2+15*x^6*y^3+5*x^5*y^4+3*x^7*y-9*x^6*y^2-6* x^5*y^3+3*x^4*y^4-x^7+x^6*y+4*x^5*y^2-8*x^4*y^3+x^6-3*x^5*y+4*x^4*y^2+8*x^3*y^3 +x^5+2*x^4*y-8*x^3*y^2-x^4+x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+1)/(x^10*y^3-x ^10*y^2-x^9*y^3+x^9*y^2-x^7*y^2-x^7*y+x^6*y^2+x^7+x^6*y-x^6+x^5*y-x^5-2*x^4*y+x ^4+x^3*y-x^2+2*x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1616056782 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 171" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 18 9 18 8 17 9 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y + x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 8 18 6 17 7 16 8 18 5 17 6 - 5 x y - 10 x y + 10 x y + 3 x y + 5 x y - 10 x y 16 7 15 8 18 4 17 5 16 6 15 7 14 8 - 14 x y + x y - x y + 5 x y + 26 x y - 6 x y - x y 17 4 16 5 15 6 14 7 16 4 15 5 - x y - 24 x y + 14 x y + 7 x y + 11 x y - 16 x y 14 6 16 3 15 4 14 5 13 6 12 7 - 19 x y - 2 x y + 9 x y + 26 x y - 3 x y - 3 x y 15 3 14 4 13 5 12 6 11 7 14 3 - 2 x y - 19 x y + 10 x y + 13 x y - 2 x y + 7 x y 13 4 12 5 11 6 14 2 13 3 12 4 - 12 x y - 20 x y + 3 x y - x y + 6 x y + 14 x y 11 5 10 6 13 2 12 3 11 4 10 5 9 6 + x y - 6 x y - x y - 5 x y - x y + 12 x y - x y 12 2 11 3 10 4 9 5 8 6 11 2 10 3 + x y - 3 x y - 5 x y + 3 x y + 2 x y + 2 x y - 3 x y 9 4 8 5 10 2 9 3 8 4 7 5 9 2 - 8 x y - 7 x y + 2 x y + 10 x y + 8 x y + 2 x y - 4 x y 8 3 7 4 6 5 8 2 7 3 6 4 8 - 6 x y + 7 x y + 3 x y + 5 x y - 11 x y - 12 x y - 2 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 7 + x y + 15 x y + 5 x y + 3 x y - 9 x y - 6 x y + 3 x y - x 6 5 2 4 3 6 5 4 2 3 3 5 4 + x y + 4 x y - 8 x y + x - 3 x y + 4 x y + 8 x y + x + 2 x y 3 2 4 3 2 2 2 2 / - 8 x y - x + x y + 4 x y - 4 x y + x + 2 x y - 2 x + 1) / ( / 10 3 10 2 9 3 9 2 7 2 7 6 2 7 6 6 x y - x y - x y + x y - x y - x y + x y + x + x y - x 5 5 4 4 3 2 + x y - x - 2 x y + x + x y - x + 2 x - 1) and in Maple notation -(x^18*y^9-5*x^18*y^8+x^17*y^9+10*x^18*y^7-5*x^17*y^8-10*x^18*y^6+10*x^17*y^7+3 *x^16*y^8+5*x^18*y^5-10*x^17*y^6-14*x^16*y^7+x^15*y^8-x^18*y^4+5*x^17*y^5+26*x^ 16*y^6-6*x^15*y^7-x^14*y^8-x^17*y^4-24*x^16*y^5+14*x^15*y^6+7*x^14*y^7+11*x^16* y^4-16*x^15*y^5-19*x^14*y^6-2*x^16*y^3+9*x^15*y^4+26*x^14*y^5-3*x^13*y^6-3*x^12 *y^7-2*x^15*y^3-19*x^14*y^4+10*x^13*y^5+13*x^12*y^6-2*x^11*y^7+7*x^14*y^3-12*x^ 13*y^4-20*x^12*y^5+3*x^11*y^6-x^14*y^2+6*x^13*y^3+14*x^12*y^4+x^11*y^5-6*x^10*y ^6-x^13*y^2-5*x^12*y^3-x^11*y^4+12*x^10*y^5-x^9*y^6+x^12*y^2-3*x^11*y^3-5*x^10* y^4+3*x^9*y^5+2*x^8*y^6+2*x^11*y^2-3*x^10*y^3-8*x^9*y^4-7*x^8*y^5+2*x^10*y^2+10 *x^9*y^3+8*x^8*y^4+2*x^7*y^5-4*x^9*y^2-6*x^8*y^3+7*x^7*y^4+3*x^6*y^5+5*x^8*y^2-\ 11*x^7*y^3-12*x^6*y^4-2*x^8*y+x^7*y^2+15*x^6*y^3+5*x^5*y^4+3*x^7*y-9*x^6*y^2-6* x^5*y^3+3*x^4*y^4-x^7+x^6*y+4*x^5*y^2-8*x^4*y^3+x^6-3*x^5*y+4*x^4*y^2+8*x^3*y^3 +x^5+2*x^4*y-8*x^3*y^2-x^4+x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+1)/(x^10*y^3-x ^10*y^2-x^9*y^3+x^9*y^2-x^7*y^2-x^7*y+x^6*y^2+x^7+x^6*y-x^6+x^5*y-x^5-2*x^4*y+x ^4+x^3*y-x^2+2*x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1616056782 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 172" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 17 8 17 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 6 15 8 17 5 15 7 14 8 17 4 15 6 - 10 x y + x y + 5 x y - 4 x y - x y - x y + 7 x y 14 7 15 5 14 6 15 4 14 5 13 6 + 3 x y - 8 x y - 3 x y + 7 x y + 2 x y - 2 x y 12 7 15 3 14 4 13 5 12 6 15 2 14 3 - 2 x y - 4 x y - 3 x y + 8 x y + 6 x y + x y + 3 x y 13 4 12 5 11 6 10 7 14 2 13 3 12 4 - 12 x y - 7 x y + x y + x y - x y + 8 x y + 6 x y 11 5 10 6 9 7 13 2 12 3 11 4 10 5 - 6 x y - 3 x y - x y - 2 x y - 6 x y + 11 x y + 3 x y 9 6 12 2 11 3 10 4 9 5 8 6 12 + 3 x y + 4 x y - 7 x y - 4 x y + 2 x y - 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 9 3 + 8 x y - 12 x y + 4 x y - x y + x y - 7 x y + 8 x y 8 4 7 5 6 6 10 9 2 8 3 7 4 + 6 x y - 3 x y + x y + 2 x y + 4 x y - 17 x y + 9 x y 9 8 2 7 3 6 4 5 5 9 8 7 2 - 5 x y + 8 x y - 2 x y - 10 x y + 2 x y + x + 3 x y - 6 x y 6 3 5 4 8 7 6 2 5 3 4 4 + 19 x y - 2 x y - 2 x + 3 x y - 8 x y - 8 x y + 5 x y 6 5 2 4 3 6 5 3 3 5 4 - 4 x y + 11 x y - 10 x y + 2 x - 2 x y + 8 x y - x + 6 x y 3 2 4 3 2 2 3 2 / - 8 x y - x - x y + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ( / 11 3 11 2 11 9 2 9 8 2 8 8 (x - 1) (x y - 2 x y + x y + x y - x y - x y + 2 x y - x 7 7 6 6 5 5 4 3 3 2 - x y + x - 2 x y + x + x y - x + x y - x y + x - x - x + 1)) and in Maple notation -(x^17*y^9-5*x^17*y^8+10*x^17*y^7-10*x^17*y^6+x^15*y^8+5*x^17*y^5-4*x^15*y^7-x^ 14*y^8-x^17*y^4+7*x^15*y^6+3*x^14*y^7-8*x^15*y^5-3*x^14*y^6+7*x^15*y^4+2*x^14*y ^5-2*x^13*y^6-2*x^12*y^7-4*x^15*y^3-3*x^14*y^4+8*x^13*y^5+6*x^12*y^6+x^15*y^2+3 *x^14*y^3-12*x^13*y^4-7*x^12*y^5+x^11*y^6+x^10*y^7-x^14*y^2+8*x^13*y^3+6*x^12*y ^4-6*x^11*y^5-3*x^10*y^6-x^9*y^7-2*x^13*y^2-6*x^12*y^3+11*x^11*y^4+3*x^10*y^5+3 *x^9*y^6+4*x^12*y^2-7*x^11*y^3-4*x^10*y^4+2*x^9*y^5-2*x^8*y^6-x^12*y+8*x^10*y^3 -12*x^9*y^4+4*x^8*y^5-x^7*y^6+x^11*y-7*x^10*y^2+8*x^9*y^3+6*x^8*y^4-3*x^7*y^5+x ^6*y^6+2*x^10*y+4*x^9*y^2-17*x^8*y^3+9*x^7*y^4-5*x^9*y+8*x^8*y^2-2*x^7*y^3-10*x ^6*y^4+2*x^5*y^5+x^9+3*x^8*y-6*x^7*y^2+19*x^6*y^3-2*x^5*y^4-2*x^8+3*x^7*y-8*x^6 *y^2-8*x^5*y^3+5*x^4*y^4-4*x^6*y+11*x^5*y^2-10*x^4*y^3+2*x^6-2*x^5*y+8*x^3*y^3- x^5+6*x^4*y-8*x^3*y^2-x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x-1)/(x^ 11*y^3-2*x^11*y^2+x^11*y+x^9*y^2-x^9*y-x^8*y^2+2*x^8*y-x^8-x^7*y+x^7-2*x^6*y+x^ 6+x^5*y-x^5+x^4*y-x^3*y+x^3-x^2-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1405334701 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 173" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 9 17 8 17 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y / | / | ----- | ----- | m = 0 \ n = 0 / 17 6 15 8 17 5 15 7 14 8 17 4 15 6 - 10 x y + x y + 5 x y - 4 x y - x y - x y + 7 x y 14 7 15 5 14 6 15 4 14 5 13 6 + 3 x y - 8 x y - 3 x y + 7 x y + 2 x y - 2 x y 12 7 15 3 14 4 13 5 12 6 15 2 14 3 - 2 x y - 4 x y - 3 x y + 8 x y + 6 x y + x y + 3 x y 13 4 12 5 11 6 10 7 14 2 13 3 12 4 - 12 x y - 7 x y + x y + x y - x y + 8 x y + 6 x y 11 5 10 6 9 7 13 2 12 3 11 4 10 5 - 6 x y - 3 x y - x y - 2 x y - 6 x y + 11 x y + 3 x y 9 6 12 2 11 3 10 4 9 5 8 6 12 + 3 x y + 4 x y - 7 x y - 4 x y + 2 x y - 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 9 3 + 8 x y - 12 x y + 4 x y - x y + x y - 7 x y + 8 x y 8 4 7 5 6 6 10 9 2 8 3 7 4 + 6 x y - 3 x y + x y + 2 x y + 4 x y - 17 x y + 9 x y 9 8 2 7 3 6 4 5 5 9 8 7 2 - 5 x y + 8 x y - 2 x y - 10 x y + 2 x y + x + 3 x y - 6 x y 6 3 5 4 8 7 6 2 5 3 4 4 + 19 x y - 2 x y - 2 x + 3 x y - 8 x y - 8 x y + 5 x y 6 5 2 4 3 6 5 3 3 5 4 - 4 x y + 11 x y - 10 x y + 2 x - 2 x y + 8 x y - x + 6 x y 3 2 4 3 2 2 3 2 / - 8 x y - x - x y + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ( / 11 3 11 2 11 9 2 9 8 2 8 8 (x - 1) (x y - 2 x y + x y + x y - x y - x y + 2 x y - x 7 7 6 6 5 5 4 3 3 2 - x y + x - 2 x y + x + x y - x + x y - x y + x - x - x + 1)) and in Maple notation -(x^17*y^9-5*x^17*y^8+10*x^17*y^7-10*x^17*y^6+x^15*y^8+5*x^17*y^5-4*x^15*y^7-x^ 14*y^8-x^17*y^4+7*x^15*y^6+3*x^14*y^7-8*x^15*y^5-3*x^14*y^6+7*x^15*y^4+2*x^14*y ^5-2*x^13*y^6-2*x^12*y^7-4*x^15*y^3-3*x^14*y^4+8*x^13*y^5+6*x^12*y^6+x^15*y^2+3 *x^14*y^3-12*x^13*y^4-7*x^12*y^5+x^11*y^6+x^10*y^7-x^14*y^2+8*x^13*y^3+6*x^12*y ^4-6*x^11*y^5-3*x^10*y^6-x^9*y^7-2*x^13*y^2-6*x^12*y^3+11*x^11*y^4+3*x^10*y^5+3 *x^9*y^6+4*x^12*y^2-7*x^11*y^3-4*x^10*y^4+2*x^9*y^5-2*x^8*y^6-x^12*y+8*x^10*y^3 -12*x^9*y^4+4*x^8*y^5-x^7*y^6+x^11*y-7*x^10*y^2+8*x^9*y^3+6*x^8*y^4-3*x^7*y^5+x ^6*y^6+2*x^10*y+4*x^9*y^2-17*x^8*y^3+9*x^7*y^4-5*x^9*y+8*x^8*y^2-2*x^7*y^3-10*x ^6*y^4+2*x^5*y^5+x^9+3*x^8*y-6*x^7*y^2+19*x^6*y^3-2*x^5*y^4-2*x^8+3*x^7*y-8*x^6 *y^2-8*x^5*y^3+5*x^4*y^4-4*x^6*y+11*x^5*y^2-10*x^4*y^3+2*x^6-2*x^5*y+8*x^3*y^3- x^5+6*x^4*y-8*x^3*y^2-x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x-1)/(x^ 11*y^3-2*x^11*y^2+x^11*y+x^9*y^2-x^9*y-x^8*y^2+2*x^8*y-x^8-x^7*y+x^7-2*x^6*y+x^ 6+x^5*y-x^5+x^4*y-x^3*y+x^3-x^2-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1405334701 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 174" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 8 19 7 19 6 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 19 5 18 6 17 7 19 4 18 5 17 6 - 10 x y - 5 x y + 2 x y + 5 x y + 10 x y - 9 x y 16 7 19 3 18 4 17 5 16 6 15 7 18 3 - x y - x y - 10 x y + 16 x y + 6 x y + 2 x y + 5 x y 17 4 16 5 15 6 13 8 18 2 17 3 - 14 x y - 14 x y - 9 x y - x y - x y + 6 x y 16 4 15 5 14 6 17 2 16 3 15 4 + 16 x y + 17 x y + 2 x y - x y - 9 x y - 18 x y 14 5 13 6 12 7 16 2 15 3 14 4 - 8 x y + 6 x y - x y + 2 x y + 12 x y + 13 x y 13 5 12 6 15 2 14 3 13 4 12 5 - 13 x y + x y - 5 x y - 11 x y + 19 x y + 8 x y 11 6 10 7 15 14 2 13 3 12 4 11 5 - 2 x y + x y + x y + 5 x y - 19 x y - 20 x y - 2 x y 14 13 2 12 3 11 4 10 5 9 6 13 - x y + 10 x y + 18 x y + 11 x y - 6 x y + 2 x y - 2 x y 12 2 11 3 10 4 9 5 12 11 2 10 3 - 7 x y - 7 x y + 4 x y + x y + x y - 2 x y + 5 x y 9 4 8 5 11 10 2 9 3 8 4 7 5 - 13 x y + 6 x y + 2 x y - 5 x y + 10 x y - x y - 2 x y 10 9 2 8 3 7 4 9 8 2 7 3 + x y + 5 x y - 8 x y + 13 x y - 6 x y + x y - 9 x y 6 4 9 8 7 2 6 3 5 4 8 7 - 9 x y + x + 5 x y - 4 x y + 15 x y - x y - 2 x + 3 x y 6 2 5 3 4 4 6 5 2 4 3 6 5 - 3 x y - 7 x y + 3 x y - 6 x y + 9 x y - 8 x y + 2 x - x y 3 3 5 4 3 2 4 3 2 2 3 2 + 8 x y - x + 6 x y - 8 x y - x - x y + 4 x y + 2 x - 4 x y / 12 3 12 2 11 3 12 11 2 + 2 x y - 2 x + 1) / (x y - 2 x y - x y + x y + 2 x y / 11 10 2 10 9 2 9 8 2 9 8 8 - x y + x y - x y - 2 x y + 3 x y + x y - x - 3 x y + 2 x 7 6 6 5 4 4 3 3 - x y + 3 x y - 2 x + x - 2 x y + x + x y - 2 x + 2 x - 1) and in Maple notation -(x^19*y^8-5*x^19*y^7+10*x^19*y^6+x^18*y^7-10*x^19*y^5-5*x^18*y^6+2*x^17*y^7+5* x^19*y^4+10*x^18*y^5-9*x^17*y^6-x^16*y^7-x^19*y^3-10*x^18*y^4+16*x^17*y^5+6*x^ 16*y^6+2*x^15*y^7+5*x^18*y^3-14*x^17*y^4-14*x^16*y^5-9*x^15*y^6-x^13*y^8-x^18*y ^2+6*x^17*y^3+16*x^16*y^4+17*x^15*y^5+2*x^14*y^6-x^17*y^2-9*x^16*y^3-18*x^15*y^ 4-8*x^14*y^5+6*x^13*y^6-x^12*y^7+2*x^16*y^2+12*x^15*y^3+13*x^14*y^4-13*x^13*y^5 +x^12*y^6-5*x^15*y^2-11*x^14*y^3+19*x^13*y^4+8*x^12*y^5-2*x^11*y^6+x^10*y^7+x^ 15*y+5*x^14*y^2-19*x^13*y^3-20*x^12*y^4-2*x^11*y^5-x^14*y+10*x^13*y^2+18*x^12*y ^3+11*x^11*y^4-6*x^10*y^5+2*x^9*y^6-2*x^13*y-7*x^12*y^2-7*x^11*y^3+4*x^10*y^4+x ^9*y^5+x^12*y-2*x^11*y^2+5*x^10*y^3-13*x^9*y^4+6*x^8*y^5+2*x^11*y-5*x^10*y^2+10 *x^9*y^3-x^8*y^4-2*x^7*y^5+x^10*y+5*x^9*y^2-8*x^8*y^3+13*x^7*y^4-6*x^9*y+x^8*y^ 2-9*x^7*y^3-9*x^6*y^4+x^9+5*x^8*y-4*x^7*y^2+15*x^6*y^3-x^5*y^4-2*x^8+3*x^7*y-3* x^6*y^2-7*x^5*y^3+3*x^4*y^4-6*x^6*y+9*x^5*y^2-8*x^4*y^3+2*x^6-x^5*y+8*x^3*y^3-x ^5+6*x^4*y-8*x^3*y^2-x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^12*y^3-2 *x^12*y^2-x^11*y^3+x^12*y+2*x^11*y^2-x^11*y+x^10*y^2-x^10*y-2*x^9*y^2+3*x^9*y+x ^8*y^2-x^9-3*x^8*y+2*x^8-x^7*y+3*x^6*y-2*x^6+x^5-2*x^4*y+x^4+x^3*y-2*x^3+2*x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1399121351 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 175" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 2], [2, 2, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 19 8 19 7 19 6 18 7 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 19 5 18 6 17 7 19 4 18 5 17 6 - 10 x y - 5 x y + 2 x y + 5 x y + 10 x y - 9 x y 16 7 19 3 18 4 17 5 16 6 15 7 18 3 - x y - x y - 10 x y + 16 x y + 6 x y + 2 x y + 5 x y 17 4 16 5 15 6 13 8 18 2 17 3 - 14 x y - 14 x y - 9 x y - x y - x y + 6 x y 16 4 15 5 14 6 17 2 16 3 15 4 + 16 x y + 17 x y + 2 x y - x y - 9 x y - 18 x y 14 5 13 6 12 7 16 2 15 3 14 4 - 8 x y + 6 x y - x y + 2 x y + 12 x y + 13 x y 13 5 12 6 15 2 14 3 13 4 12 5 - 13 x y + x y - 5 x y - 11 x y + 19 x y + 8 x y 11 6 10 7 15 14 2 13 3 12 4 11 5 - 2 x y + x y + x y + 5 x y - 19 x y - 20 x y - 2 x y 14 13 2 12 3 11 4 10 5 9 6 13 - x y + 10 x y + 18 x y + 11 x y - 6 x y + 2 x y - 2 x y 12 2 11 3 10 4 9 5 12 11 2 10 3 - 7 x y - 7 x y + 4 x y + x y + x y - 2 x y + 5 x y 9 4 8 5 11 10 2 9 3 8 4 7 5 - 13 x y + 6 x y + 2 x y - 5 x y + 10 x y - x y - 2 x y 10 9 2 8 3 7 4 9 8 2 7 3 + x y + 5 x y - 8 x y + 13 x y - 6 x y + x y - 9 x y 6 4 9 8 7 2 6 3 5 4 8 7 - 9 x y + x + 5 x y - 4 x y + 15 x y - x y - 2 x + 3 x y 6 2 5 3 4 4 6 5 2 4 3 6 5 - 3 x y - 7 x y + 3 x y - 6 x y + 9 x y - 8 x y + 2 x - x y 3 3 5 4 3 2 4 3 2 2 3 2 + 8 x y - x + 6 x y - 8 x y - x - x y + 4 x y + 2 x - 4 x y / 11 3 11 2 11 9 2 9 8 2 + 2 x y - 2 x + 1) / ((x y - 2 x y + x y + x y - x y - x y / 8 8 7 7 6 6 5 5 4 3 3 + 2 x y - x - x y + x - 2 x y + x + x y - x + x y - x y + x 2 - x - x + 1) (x - 1)) and in Maple notation -(x^19*y^8-5*x^19*y^7+10*x^19*y^6+x^18*y^7-10*x^19*y^5-5*x^18*y^6+2*x^17*y^7+5* x^19*y^4+10*x^18*y^5-9*x^17*y^6-x^16*y^7-x^19*y^3-10*x^18*y^4+16*x^17*y^5+6*x^ 16*y^6+2*x^15*y^7+5*x^18*y^3-14*x^17*y^4-14*x^16*y^5-9*x^15*y^6-x^13*y^8-x^18*y ^2+6*x^17*y^3+16*x^16*y^4+17*x^15*y^5+2*x^14*y^6-x^17*y^2-9*x^16*y^3-18*x^15*y^ 4-8*x^14*y^5+6*x^13*y^6-x^12*y^7+2*x^16*y^2+12*x^15*y^3+13*x^14*y^4-13*x^13*y^5 +x^12*y^6-5*x^15*y^2-11*x^14*y^3+19*x^13*y^4+8*x^12*y^5-2*x^11*y^6+x^10*y^7+x^ 15*y+5*x^14*y^2-19*x^13*y^3-20*x^12*y^4-2*x^11*y^5-x^14*y+10*x^13*y^2+18*x^12*y ^3+11*x^11*y^4-6*x^10*y^5+2*x^9*y^6-2*x^13*y-7*x^12*y^2-7*x^11*y^3+4*x^10*y^4+x ^9*y^5+x^12*y-2*x^11*y^2+5*x^10*y^3-13*x^9*y^4+6*x^8*y^5+2*x^11*y-5*x^10*y^2+10 *x^9*y^3-x^8*y^4-2*x^7*y^5+x^10*y+5*x^9*y^2-8*x^8*y^3+13*x^7*y^4-6*x^9*y+x^8*y^ 2-9*x^7*y^3-9*x^6*y^4+x^9+5*x^8*y-4*x^7*y^2+15*x^6*y^3-x^5*y^4-2*x^8+3*x^7*y-3* x^6*y^2-7*x^5*y^3+3*x^4*y^4-6*x^6*y+9*x^5*y^2-8*x^4*y^3+2*x^6-x^5*y+8*x^3*y^3-x ^5+6*x^4*y-8*x^3*y^2-x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^11*y^3-2 *x^11*y^2+x^11*y+x^9*y^2-x^9*y-x^8*y^2+2*x^8*y-x^8-x^7*y+x^7-2*x^6*y+x^6+x^5*y- x^5+x^4*y-x^3*y+x^3-x^2-x+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1399121351 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 176" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 8 14 7 13 8 14 6 ) | ) C(m, n) x y | = (x y - 3 x y - x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 7 14 5 13 6 12 7 14 4 13 5 + 5 x y + 2 x y - 9 x y - 2 x y - 3 x y + 7 x y 12 6 11 7 14 3 13 4 12 5 11 6 12 4 + 11 x y - x y + x y - 2 x y - 18 x y - x y + 9 x y 11 5 10 6 12 3 11 4 10 5 9 6 8 7 + 13 x y - 4 x y + 2 x y - 20 x y + 3 x y + 2 x y - x y 12 2 11 3 10 4 9 5 7 7 11 2 10 3 - 2 x y + 9 x y + 19 x y - 9 x y + x y + x y - 29 x y 9 4 8 5 7 6 11 10 2 9 3 8 4 + 3 x y + 5 x y - 3 x y - x y + 11 x y + 16 x y - 13 x y 7 5 6 6 9 2 8 3 7 4 6 5 9 + x y + 3 x y - 17 x y + 11 x y + 7 x y - 6 x y + 5 x y 8 2 7 3 6 4 5 5 8 7 2 6 3 + 4 x y - 18 x y + x y + 5 x y - 7 x y + 11 x y + 12 x y 5 4 8 7 6 2 5 3 4 4 7 6 - 8 x y + x + 2 x y - 13 x y + x y + 6 x y - 2 x + 4 x y 5 2 4 3 5 4 2 3 3 5 4 + 6 x y - 13 x y - 6 x y + 3 x y + 8 x y + 2 x + 5 x y 3 2 4 2 2 3 2 / 9 2 - 8 x y - 2 x + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / (x y / 9 8 2 9 8 8 7 7 6 6 5 - 2 x y - x y + x + 2 x y - x + 2 x y - 2 x - 2 x y + 2 x + x 4 3 - 2 x + 2 x - 2 x + 1) and in Maple notation (x^14*y^8-3*x^14*y^7-x^13*y^8+2*x^14*y^6+5*x^13*y^7+2*x^14*y^5-9*x^13*y^6-2*x^ 12*y^7-3*x^14*y^4+7*x^13*y^5+11*x^12*y^6-x^11*y^7+x^14*y^3-2*x^13*y^4-18*x^12*y ^5-x^11*y^6+9*x^12*y^4+13*x^11*y^5-4*x^10*y^6+2*x^12*y^3-20*x^11*y^4+3*x^10*y^5 +2*x^9*y^6-x^8*y^7-2*x^12*y^2+9*x^11*y^3+19*x^10*y^4-9*x^9*y^5+x^7*y^7+x^11*y^2 -29*x^10*y^3+3*x^9*y^4+5*x^8*y^5-3*x^7*y^6-x^11*y+11*x^10*y^2+16*x^9*y^3-13*x^8 *y^4+x^7*y^5+3*x^6*y^6-17*x^9*y^2+11*x^8*y^3+7*x^7*y^4-6*x^6*y^5+5*x^9*y+4*x^8* y^2-18*x^7*y^3+x^6*y^4+5*x^5*y^5-7*x^8*y+11*x^7*y^2+12*x^6*y^3-8*x^5*y^4+x^8+2* x^7*y-13*x^6*y^2+x^5*y^3+6*x^4*y^4-2*x^7+4*x^6*y+6*x^5*y^2-13*x^4*y^3-6*x^5*y+3 *x^4*y^2+8*x^3*y^3+2*x^5+5*x^4*y-8*x^3*y^2-2*x^4+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-\ 2*x+1)/(x^9*y^2-2*x^9*y-x^8*y^2+x^9+2*x^8*y-x^8+2*x^7*y-2*x^7-2*x^6*y+2*x^6+x^5 -2*x^4+2*x^3-2*x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.1216192357 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 177" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 7 13 6 12 7 13 5 ) | ) C(m, n) x y | = (2 x y - 7 x y - 2 x y + 8 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 13 4 12 5 11 6 13 3 12 4 + 10 x y - 2 x y - 18 x y - 2 x y - 2 x y + 13 x y 11 5 10 6 9 7 13 2 12 3 11 4 10 5 + 12 x y - 3 x y - x y + x y - x y - 22 x y + 4 x y 9 6 8 7 12 2 11 3 10 4 9 5 8 6 + 5 x y + x y - 3 x y + 13 x y + 8 x y - 9 x y - 5 x y 12 11 2 10 3 9 4 8 5 11 10 2 + x y + 3 x y - 15 x y + 6 x y + 9 x y - 5 x y + 3 x y 9 3 8 4 7 5 6 6 11 10 9 2 + 3 x y - 11 x y - 6 x y + 2 x y + x + 5 x y - 7 x y 8 3 7 4 6 5 10 9 8 2 7 3 + 7 x y + 17 x y - 4 x y - 2 x + 3 x y + 5 x y - 14 x y 6 4 5 5 8 7 2 6 3 5 4 8 - 8 x y + 5 x y - 9 x y - 2 x y + 17 x y - 7 x y + 3 x 7 6 2 5 3 4 4 7 6 5 2 + 9 x y - 5 x y - 3 x y + 6 x y - 4 x - 4 x y + 8 x y 4 3 6 5 4 2 3 3 5 4 3 2 - 13 x y + 2 x - 4 x y + x y + 8 x y + x + 7 x y - 8 x y 4 3 2 2 3 2 / 10 2 - 2 x - x y + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ((x y / 10 9 2 10 9 9 8 8 7 7 6 - 2 x y - x y + x + 2 x y - x + x y - x - 2 x y + 2 x + 2 x y 6 5 4 4 3 3 2 - 2 x - x y - x y + x + x y - x + x + x - 1) (x - 1)) and in Maple notation (2*x^13*y^7-7*x^13*y^6-2*x^12*y^7+8*x^13*y^5+10*x^12*y^6-2*x^13*y^4-18*x^12*y^5 -2*x^11*y^6-2*x^13*y^3+13*x^12*y^4+12*x^11*y^5-3*x^10*y^6-x^9*y^7+x^13*y^2-x^12 *y^3-22*x^11*y^4+4*x^10*y^5+5*x^9*y^6+x^8*y^7-3*x^12*y^2+13*x^11*y^3+8*x^10*y^4 -9*x^9*y^5-5*x^8*y^6+x^12*y+3*x^11*y^2-15*x^10*y^3+6*x^9*y^4+9*x^8*y^5-5*x^11*y +3*x^10*y^2+3*x^9*y^3-11*x^8*y^4-6*x^7*y^5+2*x^6*y^6+x^11+5*x^10*y-7*x^9*y^2+7* x^8*y^3+17*x^7*y^4-4*x^6*y^5-2*x^10+3*x^9*y+5*x^8*y^2-14*x^7*y^3-8*x^6*y^4+5*x^ 5*y^5-9*x^8*y-2*x^7*y^2+17*x^6*y^3-7*x^5*y^4+3*x^8+9*x^7*y-5*x^6*y^2-3*x^5*y^3+ 6*x^4*y^4-4*x^7-4*x^6*y+8*x^5*y^2-13*x^4*y^3+2*x^6-4*x^5*y+x^4*y^2+8*x^3*y^3+x^ 5+7*x^4*y-8*x^3*y^2-2*x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^10*y^2-\ 2*x^10*y-x^9*y^2+x^10+2*x^9*y-x^9+x^8*y-x^8-2*x^7*y+2*x^7+2*x^6*y-2*x^6-x^5*y-x ^4*y+x^4+x^3*y-x^3+x^2+x-1)/(x-1) ------------------------------------------------ "Theorem Number 178" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 7 13 6 12 7 13 5 ) | ) C(m, n) x y | = (2 x y - 7 x y - 2 x y + 8 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 13 4 12 5 11 6 13 3 12 4 + 10 x y - 2 x y - 18 x y - 2 x y - 2 x y + 13 x y 11 5 10 6 9 7 13 2 12 3 11 4 10 5 + 12 x y - 3 x y - x y + x y - x y - 22 x y + 4 x y 9 6 8 7 12 2 11 3 10 4 9 5 8 6 + 5 x y + x y - 3 x y + 13 x y + 8 x y - 9 x y - 5 x y 12 11 2 10 3 9 4 8 5 11 10 2 + x y + 3 x y - 15 x y + 6 x y + 9 x y - 5 x y + 3 x y 9 3 8 4 7 5 6 6 11 10 9 2 + 3 x y - 11 x y - 6 x y + 2 x y + x + 5 x y - 7 x y 8 3 7 4 6 5 10 9 8 2 7 3 + 7 x y + 17 x y - 4 x y - 2 x + 3 x y + 5 x y - 14 x y 6 4 5 5 8 7 2 6 3 5 4 8 - 8 x y + 5 x y - 9 x y - 2 x y + 17 x y - 7 x y + 3 x 7 6 2 5 3 4 4 7 6 5 2 + 9 x y - 5 x y - 3 x y + 6 x y - 4 x - 4 x y + 8 x y 4 3 6 5 4 2 3 3 5 4 3 2 - 13 x y + 2 x - 4 x y + x y + 8 x y + x + 7 x y - 8 x y 4 3 2 2 3 2 / 10 2 - 2 x - x y + 4 x y + 2 x - 4 x y + 2 x y - 2 x + 1) / ((x y / 10 9 2 10 9 9 8 8 7 7 6 - 2 x y - x y + x + 2 x y - x + x y - x - 2 x y + 2 x + 2 x y 6 5 4 4 3 3 2 - 2 x - x y - x y + x + x y - x + x + x - 1) (x - 1)) and in Maple notation (2*x^13*y^7-7*x^13*y^6-2*x^12*y^7+8*x^13*y^5+10*x^12*y^6-2*x^13*y^4-18*x^12*y^5 -2*x^11*y^6-2*x^13*y^3+13*x^12*y^4+12*x^11*y^5-3*x^10*y^6-x^9*y^7+x^13*y^2-x^12 *y^3-22*x^11*y^4+4*x^10*y^5+5*x^9*y^6+x^8*y^7-3*x^12*y^2+13*x^11*y^3+8*x^10*y^4 -9*x^9*y^5-5*x^8*y^6+x^12*y+3*x^11*y^2-15*x^10*y^3+6*x^9*y^4+9*x^8*y^5-5*x^11*y +3*x^10*y^2+3*x^9*y^3-11*x^8*y^4-6*x^7*y^5+2*x^6*y^6+x^11+5*x^10*y-7*x^9*y^2+7* x^8*y^3+17*x^7*y^4-4*x^6*y^5-2*x^10+3*x^9*y+5*x^8*y^2-14*x^7*y^3-8*x^6*y^4+5*x^ 5*y^5-9*x^8*y-2*x^7*y^2+17*x^6*y^3-7*x^5*y^4+3*x^8+9*x^7*y-5*x^6*y^2-3*x^5*y^3+ 6*x^4*y^4-4*x^7-4*x^6*y+8*x^5*y^2-13*x^4*y^3+2*x^6-4*x^5*y+x^4*y^2+8*x^3*y^3+x^ 5+7*x^4*y-8*x^3*y^2-2*x^4-x^3*y+4*x^2*y^2+2*x^3-4*x^2*y+2*x*y-2*x+1)/(x^10*y^2-\ 2*x^10*y-x^9*y^2+x^10+2*x^9*y-x^9+x^8*y-x^8-2*x^7*y+2*x^7+2*x^6*y-2*x^6-x^5*y-x ^4*y+x^4+x^3*y-x^3+x^2+x-1)/(x-1) ------------------------------------------------ "Theorem Number 179" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 5 12 7 14 4 13 5 ) | ) C(m, n) x y | = - (x y - x y - 4 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 14 3 13 4 12 5 11 6 14 2 + 3 x y + 6 x y + 8 x y - x y - 2 x y - 4 x y 13 3 12 4 11 5 14 13 2 12 3 11 4 - 12 x y - 6 x y + 6 x y + x y + 8 x y + 9 x y - 5 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 - 5 x y + x y - 2 x y - 5 x y - x y + 15 x y + x y 8 6 12 11 2 10 3 9 4 8 5 11 - 2 x y + x y + 3 x y - 13 x y - 11 x y + 3 x y - x y 9 3 8 4 7 5 6 6 10 9 2 8 3 + 16 x y + x y - 3 x y + x y + 4 x y - 7 x y - 7 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 + 9 x y - x y - x - x y + 9 x y - 7 x y - 9 x y + 3 x y 9 8 7 2 6 3 5 4 8 7 6 2 + x - 5 x y - 4 x y + 13 x y - x y + x + 7 x y - 4 x y 5 3 4 4 7 6 5 2 4 3 6 5 - 7 x y + 5 x y - 2 x - 2 x y + 7 x y - 4 x y + 2 x - x y 4 2 3 3 4 3 2 4 3 2 2 3 - 2 x y + 8 x y + 3 x y - 4 x y - x - 3 x y + 4 x y + x 2 2 / 9 2 9 9 7 - 2 x y - x + 2 x y - x + 1) / ((x - 1) (x y - 2 x y + x + x y / 7 6 6 5 5 4 3 2 - x - x y + x + x y - x - x - x y - x + 1)) and in Maple notation -(x^14*y^5-x^12*y^7-4*x^14*y^4-2*x^13*y^5+3*x^12*y^6+6*x^14*y^3+8*x^13*y^4-x^12 *y^5-2*x^11*y^6-4*x^14*y^2-12*x^13*y^3-6*x^12*y^4+6*x^11*y^5+x^14*y+8*x^13*y^2+ 9*x^12*y^3-5*x^11*y^4-5*x^10*y^5+x^9*y^6-2*x^13*y-5*x^12*y^2-x^11*y^3+15*x^10*y ^4+x^9*y^5-2*x^8*y^6+x^12*y+3*x^11*y^2-13*x^10*y^3-11*x^9*y^4+3*x^8*y^5-x^11*y+ 16*x^9*y^3+x^8*y^4-3*x^7*y^5+x^6*y^6+4*x^10*y-7*x^9*y^2-7*x^8*y^3+9*x^7*y^4-x^6 *y^5-x^10-x^9*y+9*x^8*y^2-7*x^7*y^3-9*x^6*y^4+3*x^5*y^5+x^9-5*x^8*y-4*x^7*y^2+ 13*x^6*y^3-x^5*y^4+x^8+7*x^7*y-4*x^6*y^2-7*x^5*y^3+5*x^4*y^4-2*x^7-2*x^6*y+7*x^ 5*y^2-4*x^4*y^3+2*x^6-x^5*y-2*x^4*y^2+8*x^3*y^3+3*x^4*y-4*x^3*y^2-x^4-3*x^3*y+4 *x^2*y^2+x^3-2*x^2*y-x^2+2*x*y-x+1)/(x-1)/(x^9*y^2-2*x^9*y+x^9+x^7*y-x^7-x^6*y+ x^6+x^5*y-x^5-x^4-x^3*y-x^2+1) ------------------------------------------------ "Theorem Number 180" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 5 12 7 14 4 13 5 ) | ) C(m, n) x y | = - (x y - x y - 4 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 14 3 13 4 12 5 11 6 14 2 + 3 x y + 6 x y + 8 x y - x y - 2 x y - 4 x y 13 3 12 4 11 5 14 13 2 12 3 11 4 - 12 x y - 6 x y + 6 x y + x y + 8 x y + 9 x y - 5 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 - 5 x y + x y - 2 x y - 5 x y - x y + 15 x y + x y 8 6 12 11 2 10 3 9 4 8 5 11 - 2 x y + x y + 3 x y - 13 x y - 11 x y + 3 x y - x y 9 3 8 4 7 5 6 6 10 9 2 8 3 + 16 x y + x y - 3 x y + x y + 4 x y - 7 x y - 7 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 + 9 x y - x y - x - x y + 9 x y - 7 x y - 9 x y + 3 x y 9 8 7 2 6 3 5 4 8 7 6 2 + x - 5 x y - 4 x y + 13 x y - x y + x + 7 x y - 4 x y 5 3 4 4 7 6 5 2 4 3 6 5 - 7 x y + 5 x y - 2 x - 2 x y + 7 x y - 4 x y + 2 x - x y 4 2 3 3 4 3 2 4 3 2 2 3 - 2 x y + 8 x y + 3 x y - 4 x y - x - 3 x y + 4 x y + x 2 2 / 9 2 9 9 7 - 2 x y - x + 2 x y - x + 1) / ((x - 1) (x y - 2 x y + x + x y / 7 6 6 5 5 4 3 2 - x - x y + x + x y - x - x - x y - x + 1)) and in Maple notation -(x^14*y^5-x^12*y^7-4*x^14*y^4-2*x^13*y^5+3*x^12*y^6+6*x^14*y^3+8*x^13*y^4-x^12 *y^5-2*x^11*y^6-4*x^14*y^2-12*x^13*y^3-6*x^12*y^4+6*x^11*y^5+x^14*y+8*x^13*y^2+ 9*x^12*y^3-5*x^11*y^4-5*x^10*y^5+x^9*y^6-2*x^13*y-5*x^12*y^2-x^11*y^3+15*x^10*y ^4+x^9*y^5-2*x^8*y^6+x^12*y+3*x^11*y^2-13*x^10*y^3-11*x^9*y^4+3*x^8*y^5-x^11*y+ 16*x^9*y^3+x^8*y^4-3*x^7*y^5+x^6*y^6+4*x^10*y-7*x^9*y^2-7*x^8*y^3+9*x^7*y^4-x^6 *y^5-x^10-x^9*y+9*x^8*y^2-7*x^7*y^3-9*x^6*y^4+3*x^5*y^5+x^9-5*x^8*y-4*x^7*y^2+ 13*x^6*y^3-x^5*y^4+x^8+7*x^7*y-4*x^6*y^2-7*x^5*y^3+5*x^4*y^4-2*x^7-2*x^6*y+7*x^ 5*y^2-4*x^4*y^3+2*x^6-x^5*y-2*x^4*y^2+8*x^3*y^3+3*x^4*y-4*x^3*y^2-x^4-3*x^3*y+4 *x^2*y^2+x^3-2*x^2*y-x^2+2*x*y-x+1)/(x-1)/(x^9*y^2-2*x^9*y+x^9+x^7*y-x^7-x^6*y+ x^6+x^5*y-x^5-x^4-x^3*y-x^2+1) ------------------------------------------------ "Theorem Number 181" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 6 16 7 17 5 16 6 ) | ) C(m, n) x y | = - (x y + x y - 5 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 7 17 4 16 5 15 6 17 3 16 4 - x y + 10 x y + 10 x y + 5 x y - 10 x y - 10 x y 15 5 14 6 13 7 17 2 16 3 15 4 14 5 - 8 x y + x y - 2 x y + 5 x y + 5 x y + 2 x y - 5 x y 13 6 12 7 17 16 2 15 3 14 4 13 5 + 4 x y + 2 x y - x y - x y + 7 x y + 10 x y + 6 x y 12 6 11 7 15 2 14 3 13 4 12 5 - 8 x y + x y - 7 x y - 10 x y - 21 x y + 9 x y 11 6 10 7 15 14 2 13 3 12 4 11 5 - 2 x y - x y + 2 x y + 5 x y + 17 x y + x y - 9 x y 10 6 14 13 2 12 3 11 4 10 5 9 6 + 6 x y - x y - 3 x y - 7 x y + 26 x y - 4 x y - 4 x y 13 12 2 11 3 10 4 9 5 8 6 11 2 - x y + 3 x y - 19 x y - 18 x y + 11 x y + 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 + 29 x y - 5 x y - 10 x y + 2 x y + 5 x y - 10 x y 9 3 8 4 7 5 6 6 11 10 9 2 - 11 x y + 17 x y + 3 x y - 2 x y - x - 4 x y + 13 x y 8 3 7 4 6 5 10 9 8 2 7 3 - 12 x y - 20 x y + 4 x y + 2 x - 4 x y - x y + 24 x y 6 4 5 5 8 7 2 6 3 5 4 8 + 8 x y - 3 x y + 7 x y - 9 x y - 24 x y + 8 x y - 3 x 7 6 2 5 3 4 4 7 6 5 2 - 4 x y + 15 x y + 5 x y - 6 x y + 4 x + x y - 12 x y 4 3 6 5 4 2 3 3 5 4 3 2 + 13 x y - 2 x + 4 x y - x y - 8 x y - x - 7 x y + 8 x y 4 3 2 2 3 2 / 10 2 + 2 x + x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x - 1) / ((x y / 10 9 2 10 9 9 8 8 7 7 6 - 2 x y - x y + x + 2 x y - x + x y - x - 2 x y + 2 x + 2 x y 6 5 4 4 3 3 2 - 2 x - x y - x y + x + x y - x + x + x - 1) (x - 1)) and in Maple notation -(x^17*y^6+x^16*y^7-5*x^17*y^5-5*x^16*y^6-x^15*y^7+10*x^17*y^4+10*x^16*y^5+5*x^ 15*y^6-10*x^17*y^3-10*x^16*y^4-8*x^15*y^5+x^14*y^6-2*x^13*y^7+5*x^17*y^2+5*x^16 *y^3+2*x^15*y^4-5*x^14*y^5+4*x^13*y^6+2*x^12*y^7-x^17*y-x^16*y^2+7*x^15*y^3+10* x^14*y^4+6*x^13*y^5-8*x^12*y^6+x^11*y^7-7*x^15*y^2-10*x^14*y^3-21*x^13*y^4+9*x^ 12*y^5-2*x^11*y^6-x^10*y^7+2*x^15*y+5*x^14*y^2+17*x^13*y^3+x^12*y^4-9*x^11*y^5+ 6*x^10*y^6-x^14*y-3*x^13*y^2-7*x^12*y^3+26*x^11*y^4-4*x^10*y^5-4*x^9*y^6-x^13*y +3*x^12*y^2-19*x^11*y^3-18*x^10*y^4+11*x^9*y^5+2*x^8*y^6-x^11*y^2+29*x^10*y^3-5 *x^9*y^4-10*x^8*y^5+2*x^7*y^6+5*x^11*y-10*x^10*y^2-11*x^9*y^3+17*x^8*y^4+3*x^7* y^5-2*x^6*y^6-x^11-4*x^10*y+13*x^9*y^2-12*x^8*y^3-20*x^7*y^4+4*x^6*y^5+2*x^10-4 *x^9*y-x^8*y^2+24*x^7*y^3+8*x^6*y^4-3*x^5*y^5+7*x^8*y-9*x^7*y^2-24*x^6*y^3+8*x^ 5*y^4-3*x^8-4*x^7*y+15*x^6*y^2+5*x^5*y^3-6*x^4*y^4+4*x^7+x^6*y-12*x^5*y^2+13*x^ 4*y^3-2*x^6+4*x^5*y-x^4*y^2-8*x^3*y^3-x^5-7*x^4*y+8*x^3*y^2+2*x^4+x^3*y-4*x^2*y ^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^10*y^2-2*x^10*y-x^9*y^2+x^10+2*x^9*y-x^9+x^8*y -x^8-2*x^7*y+2*x^7+2*x^6*y-2*x^6-x^5*y-x^4*y+x^4+x^3*y-x^3+x^2+x-1)/(x-1) ------------------------------------------------ "Theorem Number 182" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 17 6 16 7 17 5 16 6 ) | ) C(m, n) x y | = - (x y + x y - 5 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 7 17 4 16 5 15 6 17 3 16 4 - x y + 10 x y + 10 x y + 5 x y - 10 x y - 10 x y 15 5 14 6 13 7 17 2 16 3 15 4 14 5 - 8 x y + x y - 2 x y + 5 x y + 5 x y + 2 x y - 5 x y 13 6 12 7 17 16 2 15 3 14 4 13 5 + 4 x y + 2 x y - x y - x y + 7 x y + 10 x y + 6 x y 12 6 11 7 15 2 14 3 13 4 12 5 - 8 x y + x y - 7 x y - 10 x y - 21 x y + 9 x y 11 6 10 7 15 14 2 13 3 12 4 11 5 - 2 x y - x y + 2 x y + 5 x y + 17 x y + x y - 9 x y 10 6 14 13 2 12 3 11 4 10 5 9 6 + 6 x y - x y - 3 x y - 7 x y + 26 x y - 4 x y - 4 x y 13 12 2 11 3 10 4 9 5 8 6 11 2 - x y + 3 x y - 19 x y - 18 x y + 11 x y + 2 x y - x y 10 3 9 4 8 5 7 6 11 10 2 + 29 x y - 5 x y - 10 x y + 2 x y + 5 x y - 10 x y 9 3 8 4 7 5 6 6 11 10 9 2 - 11 x y + 17 x y + 3 x y - 2 x y - x - 4 x y + 13 x y 8 3 7 4 6 5 10 9 8 2 7 3 - 12 x y - 20 x y + 4 x y + 2 x - 4 x y - x y + 24 x y 6 4 5 5 8 7 2 6 3 5 4 8 + 8 x y - 3 x y + 7 x y - 9 x y - 24 x y + 8 x y - 3 x 7 6 2 5 3 4 4 7 6 5 2 - 4 x y + 15 x y + 5 x y - 6 x y + 4 x + x y - 12 x y 4 3 6 5 4 2 3 3 5 4 3 2 + 13 x y - 2 x + 4 x y - x y - 8 x y - x - 7 x y + 8 x y 4 3 2 2 3 2 / + 2 x + x y - 4 x y - 2 x + 4 x y - 2 x y + 2 x - 1) / ( / 2 9 2 9 9 7 7 6 6 5 5 (x - 2 x + 1) (x y - 2 x y + x + x y - x - x y + x + x y - x 4 3 2 - x - x y - x + 1)) and in Maple notation -(x^17*y^6+x^16*y^7-5*x^17*y^5-5*x^16*y^6-x^15*y^7+10*x^17*y^4+10*x^16*y^5+5*x^ 15*y^6-10*x^17*y^3-10*x^16*y^4-8*x^15*y^5+x^14*y^6-2*x^13*y^7+5*x^17*y^2+5*x^16 *y^3+2*x^15*y^4-5*x^14*y^5+4*x^13*y^6+2*x^12*y^7-x^17*y-x^16*y^2+7*x^15*y^3+10* x^14*y^4+6*x^13*y^5-8*x^12*y^6+x^11*y^7-7*x^15*y^2-10*x^14*y^3-21*x^13*y^4+9*x^ 12*y^5-2*x^11*y^6-x^10*y^7+2*x^15*y+5*x^14*y^2+17*x^13*y^3+x^12*y^4-9*x^11*y^5+ 6*x^10*y^6-x^14*y-3*x^13*y^2-7*x^12*y^3+26*x^11*y^4-4*x^10*y^5-4*x^9*y^6-x^13*y +3*x^12*y^2-19*x^11*y^3-18*x^10*y^4+11*x^9*y^5+2*x^8*y^6-x^11*y^2+29*x^10*y^3-5 *x^9*y^4-10*x^8*y^5+2*x^7*y^6+5*x^11*y-10*x^10*y^2-11*x^9*y^3+17*x^8*y^4+3*x^7* y^5-2*x^6*y^6-x^11-4*x^10*y+13*x^9*y^2-12*x^8*y^3-20*x^7*y^4+4*x^6*y^5+2*x^10-4 *x^9*y-x^8*y^2+24*x^7*y^3+8*x^6*y^4-3*x^5*y^5+7*x^8*y-9*x^7*y^2-24*x^6*y^3+8*x^ 5*y^4-3*x^8-4*x^7*y+15*x^6*y^2+5*x^5*y^3-6*x^4*y^4+4*x^7+x^6*y-12*x^5*y^2+13*x^ 4*y^3-2*x^6+4*x^5*y-x^4*y^2-8*x^3*y^3-x^5-7*x^4*y+8*x^3*y^2+2*x^4+x^3*y-4*x^2*y ^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^2-2*x+1)/(x^9*y^2-2*x^9*y+x^9+x^7*y-x^7-x^6*y+ x^6+x^5*y-x^5-x^4-x^3*y-x^2+1) ------------------------------------------------ "Theorem Number 183" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 4 15 5 14 6 16 3 ) | ) C(m, n) x y | = (x y + 2 x y + x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 4 14 5 13 6 12 7 16 2 15 3 14 4 - 8 x y - 5 x y - x y + x y + 6 x y + 12 x y + 10 x y 13 5 12 6 16 15 2 14 3 13 4 12 5 + 4 x y - 3 x y - 4 x y - 8 x y - 9 x y - 6 x y + x y 11 6 16 15 14 2 13 3 12 4 11 5 + 2 x y + x + 2 x y + 2 x y + 2 x y + x y - 6 x y 10 6 14 13 2 12 3 11 4 10 5 14 + x y + 2 x y + 5 x y + 8 x y + 12 x y + 4 x y - x 13 12 2 11 3 10 4 9 5 8 6 13 - 6 x y - 14 x y - 17 x y - 17 x y - 2 x y + x y + 2 x 12 11 2 10 3 9 4 8 5 11 10 2 + 6 x y + 11 x y + 15 x y + 7 x y - 4 x y - x y + x y 9 3 8 4 7 5 6 6 11 10 9 2 8 3 - 9 x y - 2 x y + 3 x y - x y - x - 6 x y + x y + 6 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 - 6 x y + x y + 2 x + 4 x y - 3 x y + 3 x y + 9 x y - 2 x y 9 8 7 2 6 3 5 4 8 7 6 2 - x + 4 x y + 6 x y - 8 x y + x y - 2 x - 7 x y - x y 5 3 4 4 7 6 5 2 4 3 6 5 + 8 x y - 5 x y + 2 x + 3 x y - 6 x y + 4 x y - 2 x - x y 4 2 3 3 4 3 2 4 3 2 2 3 + 2 x y - 8 x y - 3 x y + 4 x y + x + 3 x y - 4 x y - x 2 2 / 9 2 9 9 7 + 2 x y + x - 2 x y + x - 1) / ((x - 1) (x y - 2 x y + x + x y / 7 6 6 5 5 4 3 2 - x - x y + x + x y - x - x - x y - x + 1)) and in Maple notation (x^16*y^4+2*x^15*y^5+x^14*y^6-4*x^16*y^3-8*x^15*y^4-5*x^14*y^5-x^13*y^6+x^12*y^ 7+6*x^16*y^2+12*x^15*y^3+10*x^14*y^4+4*x^13*y^5-3*x^12*y^6-4*x^16*y-8*x^15*y^2-\ 9*x^14*y^3-6*x^13*y^4+x^12*y^5+2*x^11*y^6+x^16+2*x^15*y+2*x^14*y^2+2*x^13*y^3+x ^12*y^4-6*x^11*y^5+x^10*y^6+2*x^14*y+5*x^13*y^2+8*x^12*y^3+12*x^11*y^4+4*x^10*y ^5-x^14-6*x^13*y-14*x^12*y^2-17*x^11*y^3-17*x^10*y^4-2*x^9*y^5+x^8*y^6+2*x^13+6 *x^12*y+11*x^11*y^2+15*x^10*y^3+7*x^9*y^4-4*x^8*y^5-x^11*y+x^10*y^2-9*x^9*y^3-2 *x^8*y^4+3*x^7*y^5-x^6*y^6-x^11-6*x^10*y+x^9*y^2+6*x^8*y^3-6*x^7*y^4+x^6*y^5+2* x^10+4*x^9*y-3*x^8*y^2+3*x^7*y^3+9*x^6*y^4-2*x^5*y^5-x^9+4*x^8*y+6*x^7*y^2-8*x^ 6*y^3+x^5*y^4-2*x^8-7*x^7*y-x^6*y^2+8*x^5*y^3-5*x^4*y^4+2*x^7+3*x^6*y-6*x^5*y^2 +4*x^4*y^3-2*x^6-x^5*y+2*x^4*y^2-8*x^3*y^3-3*x^4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2* y^2-x^3+2*x^2*y+x^2-2*x*y+x-1)/(x-1)/(x^9*y^2-2*x^9*y+x^9+x^7*y-x^7-x^6*y+x^6+x ^5*y-x^5-x^4-x^3*y-x^2+1) ------------------------------------------------ "Theorem Number 184" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 2], [2, 1, 1, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 16 4 15 5 14 6 16 3 ) | ) C(m, n) x y | = (x y + 2 x y + x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 15 4 14 5 13 6 12 7 16 2 15 3 14 4 - 8 x y - 5 x y - x y + x y + 6 x y + 12 x y + 10 x y 13 5 12 6 16 15 2 14 3 13 4 12 5 + 4 x y - 3 x y - 4 x y - 8 x y - 9 x y - 6 x y + x y 11 6 16 15 14 2 13 3 12 4 11 5 + 2 x y + x + 2 x y + 2 x y + 2 x y + x y - 6 x y 10 6 14 13 2 12 3 11 4 10 5 14 + x y + 2 x y + 5 x y + 8 x y + 12 x y + 4 x y - x 13 12 2 11 3 10 4 9 5 8 6 13 - 6 x y - 14 x y - 17 x y - 17 x y - 2 x y + x y + 2 x 12 11 2 10 3 9 4 8 5 11 10 2 + 6 x y + 11 x y + 15 x y + 7 x y - 4 x y - x y + x y 9 3 8 4 7 5 6 6 11 10 9 2 8 3 - 9 x y - 2 x y + 3 x y - x y - x - 6 x y + x y + 6 x y 7 4 6 5 10 9 8 2 7 3 6 4 5 5 - 6 x y + x y + 2 x + 4 x y - 3 x y + 3 x y + 9 x y - 2 x y 9 8 7 2 6 3 5 4 8 7 6 2 - x + 4 x y + 6 x y - 8 x y + x y - 2 x - 7 x y - x y 5 3 4 4 7 6 5 2 4 3 6 5 + 8 x y - 5 x y + 2 x + 3 x y - 6 x y + 4 x y - 2 x - x y 4 2 3 3 4 3 2 4 3 2 2 3 + 2 x y - 8 x y - 3 x y + 4 x y + x + 3 x y - 4 x y - x 2 2 / 10 2 10 9 2 10 + 2 x y + x - 2 x y + x - 1) / (x y - 2 x y - x y + x / 9 9 8 8 7 7 6 6 5 4 + 2 x y - x + x y - x - 2 x y + 2 x + 2 x y - 2 x - x y - x y 4 3 3 2 + x + x y - x + x + x - 1) and in Maple notation (x^16*y^4+2*x^15*y^5+x^14*y^6-4*x^16*y^3-8*x^15*y^4-5*x^14*y^5-x^13*y^6+x^12*y^ 7+6*x^16*y^2+12*x^15*y^3+10*x^14*y^4+4*x^13*y^5-3*x^12*y^6-4*x^16*y-8*x^15*y^2-\ 9*x^14*y^3-6*x^13*y^4+x^12*y^5+2*x^11*y^6+x^16+2*x^15*y+2*x^14*y^2+2*x^13*y^3+x ^12*y^4-6*x^11*y^5+x^10*y^6+2*x^14*y+5*x^13*y^2+8*x^12*y^3+12*x^11*y^4+4*x^10*y ^5-x^14-6*x^13*y-14*x^12*y^2-17*x^11*y^3-17*x^10*y^4-2*x^9*y^5+x^8*y^6+2*x^13+6 *x^12*y+11*x^11*y^2+15*x^10*y^3+7*x^9*y^4-4*x^8*y^5-x^11*y+x^10*y^2-9*x^9*y^3-2 *x^8*y^4+3*x^7*y^5-x^6*y^6-x^11-6*x^10*y+x^9*y^2+6*x^8*y^3-6*x^7*y^4+x^6*y^5+2* x^10+4*x^9*y-3*x^8*y^2+3*x^7*y^3+9*x^6*y^4-2*x^5*y^5-x^9+4*x^8*y+6*x^7*y^2-8*x^ 6*y^3+x^5*y^4-2*x^8-7*x^7*y-x^6*y^2+8*x^5*y^3-5*x^4*y^4+2*x^7+3*x^6*y-6*x^5*y^2 +4*x^4*y^3-2*x^6-x^5*y+2*x^4*y^2-8*x^3*y^3-3*x^4*y+4*x^3*y^2+x^4+3*x^3*y-4*x^2* y^2-x^3+2*x^2*y+x^2-2*x*y+x-1)/(x^10*y^2-2*x^10*y-x^9*y^2+x^10+2*x^9*y-x^9+x^8* y-x^8-2*x^7*y+2*x^7+2*x^6*y-2*x^6-x^5*y-x^4*y+x^4+x^3*y-x^3+x^2+x-1) ------------------------------------------------ "Theorem Number 185" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 7 13 6 12 7 13 5 ) | ) C(m, n) x y | = - (2 x y - 9 x y - x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 11 7 13 4 12 5 11 6 10 7 + 8 x y - 2 x y - 10 x y - 20 x y + 4 x y + x y 12 4 11 5 10 6 13 2 12 3 11 4 + 21 x y + 6 x y - 8 x y + 3 x y - 8 x y - 22 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 + 12 x y + 3 x y - x y - x y + 20 x y + 3 x y - 13 x y 8 6 12 11 2 10 3 9 4 8 5 7 6 - x y + x y - 6 x y - 17 x y + 14 x y + 5 x y + x y 10 2 9 3 8 4 7 5 6 6 10 9 2 + 11 x y + 2 x y - 12 x y - 4 x y + x y - 2 x y - 9 x y 8 3 7 4 6 5 9 8 2 7 3 6 4 + 12 x y + 8 x y + 2 x y + 3 x y - x y - 11 x y - 7 x y 5 5 8 7 2 6 3 5 4 8 7 6 2 + 3 x y - 5 x y + 7 x y + 5 x y - x y + 2 x + x y - 5 x y 5 3 4 4 7 6 5 2 4 3 6 5 - 5 x y + 6 x y - 2 x + 5 x y + 3 x y - 5 x y - x - 2 x y 4 2 3 3 5 4 3 2 3 2 2 2 2 - x y + 8 x y + 2 x + x y - 4 x y - 2 x y + 4 x y - 2 x y - x / + 2 x y - x + 1) / ( / 8 8 7 7 6 6 5 5 2 2 x y - 2 x - 2 x y + 2 x - 2 x y + x + 2 x y - 2 x + x + x - 1) and in Maple notation -(2*x^13*y^7-9*x^13*y^6-x^12*y^7+15*x^13*y^5+8*x^12*y^6-2*x^11*y^7-10*x^13*y^4-\ 20*x^12*y^5+4*x^11*y^6+x^10*y^7+21*x^12*y^4+6*x^11*y^5-8*x^10*y^6+3*x^13*y^2-8* x^12*y^3-22*x^11*y^4+12*x^10*y^5+3*x^9*y^6-x^13*y-x^12*y^2+20*x^11*y^3+3*x^10*y ^4-13*x^9*y^5-x^8*y^6+x^12*y-6*x^11*y^2-17*x^10*y^3+14*x^9*y^4+5*x^8*y^5+x^7*y^ 6+11*x^10*y^2+2*x^9*y^3-12*x^8*y^4-4*x^7*y^5+x^6*y^6-2*x^10*y-9*x^9*y^2+12*x^8* y^3+8*x^7*y^4+2*x^6*y^5+3*x^9*y-x^8*y^2-11*x^7*y^3-7*x^6*y^4+3*x^5*y^5-5*x^8*y+ 7*x^7*y^2+5*x^6*y^3-x^5*y^4+2*x^8+x^7*y-5*x^6*y^2-5*x^5*y^3+6*x^4*y^4-2*x^7+5*x ^6*y+3*x^5*y^2-5*x^4*y^3-x^6-2*x^5*y-x^4*y^2+8*x^3*y^3+2*x^5+x^4*y-4*x^3*y^2-2* x^3*y+4*x^2*y^2-2*x^2*y-x^2+2*x*y-x+1)/(2*x^8*y-2*x^8-2*x^7*y+2*x^7-2*x^6*y+x^6 +2*x^5*y-2*x^5+x^2+x-1) ------------------------------------------------ "Theorem Number 186" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 1, 1], [1, 2, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 7 13 6 12 7 13 5 ) | ) C(m, n) x y | = - (2 x y - 9 x y - x y + 15 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 6 11 7 13 4 12 5 11 6 10 7 + 8 x y - 2 x y - 10 x y - 20 x y + 4 x y + x y 12 4 11 5 10 6 13 2 12 3 11 4 + 21 x y + 6 x y - 8 x y + 3 x y - 8 x y - 22 x y 10 5 9 6 13 12 2 11 3 10 4 9 5 + 12 x y + 3 x y - x y - x y + 20 x y + 3 x y - 13 x y 8 6 12 11 2 10 3 9 4 8 5 7 6 - x y + x y - 6 x y - 17 x y + 14 x y + 5 x y + x y 10 2 9 3 8 4 7 5 6 6 10 9 2 + 11 x y + 2 x y - 12 x y - 4 x y + x y - 2 x y - 9 x y 8 3 7 4 6 5 9 8 2 7 3 6 4 + 12 x y + 8 x y + 2 x y + 3 x y - x y - 11 x y - 7 x y 5 5 8 7 2 6 3 5 4 8 7 6 2 + 3 x y - 5 x y + 7 x y + 5 x y - x y + 2 x + x y - 5 x y 5 3 4 4 7 6 5 2 4 3 6 5 - 5 x y + 6 x y - 2 x + 5 x y + 3 x y - 5 x y - x - 2 x y 4 2 3 3 5 4 3 2 3 2 2 2 2 - x y + 8 x y + 2 x + x y - 4 x y - 2 x y + 4 x y - 2 x y - x / + 2 x y - x + 1) / ( / 8 8 7 7 6 6 5 5 2 2 x y - 2 x - 2 x y + 2 x - 2 x y + x + 2 x y - 2 x + x + x - 1) and in Maple notation -(2*x^13*y^7-9*x^13*y^6-x^12*y^7+15*x^13*y^5+8*x^12*y^6-2*x^11*y^7-10*x^13*y^4-\ 20*x^12*y^5+4*x^11*y^6+x^10*y^7+21*x^12*y^4+6*x^11*y^5-8*x^10*y^6+3*x^13*y^2-8* x^12*y^3-22*x^11*y^4+12*x^10*y^5+3*x^9*y^6-x^13*y-x^12*y^2+20*x^11*y^3+3*x^10*y ^4-13*x^9*y^5-x^8*y^6+x^12*y-6*x^11*y^2-17*x^10*y^3+14*x^9*y^4+5*x^8*y^5+x^7*y^ 6+11*x^10*y^2+2*x^9*y^3-12*x^8*y^4-4*x^7*y^5+x^6*y^6-2*x^10*y-9*x^9*y^2+12*x^8* y^3+8*x^7*y^4+2*x^6*y^5+3*x^9*y-x^8*y^2-11*x^7*y^3-7*x^6*y^4+3*x^5*y^5-5*x^8*y+ 7*x^7*y^2+5*x^6*y^3-x^5*y^4+2*x^8+x^7*y-5*x^6*y^2-5*x^5*y^3+6*x^4*y^4-2*x^7+5*x ^6*y+3*x^5*y^2-5*x^4*y^3-x^6-2*x^5*y-x^4*y^2+8*x^3*y^3+2*x^5+x^4*y-4*x^3*y^2-2* x^3*y+4*x^2*y^2-2*x^2*y-x^2+2*x*y-x+1)/(2*x^8*y-2*x^8-2*x^7*y+2*x^7-2*x^6*y+x^6 +2*x^5*y-2*x^5+x^2+x-1) ------------------------------------------------ "Theorem Number 187" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 4 6 6 8 3 7 4 ) | ) C(m, n) x y | = (x y - x y - 2 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 5 8 2 7 3 6 4 5 5 7 2 6 3 + 2 x y + x y + 4 x y + 3 x y - 4 x y - 2 x y - 8 x y 5 4 6 2 5 3 4 4 5 2 4 3 5 + 4 x y + 4 x y + 4 x y - 6 x y - 6 x y + 10 x y + 2 x y 4 2 3 3 4 3 2 4 2 2 3 2 - x y - 8 x y - 4 x y + 8 x y + x - 4 x y - 2 x + 4 x y / 3 2 - 2 x y + 2 x - 1) / ((x - x - x + 1) (x - 1)) / and in Maple notation (x^8*y^4-x^6*y^6-2*x^8*y^3-2*x^7*y^4+2*x^6*y^5+x^8*y^2+4*x^7*y^3+3*x^6*y^4-4*x^ 5*y^5-2*x^7*y^2-8*x^6*y^3+4*x^5*y^4+4*x^6*y^2+4*x^5*y^3-6*x^4*y^4-6*x^5*y^2+10* x^4*y^3+2*x^5*y-x^4*y^2-8*x^3*y^3-4*x^4*y+8*x^3*y^2+x^4-4*x^2*y^2-2*x^3+4*x^2*y -2*x*y+2*x-1)/(x^3-x^2-x+1)/(x-1) ------------------------------------------------ "Theorem Number 188" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [1, 2, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 5 6 4 5 5 6 3 5 4 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 3 4 4 5 2 4 3 4 2 3 3 4 + 5 x y + 3 x y - 4 x y - 14 x y + 12 x y + 8 x y - 2 x y 3 2 3 2 2 3 2 2 / - 12 x y + 6 x y + 4 x y - x - 6 x y + 3 x + 2 x y - 3 x + 1) / / 3 2 (x - 3 x + 3 x - 1) and in Maple notation -(x^6*y^5-2*x^6*y^4+x^5*y^5+x^6*y^3-x^5*y^4+5*x^5*y^3+3*x^4*y^4-4*x^5*y^2-14*x^ 4*y^3+12*x^4*y^2+8*x^3*y^3-2*x^4*y-12*x^3*y^2+6*x^3*y+4*x^2*y^2-x^3-6*x^2*y+3*x ^2+2*x*y-3*x+1)/(x^3-3*x^2+3*x-1) ------------------------------------------------ "Theorem Number 189" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 7 8 6 8 5 7 6 ) | ) C(m, n) x y | = - (2 x y - 5 x y + 4 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 4 7 5 6 6 7 4 6 5 7 3 6 4 - x y - 9 x y + 2 x y + 6 x y - x y + 3 x y - 5 x y 5 5 7 2 6 3 5 4 6 2 5 3 4 4 + 6 x y - 3 x y + 3 x y - 6 x y + 3 x y - 4 x y + 6 x y 6 5 2 4 3 5 4 2 3 3 5 4 - 2 x y + 3 x y - 10 x y + 2 x y + x y + 8 x y - x + 2 x y 3 2 4 2 2 3 2 / - 8 x y + x + 4 x y + x - 4 x y + 2 x y - 2 x + 1) / ( / 4 2 (x - x - x + 1) (x - 1)) and in Maple notation -(2*x^8*y^7-5*x^8*y^6+4*x^8*y^5+3*x^7*y^6-x^8*y^4-9*x^7*y^5+2*x^6*y^6+6*x^7*y^4 -x^6*y^5+3*x^7*y^3-5*x^6*y^4+6*x^5*y^5-3*x^7*y^2+3*x^6*y^3-6*x^5*y^4+3*x^6*y^2-\ 4*x^5*y^3+6*x^4*y^4-2*x^6*y+3*x^5*y^2-10*x^4*y^3+2*x^5*y+x^4*y^2+8*x^3*y^3-x^5+ 2*x^4*y-8*x^3*y^2+x^4+4*x^2*y^2+x^3-4*x^2*y+2*x*y-2*x+1)/(x^4-x^2-x+1)/(x-1) ------------------------------------------------ "Theorem Number 190" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [2, 1, 1, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 5 7 4 7 3 6 4 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 5 7 2 6 3 5 4 6 2 5 3 4 4 - 3 x y - x y - 10 x y + 3 x y + 5 x y + 5 x y - 6 x y 5 2 4 3 5 4 2 3 3 4 3 2 4 - 7 x y + 10 x y + 2 x y - x y - 8 x y - 4 x y + 8 x y + x 2 2 3 2 / 4 3 - 4 x y - 2 x + 4 x y - 2 x y + 2 x - 1) / (x - 2 x + 2 x - 1) / and in Maple notation (x^7*y^5-3*x^7*y^4+3*x^7*y^3+5*x^6*y^4-3*x^5*y^5-x^7*y^2-10*x^6*y^3+3*x^5*y^4+5 *x^6*y^2+5*x^5*y^3-6*x^4*y^4-7*x^5*y^2+10*x^4*y^3+2*x^5*y-x^4*y^2-8*x^3*y^3-4*x ^4*y+8*x^3*y^2+x^4-4*x^2*y^2-2*x^3+4*x^2*y-2*x*y+2*x-1)/(x^4-2*x^3+2*x-1) ------------------------------------------------ "Theorem Number 191" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 2], [2, 1, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 5 9 4 8 5 7 6 ) | ) C(m, n) x y | = - (x y - 3 x y - 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 3 8 4 7 5 9 2 8 3 7 4 8 2 + 3 x y + 7 x y - 2 x y - x y - 8 x y - 4 x y + 3 x y 7 3 6 4 5 5 7 2 5 4 6 2 5 3 + 10 x y - x y + 3 x y - 5 x y - 4 x y + 3 x y - 3 x y 4 4 6 5 2 4 3 5 4 2 3 3 5 + 5 x y - 2 x y + 2 x y - 10 x y + 2 x y + 2 x y + 8 x y - x 4 3 2 4 2 2 3 2 / + 2 x y - 8 x y + x + 4 x y + x - 4 x y + 2 x y - 2 x + 1) / ( / 2 3 2 (x - 2 x + 1) (x + x - 1)) and in Maple notation -(x^9*y^5-3*x^9*y^4-2*x^8*y^5+x^7*y^6+3*x^9*y^3+7*x^8*y^4-2*x^7*y^5-x^9*y^2-8*x ^8*y^3-4*x^7*y^4+3*x^8*y^2+10*x^7*y^3-x^6*y^4+3*x^5*y^5-5*x^7*y^2-4*x^5*y^4+3*x ^6*y^2-3*x^5*y^3+5*x^4*y^4-2*x^6*y+2*x^5*y^2-10*x^4*y^3+2*x^5*y+2*x^4*y^2+8*x^3 *y^3-x^5+2*x^4*y-8*x^3*y^2+x^4+4*x^2*y^2+x^3-4*x^2*y+2*x*y-2*x+1)/(x^2-2*x+1)/( x^3+x^2-1) ------------------------------------------------ "Theorem Number 192" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 6 7 5 7 4 6 5 ) | ) C(m, n) x y | = (x y - 2 x y + x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 4 5 5 5 4 6 2 5 3 4 4 5 2 4 3 - 6 x y + x y - x y + 2 x y + x y + 6 x y - 3 x y - 10 x y 5 4 2 3 3 4 3 2 4 3 2 2 + 2 x y + 5 x y + 8 x y - 2 x y - 8 x y + x + 2 x y + 4 x y 3 2 2 / 3 - x - 4 x y + x + 2 x y - 2 x + 1) / ((x - 1) (x + x - 1)) / and in Maple notation (x^7*y^6-2*x^7*y^5+x^7*y^4+4*x^6*y^5-6*x^6*y^4+x^5*y^5-x^5*y^4+2*x^6*y^2+x^5*y^ 3+6*x^4*y^4-3*x^5*y^2-10*x^4*y^3+2*x^5*y+5*x^4*y^2+8*x^3*y^3-2*x^4*y-8*x^3*y^2+ x^4+2*x^3*y+4*x^2*y^2-x^3-4*x^2*y+x^2+2*x*y-2*x+1)/(x-1)/(x^3+x-1) ------------------------------------------------ "Theorem Number 193" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 1, 2, 1], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 14 6 13 7 14 5 ) | ) C(m, n) x y | = (x y - 3 x y - 2 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 6 12 7 14 4 13 5 12 6 14 3 13 4 + 6 x y + x y + 2 x y - 3 x y - 4 x y - 3 x y - 7 x y 12 5 10 7 14 2 13 3 12 4 11 5 10 6 + 3 x y - x y + x y + 9 x y + 5 x y + x y + 3 x y 9 7 13 2 12 3 11 4 10 5 9 6 8 7 + 2 x y - 3 x y - 8 x y - 2 x y - 7 x y - 4 x y - x y 12 2 10 4 9 5 8 6 11 2 10 3 9 4 + 3 x y + 8 x y + 6 x y + 3 x y + 2 x y + x y - 11 x y 8 5 11 10 2 9 3 8 4 7 5 6 6 10 - 2 x y - x y - 7 x y + 7 x y + 5 x y - x y - x y + 3 x y 9 2 8 3 7 4 6 5 9 8 2 7 3 + 3 x y - 10 x y + x y + 4 x y - 3 x y + 4 x y + 2 x y 6 4 5 5 8 7 2 6 3 5 4 7 6 2 - 6 x y - 3 x y + x y - 3 x y + 10 x y + 9 x y + x y - 5 x y 5 3 4 4 6 5 2 4 3 6 5 4 2 - 20 x y - 5 x y - 2 x y + 13 x y + 20 x y + x + x y - 16 x y 3 3 5 4 3 2 4 3 2 2 3 2 - 8 x y - 2 x + 2 x y + 12 x y + x - 6 x y - 4 x y + x + 6 x y 2 / 6 5 4 3 2 - 3 x - 2 x y + 3 x - 1) / (x - 2 x + x + x - 3 x + 3 x - 1) / and in Maple notation (x^14*y^7-3*x^14*y^6-2*x^13*y^7+2*x^14*y^5+6*x^13*y^6+x^12*y^7+2*x^14*y^4-3*x^ 13*y^5-4*x^12*y^6-3*x^14*y^3-7*x^13*y^4+3*x^12*y^5-x^10*y^7+x^14*y^2+9*x^13*y^3 +5*x^12*y^4+x^11*y^5+3*x^10*y^6+2*x^9*y^7-3*x^13*y^2-8*x^12*y^3-2*x^11*y^4-7*x^ 10*y^5-4*x^9*y^6-x^8*y^7+3*x^12*y^2+8*x^10*y^4+6*x^9*y^5+3*x^8*y^6+2*x^11*y^2+x ^10*y^3-11*x^9*y^4-2*x^8*y^5-x^11*y-7*x^10*y^2+7*x^9*y^3+5*x^8*y^4-x^7*y^5-x^6* y^6+3*x^10*y+3*x^9*y^2-10*x^8*y^3+x^7*y^4+4*x^6*y^5-3*x^9*y+4*x^8*y^2+2*x^7*y^3 -6*x^6*y^4-3*x^5*y^5+x^8*y-3*x^7*y^2+10*x^6*y^3+9*x^5*y^4+x^7*y-5*x^6*y^2-20*x^ 5*y^3-5*x^4*y^4-2*x^6*y+13*x^5*y^2+20*x^4*y^3+x^6+x^5*y-16*x^4*y^2-8*x^3*y^3-2* x^5+2*x^4*y+12*x^3*y^2+x^4-6*x^3*y-4*x^2*y^2+x^3+6*x^2*y-3*x^2-2*x*y+3*x-1)/(x^ 6-2*x^5+x^4+x^3-3*x^2+3*x-1) ------------------------------------------------ "Theorem Number 194" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1], [1, 2, 2, 2], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 14 7 14 6 13 7 14 5 ) | ) C(m, n) x y | = (x y - 3 x y - 2 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 6 12 7 14 4 13 5 12 6 14 3 13 4 + 6 x y + x y + 2 x y - 3 x y - 4 x y - 3 x y - 7 x y 12 5 10 7 14 2 13 3 12 4 11 5 10 6 + 3 x y - x y + x y + 9 x y + 5 x y + x y + 3 x y 9 7 13 2 12 3 11 4 10 5 9 6 8 7 + 2 x y - 3 x y - 8 x y - 2 x y - 7 x y - 4 x y - x y 12 2 10 4 9 5 8 6 11 2 10 3 9 4 + 3 x y + 8 x y + 6 x y + 3 x y + 2 x y + x y - 11 x y 8 5 11 10 2 9 3 8 4 7 5 6 6 10 - 2 x y - x y - 7 x y + 7 x y + 5 x y - x y - x y + 3 x y 9 2 8 3 7 4 6 5 9 8 2 7 3 + 3 x y - 10 x y + x y + 4 x y - 3 x y + 4 x y + 2 x y 6 4 5 5 8 7 2 6 3 5 4 7 6 2 - 6 x y - 3 x y + x y - 3 x y + 10 x y + 9 x y + x y - 5 x y 5 3 4 4 6 5 2 4 3 6 5 4 2 - 20 x y - 5 x y - 2 x y + 13 x y + 20 x y + x + x y - 16 x y 3 3 5 4 3 2 4 3 2 2 3 2 - 8 x y - 2 x + 2 x y + 12 x y + x - 6 x y - 4 x y + x + 6 x y 2 / 2 4 - 3 x - 2 x y + 3 x - 1) / ((x - 2 x + 1) (x + x - 1)) / and in Maple notation (x^14*y^7-3*x^14*y^6-2*x^13*y^7+2*x^14*y^5+6*x^13*y^6+x^12*y^7+2*x^14*y^4-3*x^ 13*y^5-4*x^12*y^6-3*x^14*y^3-7*x^13*y^4+3*x^12*y^5-x^10*y^7+x^14*y^2+9*x^13*y^3 +5*x^12*y^4+x^11*y^5+3*x^10*y^6+2*x^9*y^7-3*x^13*y^2-8*x^12*y^3-2*x^11*y^4-7*x^ 10*y^5-4*x^9*y^6-x^8*y^7+3*x^12*y^2+8*x^10*y^4+6*x^9*y^5+3*x^8*y^6+2*x^11*y^2+x ^10*y^3-11*x^9*y^4-2*x^8*y^5-x^11*y-7*x^10*y^2+7*x^9*y^3+5*x^8*y^4-x^7*y^5-x^6* y^6+3*x^10*y+3*x^9*y^2-10*x^8*y^3+x^7*y^4+4*x^6*y^5-3*x^9*y+4*x^8*y^2+2*x^7*y^3 -6*x^6*y^4-3*x^5*y^5+x^8*y-3*x^7*y^2+10*x^6*y^3+9*x^5*y^4+x^7*y-5*x^6*y^2-20*x^ 5*y^3-5*x^4*y^4-2*x^6*y+13*x^5*y^2+20*x^4*y^3+x^6+x^5*y-16*x^4*y^2-8*x^3*y^3-2* x^5+2*x^4*y+12*x^3*y^2+x^4-6*x^3*y-4*x^2*y^2+x^3+6*x^2*y-3*x^2-2*x*y+3*x-1)/(x^ 2-2*x+1)/(x^4+x-1) ------------------------------------------------ "Theorem Number 195" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 2, 1], [2, 2, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 5 8 4 7 5 8 3 ) | ) C(m, n) x y | = - (x y - 2 x y - x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 4 7 3 6 4 5 5 6 3 5 4 6 2 + 2 x y - x y - 2 x y - x y + 6 x y - 3 x y - 4 x y 5 3 4 4 4 3 5 4 2 3 3 3 2 4 + 5 x y - 3 x y - 2 x y - 2 x y + 4 x y - 8 x y + 4 x y - x 3 2 2 2 2 / + 2 x y - 4 x y + 2 x y + x - 2 x y + x - 1) / ((x - 1) / 3 2 (x + x - 1)) and in Maple notation -(x^8*y^5-2*x^8*y^4-x^7*y^5+x^8*y^3+2*x^7*y^4-x^7*y^3-2*x^6*y^4-x^5*y^5+6*x^6*y ^3-3*x^5*y^4-4*x^6*y^2+5*x^5*y^3-3*x^4*y^4-2*x^4*y^3-2*x^5*y+4*x^4*y^2-8*x^3*y^ 3+4*x^3*y^2-x^4+2*x^3*y-4*x^2*y^2+2*x^2*y+x^2-2*x*y+x-1)/(x-1)/(x^3+x^2-1) ------------------------------------------------ "Theorem Number 196" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2], [2, 1, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 5 8 4 7 5 8 3 ) | ) C(m, n) x y | = - (2 x y - 5 x y - 2 x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 4 8 2 7 3 6 4 5 5 7 2 6 3 5 4 + 6 x y - x y - 6 x y - x y - 2 x y + 2 x y + 4 x y - x y 6 2 5 3 4 4 4 3 5 4 2 3 3 - 3 x y + 4 x y - 5 x y + 2 x y - 2 x y + 2 x y - 8 x y 3 2 4 3 2 2 2 2 / + 4 x y - x + 2 x y - 4 x y + 2 x y + x - 2 x y + x - 1) / ( / 4 2 x - x - x + 1) and in Maple notation -(2*x^8*y^5-5*x^8*y^4-2*x^7*y^5+4*x^8*y^3+6*x^7*y^4-x^8*y^2-6*x^7*y^3-x^6*y^4-2 *x^5*y^5+2*x^7*y^2+4*x^6*y^3-x^5*y^4-3*x^6*y^2+4*x^5*y^3-5*x^4*y^4+2*x^4*y^3-2* x^5*y+2*x^4*y^2-8*x^3*y^3+4*x^3*y^2-x^4+2*x^3*y-4*x^2*y^2+2*x^2*y+x^2-2*x*y+x-1 )/(x^4-x^2-x+1) ------------------------------------------------ "Theorem Number 197" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 2], [2, 1, 2, 1]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 5 9 4 8 5 9 3 8 4 ) | ) C(m, n) x y | = (x y - 2 x y - x y + x y + 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 3 7 4 6 5 8 2 7 3 6 4 5 5 - 5 x y - 4 x y - x y + 2 x y + 9 x y + 2 x y + 2 x y 7 2 6 3 5 4 7 6 2 5 3 4 4 - 6 x y - 9 x y - 3 x y + x y + 9 x y + 5 x y + 5 x y 6 5 2 4 3 5 4 2 3 3 4 - 2 x y - 7 x y - 10 x y + 3 x y + 6 x y + 8 x y - 2 x y 3 2 4 3 2 2 3 2 2 - 8 x y + x + 2 x y + 4 x y - x - 4 x y + x + 2 x y - 2 x + 1) / 4 3 2 / (x - x + x - 2 x + 1) / and in Maple notation (x^9*y^5-2*x^9*y^4-x^8*y^5+x^9*y^3+4*x^8*y^4-5*x^8*y^3-4*x^7*y^4-x^6*y^5+2*x^8* y^2+9*x^7*y^3+2*x^6*y^4+2*x^5*y^5-6*x^7*y^2-9*x^6*y^3-3*x^5*y^4+x^7*y+9*x^6*y^2 +5*x^5*y^3+5*x^4*y^4-2*x^6*y-7*x^5*y^2-10*x^4*y^3+3*x^5*y+6*x^4*y^2+8*x^3*y^3-2 *x^4*y-8*x^3*y^2+x^4+2*x^3*y+4*x^2*y^2-x^3-4*x^2*y+x^2+2*x*y-2*x+1)/(x^4-x^3+x^ 2-2*x+1) ------------------------------------------------ "Theorem Number 198" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2], [1, 2, 1, 1], [2, 1, 2, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 5 12 6 13 4 12 5 ) | ) C(m, n) x y | = (x y + 2 x y - 3 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 6 13 3 12 4 11 5 10 6 13 2 12 3 - x y + 3 x y + 4 x y + 5 x y - x y - x y - 2 x y 11 4 12 2 11 3 10 4 9 5 8 6 12 - 7 x y + 2 x y + 2 x y + 4 x y - 6 x y - 2 x y - x y 11 2 10 3 9 4 8 5 7 6 11 10 2 9 3 + 2 x y - 3 x y + 9 x y + 4 x y + x y - x y - x y + x y 8 4 7 5 6 6 10 9 2 8 3 7 4 6 5 - 9 x y - 2 x y + x y + x y - 5 x y + 12 x y + 5 x y + x y 9 8 2 7 3 6 4 5 5 7 2 6 3 + x y - 5 x y - 12 x y - x y + 2 x y + 10 x y + 2 x y 5 4 7 6 2 5 3 4 4 6 5 2 - 4 x y - 2 x y - 3 x y + 6 x y + 5 x y + x y - 4 x y 4 3 4 2 3 3 5 3 2 4 3 2 2 - 12 x y + 8 x y + 8 x y + x - 8 x y - x + 2 x y + 4 x y 2 2 / 5 4 2 - 4 x y + x + 2 x y - 2 x + 1) / (x - x + x - 2 x + 1) / and in Maple notation (x^13*y^5+2*x^12*y^6-3*x^13*y^4-5*x^12*y^5-x^11*y^6+3*x^13*y^3+4*x^12*y^4+5*x^ 11*y^5-x^10*y^6-x^13*y^2-2*x^12*y^3-7*x^11*y^4+2*x^12*y^2+2*x^11*y^3+4*x^10*y^4 -6*x^9*y^5-2*x^8*y^6-x^12*y+2*x^11*y^2-3*x^10*y^3+9*x^9*y^4+4*x^8*y^5+x^7*y^6-x ^11*y-x^10*y^2+x^9*y^3-9*x^8*y^4-2*x^7*y^5+x^6*y^6+x^10*y-5*x^9*y^2+12*x^8*y^3+ 5*x^7*y^4+x^6*y^5+x^9*y-5*x^8*y^2-12*x^7*y^3-x^6*y^4+2*x^5*y^5+10*x^7*y^2+2*x^6 *y^3-4*x^5*y^4-2*x^7*y-3*x^6*y^2+6*x^5*y^3+5*x^4*y^4+x^6*y-4*x^5*y^2-12*x^4*y^3 +8*x^4*y^2+8*x^3*y^3+x^5-8*x^3*y^2-x^4+2*x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+ 1)/(x^5-x^4+x^2-2*x+1) ------------------------------------------------ "Theorem Number 199" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1], [1, 2, 2, 2], [2, 2, 1, 2]}, and having m neighbors (i.e. \ Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 13 5 12 6 13 4 12 5 ) | ) C(m, n) x y | = (x y + 2 x y - 3 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 6 13 3 12 4 11 5 10 6 13 2 12 3 - x y + 3 x y + 4 x y + 5 x y - x y - x y - 2 x y 11 4 12 2 11 3 10 4 9 5 8 6 12 - 7 x y + 2 x y + 2 x y + 4 x y - 6 x y - 2 x y - x y 11 2 10 3 9 4 8 5 7 6 11 10 2 9 3 + 2 x y - 3 x y + 9 x y + 4 x y + x y - x y - x y + x y 8 4 7 5 6 6 10 9 2 8 3 7 4 6 5 - 9 x y - 2 x y + x y + x y - 5 x y + 12 x y + 5 x y + x y 9 8 2 7 3 6 4 5 5 7 2 6 3 + x y - 5 x y - 12 x y - x y + 2 x y + 10 x y + 2 x y 5 4 7 6 2 5 3 4 4 6 5 2 - 4 x y - 2 x y - 3 x y + 6 x y + 5 x y + x y - 4 x y 4 3 4 2 3 3 5 3 2 4 3 2 2 - 12 x y + 8 x y + 8 x y + x - 8 x y - x + 2 x y + 4 x y 2 2 / 5 4 2 - 4 x y + x + 2 x y - 2 x + 1) / (x - x + x - 2 x + 1) / and in Maple notation (x^13*y^5+2*x^12*y^6-3*x^13*y^4-5*x^12*y^5-x^11*y^6+3*x^13*y^3+4*x^12*y^4+5*x^ 11*y^5-x^10*y^6-x^13*y^2-2*x^12*y^3-7*x^11*y^4+2*x^12*y^2+2*x^11*y^3+4*x^10*y^4 -6*x^9*y^5-2*x^8*y^6-x^12*y+2*x^11*y^2-3*x^10*y^3+9*x^9*y^4+4*x^8*y^5+x^7*y^6-x ^11*y-x^10*y^2+x^9*y^3-9*x^8*y^4-2*x^7*y^5+x^6*y^6+x^10*y-5*x^9*y^2+12*x^8*y^3+ 5*x^7*y^4+x^6*y^5+x^9*y-5*x^8*y^2-12*x^7*y^3-x^6*y^4+2*x^5*y^5+10*x^7*y^2+2*x^6 *y^3-4*x^5*y^4-2*x^7*y-3*x^6*y^2+6*x^5*y^3+5*x^4*y^4+x^6*y-4*x^5*y^2-12*x^4*y^3 +8*x^4*y^2+8*x^3*y^3+x^5-8*x^3*y^2-x^4+2*x^3*y+4*x^2*y^2-4*x^2*y+x^2+2*x*y-2*x+ 1)/(x^5-x^4+x^2-2*x+1) ---------------------------------- This ends this paper that took, 219.485, seconds to produce this took, 219.485, seconds.