Counting Words in the Alphabet, {1, 2, 3}, That Avoid A Certain set of , 1, consecutive subwords of length, 2, 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, 3}, avoiding consecutive substrings in the set, {[1, 1]}, and having m neighb\ ors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | 2 4 2 3 2 \ | \ n m| 2 x y - 2 x y - x y - 1 ) | ) C(m, n) x y | = - ---------------------------------------- / | / | 3 5 3 4 2 3 2 ----- | ----- | 4 x y - 4 x y - 2 x y - 2 x y + 1 m = 0 \ n = 0 / and in Maple notation -(2*x^2*y^4-2*x^2*y^3-x*y^2-1)/(4*x^3*y^5-4*x^3*y^4-2*x^2*y^3-2*x*y^2+1) As the length of the word goes to infinity, the average number of good neigh\ 2 a + 2 a + 3 bors of a random word of length n tends to n times, -------------- 2 2 a + 3 a + 1 2 where a is the root of the polynomial, 2 x + 2 x - 1, and in decimals this is, 1.633974595 BTW the ratio for words with, 500, letters is, 1.631247999 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2]}, and having m neighb\ ors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = / | / | ----- | ----- | m = 0 \ n = 0 / 2 4 2 3 2 2 2 x y - 2 x y + x y + 2 x y - 2 x y + 1 -------------------------------------------- 2 2 2 x y - x y - 2 x y + 1 and in Maple notation (x^2*y^4-2*x^2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(x^2*y^2-x*y^2-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ 2 (a - 1) (a - 3) bors of a random word of length n tends to n times, - ----------------- 2 a - 3 2 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.447213593 BTW the ratio for words with, 500, letters is, 1.445544797 ---------------------------------- This ends this paper that took, 0.914, seconds to produce ------------------------------ Counting Words in the Alphabet, {1, 2, 3}, That Avoid A Certain set of , 1, 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, 3}, avoiding consecutive substrings in the set, {[1, 1, 1]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 3 6 ) | ) C(m, n) x y | = - (4 x y - 8 x y + 4 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 5 2 4 2 3 2 / 6 11 6 10 - 2 x y - 3 x y + 2 x y - x y - 1) / (8 x y - 16 x y / 6 9 4 7 4 6 3 6 3 5 2 3 2 + 8 x y + 4 x y - 4 x y - 4 x y + 2 x y - 2 x y - 2 x y + 1) and in Maple notation -(4*x^5*y^10-8*x^5*y^9+4*x^5*y^8+2*x^3*y^6-2*x^3*y^5-3*x^2*y^4+2*x^2*y^3-x*y^2-\ 1)/(8*x^6*y^11-16*x^6*y^10+8*x^6*y^9+4*x^4*y^7-4*x^4*y^6-4*x^3*y^6+2*x^3*y^5-2* x^2*y^3-2*x*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 3 2 a + 2 a + 3 a + 6 a + 3 ---------------------------- 4 3 2 3 a + 5 a + 6 a + 3 a + 1 3 2 where a is the root of the polynomial, 2 x + 2 x + 2 x - 1, and in decimals this is, 1.849993118 BTW the ratio for words with, 500, letters is, 1.846767610 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 1]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 4 8 ) | ) C(m, n) x y | = - (2 x y - 4 x y + 2 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 7 4 6 4 5 3 6 3 5 2 4 2 3 2 - 5 x y + 3 x y - x y - x y + x y - 3 x y + 2 x y - x y / 6 11 6 10 6 9 5 9 5 7 4 8 + x y - 1) / (4 x y - 8 x y + 4 x y + 2 x y - 2 x y - x y / 4 7 4 6 4 5 3 6 3 5 2 3 2 - x y + x y + x y - x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(2*x^5*y^10-4*x^5*y^9+2*x^5*y^8+3*x^4*y^8-5*x^4*y^7+3*x^4*y^6-x^4*y^5-x^3*y^6+ x^3*y^5-3*x^2*y^4+2*x^2*y^3-x*y^2+x*y-1)/(4*x^6*y^11-8*x^6*y^10+4*x^6*y^9+2*x^5 *y^9-2*x^5*y^7-x^4*y^8-x^4*y^7+x^4*y^6+x^4*y^5-x^3*y^6-x^3*y^5+x^2*y^3-2*x*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, 4 2 (a + 3) ------------------------- 2 2 (6 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, 2 x - x + 3 x - 1, and in decimals this is, 1.779950118 BTW the ratio for words with, 500, letters is, 1.777201655 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 1, 2]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 4 7 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 6 4 5 2 4 2 3 2 / + 2 x y - x y - 2 x y + 2 x y - x y + x y - 1) / ( / 4 6 4 5 3 6 2 4 2 3 2 x y - x y - x y + x y - x y + 2 x y + x y - 1) and in Maple notation (x^5*y^10-3*x^5*y^9+3*x^5*y^8-x^5*y^7-x^4*y^7+2*x^4*y^6-x^4*y^5-2*x^2*y^4+2*x^2 *y^3-x*y^2+x*y-1)/(x^4*y^6-x^4*y^5-x^3*y^6+x^2*y^4-x^2*y^3+2*x*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, 1.765882852 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 2]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 4 7 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 6 4 5 2 4 2 3 2 / + 2 x y - x y - 2 x y + 2 x y - x y + x y - 1) / ( / 4 6 4 5 3 6 2 4 2 3 2 x y - x y - x y + x y - x y + 2 x y + x y - 1) and in Maple notation (x^5*y^10-3*x^5*y^9+3*x^5*y^8-x^5*y^7-x^4*y^7+2*x^4*y^6-x^4*y^5-2*x^2*y^4+2*x^2 *y^3-x*y^2+x*y-1)/(x^4*y^6-x^4*y^5-x^3*y^6+x^2*y^4-x^2*y^3+2*x*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, 1.765882852 [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, 3}, avoiding consecutive substrings in the set, {[1, 2, 3]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 6 2 4 2 3 x y - 2 x y + x y + 2 x y - 2 x y + 1 - ---------------------------------------------------------------- 4 8 4 7 4 6 3 6 2 4 2 3 2 x y - 2 x y + x y - x y - 2 x y + 2 x y + 3 x y - 1 and in Maple notation -(x^4*y^8-2*x^4*y^7+x^4*y^6+2*x^2*y^4-2*x^2*y^3+1)/(x^4*y^8-2*x^4*y^7+x^4*y^6-x ^3*y^6-2*x^2*y^4+2*x^2*y^3+3*x*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, approximaley to, 1.734397457 [This estimate was obtained by looking at words of length, 500, ] ---------------------------------- This ends this paper that took, 32.105, seconds to produce ------------------------------ Counting Words in the Alphabet, {1, 2, 3}, That Avoid A Certain set of , 1, 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, 3}, avoiding consecutive substrings in the set, {[1, 1, 1, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 18 9 17 9 16 ) | ) C(m, n) x y | = - (8 x y - 24 x y + 24 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 15 7 14 7 13 7 12 5 10 5 9 - 8 x y - 4 x y + 8 x y - 4 x y - 6 x y + 10 x y 5 8 4 8 4 7 3 6 3 5 2 4 2 / - 4 x y - 4 x y + 4 x y + 3 x y - 2 x y + x y + x y + 1) / / 10 19 10 18 10 17 10 16 8 15 8 14 (16 x y - 48 x y + 48 x y - 16 x y - 8 x y + 16 x y 8 13 6 12 6 11 6 10 5 9 5 8 4 8 - 8 x y - 8 x y + 12 x y - 4 x y - 8 x y + 8 x y + 4 x y 4 7 3 5 2 4 2 - 2 x y + 2 x y + 2 x y + 2 x y - 1) and in Maple notation -(8*x^9*y^18-24*x^9*y^17+24*x^9*y^16-8*x^9*y^15-4*x^7*y^14+8*x^7*y^13-4*x^7*y^ 12-6*x^5*y^10+10*x^5*y^9-4*x^5*y^8-4*x^4*y^8+4*x^4*y^7+3*x^3*y^6-2*x^3*y^5+x^2* y^4+x*y^2+1)/(16*x^10*y^19-48*x^10*y^18+48*x^10*y^17-16*x^10*y^16-8*x^8*y^15+16 *x^8*y^14-8*x^8*y^13-8*x^6*y^12+12*x^6*y^11-4*x^6*y^10-8*x^5*y^9+8*x^5*y^8+4*x^ 4*y^8-2*x^4*y^7+2*x^3*y^5+2*x^2*y^4+2*x*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, 6 5 4 3 2 a + 2 a + 3 a + 4 a + 9 a + 6 a + 3 ------------------------------------------- 6 5 4 3 2 4 a + 7 a + 9 a + 10 a + 6 a + 3 a + 1 4 3 2 where a is the root of the polynomial, 2 x + 2 x + 2 x + 2 x - 1, and in decimals this is, 1.937729820 BTW the ratio for words with, 500, letters is, 1.934168305 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 1, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 5 10 ) | ) C(m, n) x y | = - (x y - 2 x y + x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 9 5 8 4 8 4 7 3 6 3 5 2 4 + 10 x y - 5 x y - 4 x y + 4 x y + 2 x y - 2 x y + x y + 1) / 6 12 6 11 5 10 5 9 5 8 4 8 4 7 / (4 x y - 4 x y - 7 x y + 2 x y + 5 x y + x y - 2 x y / 3 6 3 5 2 4 2 + x y + 2 x y - x y + 3 x y - 1) and in Maple notation -(x^6*y^12-2*x^6*y^11+x^6*y^10-5*x^5*y^10+10*x^5*y^9-5*x^5*y^8-4*x^4*y^8+4*x^4* y^7+2*x^3*y^6-2*x^3*y^5+x^2*y^4+1)/(4*x^6*y^12-4*x^6*y^11-7*x^5*y^10+2*x^5*y^9+ 5*x^5*y^8+x^4*y^8-2*x^4*y^7+x^3*y^6+2*x^3*y^5-x^2*y^4+3*x*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, 6 5 4 3 2 2 (a - 4 a + 3 a - 8 a + 6 a + 3) - -------------------------------------- 3 2 2 (4 a - 9 a + 2 a - 3) (a + 1) 4 3 2 where a is the root of the polynomial, x - 3 x + x - 3 x + 1, and in decimals this is, 1.909738647 BTW the ratio for words with, 500, letters is, 1.906430467 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 1, 1, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 18 9 17 9 16 9 15 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 16 9 14 8 15 8 14 8 13 7 14 8 12 + x y + x y - 4 x y + 6 x y - 4 x y + 2 x y + x y 7 13 7 12 7 11 6 12 7 10 6 11 6 10 - 7 x y + 9 x y - 5 x y + x y + x y - 4 x y + 5 x y 6 9 5 10 5 9 5 8 5 7 4 8 4 7 4 6 - 2 x y - x y + x y + x y - x y - x y + 2 x y - x y 3 6 3 5 3 4 2 4 2 3 2 / - 3 x y + 4 x y - x y - 2 x y + 2 x y - x y + x y - 1) / ( / 7 12 7 11 6 12 7 10 6 11 6 10 6 9 5 8 x y - 2 x y - x y + x y + x y + x y - x y + x y 5 7 4 8 4 7 4 6 3 6 3 5 3 4 2 4 - x y - x y - 2 x y + 2 x y + 3 x y - 2 x y - x y + x y 2 3 2 - x y + 2 x y + x y - 1) and in Maple notation (x^9*y^18-4*x^9*y^17+6*x^9*y^16-4*x^9*y^15+x^8*y^16+x^9*y^14-4*x^8*y^15+6*x^8*y ^14-4*x^8*y^13+2*x^7*y^14+x^8*y^12-7*x^7*y^13+9*x^7*y^12-5*x^7*y^11+x^6*y^12+x^ 7*y^10-4*x^6*y^11+5*x^6*y^10-2*x^6*y^9-x^5*y^10+x^5*y^9+x^5*y^8-x^5*y^7-x^4*y^8 +2*x^4*y^7-x^4*y^6-3*x^3*y^6+4*x^3*y^5-x^3*y^4-2*x^2*y^4+2*x^2*y^3-x*y^2+x*y-1) /(x^7*y^12-2*x^7*y^11-x^6*y^12+x^7*y^10+x^6*y^11+x^6*y^10-x^6*y^9+x^5*y^8-x^5*y ^7-x^4*y^8-2*x^4*y^7+2*x^4*y^6+3*x^3*y^6-2*x^3*y^5-x^3*y^4+x^2*y^4-x^2*y^3+2*x* 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, 6 5 4 3 2 (a + a + a - 8 a + 3) - --------------------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.905039195 BTW the ratio for words with, 500, letters is, 1.901799331 ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 2, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 18 9 17 9 16 9 15 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 16 9 14 8 15 8 14 8 13 7 14 8 12 + x y + x y - 4 x y + 6 x y - 4 x y + 2 x y + x y 7 13 7 12 7 11 6 12 7 10 6 11 6 10 - 7 x y + 9 x y - 5 x y + x y + x y - 4 x y + 5 x y 6 9 5 10 5 9 5 8 5 7 4 8 4 7 4 6 - 2 x y - x y + x y + x y - x y - x y + 2 x y - x y 3 6 3 5 3 4 2 4 2 3 2 / - 3 x y + 4 x y - x y - 2 x y + 2 x y - x y + x y - 1) / ( / 7 12 7 11 6 12 7 10 6 11 6 10 6 9 5 8 x y - 2 x y - x y + x y + x y + x y - x y + x y 5 7 4 8 4 7 4 6 3 6 3 5 3 4 2 4 - x y - x y - 2 x y + 2 x y + 3 x y - 2 x y - x y + x y 2 3 2 - x y + 2 x y + x y - 1) and in Maple notation (x^9*y^18-4*x^9*y^17+6*x^9*y^16-4*x^9*y^15+x^8*y^16+x^9*y^14-4*x^8*y^15+6*x^8*y ^14-4*x^8*y^13+2*x^7*y^14+x^8*y^12-7*x^7*y^13+9*x^7*y^12-5*x^7*y^11+x^6*y^12+x^ 7*y^10-4*x^6*y^11+5*x^6*y^10-2*x^6*y^9-x^5*y^10+x^5*y^9+x^5*y^8-x^5*y^7-x^4*y^8 +2*x^4*y^7-x^4*y^6-3*x^3*y^6+4*x^3*y^5-x^3*y^4-2*x^2*y^4+2*x^2*y^3-x*y^2+x*y-1) /(x^7*y^12-2*x^7*y^11-x^6*y^12+x^7*y^10+x^6*y^11+x^6*y^10-x^6*y^9+x^5*y^8-x^5*y ^7-x^4*y^8-2*x^4*y^7+2*x^4*y^6+3*x^3*y^6-2*x^3*y^5-x^3*y^4+x^2*y^4-x^2*y^3+2*x* 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, 6 5 4 3 2 (a + a + a - 8 a + 3) - --------------------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.905039195 BTW the ratio for words with, 500, letters is, 1.901799331 ------------------------------------------------ "Theorem Number 5" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 1, 2, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 14 7 13 7 12 ) | ) C(m, n) x y | = - (2 x y - 5 x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 11 6 12 7 10 6 11 6 10 6 9 5 10 - 3 x y + 2 x y + x y - 3 x y + 2 x y - x y - 2 x y 5 9 5 7 4 8 4 7 4 6 3 6 3 5 + 3 x y - x y - 4 x y + 7 x y - 3 x y - 6 x y + 6 x y 3 4 2 4 2 3 2 / 8 15 8 14 - x y - 2 x y + 2 x y - x y + x y - 1) / (2 x y - 2 x y / 8 13 8 12 7 13 7 12 7 11 6 12 7 10 - 2 x y + 2 x y + x y + x y - x y - 2 x y - x y 6 11 6 10 6 9 5 10 5 9 5 8 5 7 + x y + 2 x y - x y - 3 x y + 5 x y - 3 x y + x y 4 8 4 7 3 4 2 4 2 3 2 - 5 x y + 3 x y + x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(2*x^7*y^14-5*x^7*y^13+5*x^7*y^12-3*x^7*y^11+2*x^6*y^12+x^7*y^10-3*x^6*y^11+2* x^6*y^10-x^6*y^9-2*x^5*y^10+3*x^5*y^9-x^5*y^7-4*x^4*y^8+7*x^4*y^7-3*x^4*y^6-6*x ^3*y^6+6*x^3*y^5-x^3*y^4-2*x^2*y^4+2*x^2*y^3-x*y^2+x*y-1)/(2*x^8*y^15-2*x^8*y^ 14-2*x^8*y^13+2*x^8*y^12+x^7*y^13+x^7*y^12-x^7*y^11-2*x^6*y^12-x^7*y^10+x^6*y^ 11+2*x^6*y^10-x^6*y^9-3*x^5*y^10+5*x^5*y^9-3*x^5*y^8+x^5*y^7-5*x^4*y^8+3*x^4*y^ 7+x^3*y^4-x^2*y^4+x^2*y^3-2*x*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, 6 5 3 2 (a + a - 2 a + 3) -------------------------- 3 2 3 (8 a - 3 a + 3) (a + 1) 4 3 where a is the root of the polynomial, 2 x - x + 3 x - 1, and in decimals this is, 1.902027576 BTW the ratio for words with, 500, letters is, 1.898796400 ------------------------------------------------ "Theorem Number 6" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 1, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 14 7 13 7 12 ) | ) C(m, n) x y | = - (2 x y - 5 x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 11 6 12 7 10 6 11 6 10 6 9 5 10 - 3 x y + 2 x y + x y - 3 x y + 2 x y - x y - 2 x y 5 9 5 7 4 8 4 7 4 6 3 6 3 5 + 3 x y - x y - 4 x y + 7 x y - 3 x y - 6 x y + 6 x y 3 4 2 4 2 3 2 / 8 15 8 14 - x y - 2 x y + 2 x y - x y + x y - 1) / (2 x y - 2 x y / 8 13 8 12 7 13 7 12 7 11 6 12 7 10 - 2 x y + 2 x y + x y + x y - x y - 2 x y - x y 6 11 6 10 6 9 5 10 5 9 5 8 5 7 + x y + 2 x y - x y - 3 x y + 5 x y - 3 x y + x y 4 8 4 7 3 4 2 4 2 3 2 - 5 x y + 3 x y + x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(2*x^7*y^14-5*x^7*y^13+5*x^7*y^12-3*x^7*y^11+2*x^6*y^12+x^7*y^10-3*x^6*y^11+2* x^6*y^10-x^6*y^9-2*x^5*y^10+3*x^5*y^9-x^5*y^7-4*x^4*y^8+7*x^4*y^7-3*x^4*y^6-6*x ^3*y^6+6*x^3*y^5-x^3*y^4-2*x^2*y^4+2*x^2*y^3-x*y^2+x*y-1)/(2*x^8*y^15-2*x^8*y^ 14-2*x^8*y^13+2*x^8*y^12+x^7*y^13+x^7*y^12-x^7*y^11-2*x^6*y^12-x^7*y^10+x^6*y^ 11+2*x^6*y^10-x^6*y^9-3*x^5*y^10+5*x^5*y^9-3*x^5*y^8+x^5*y^7-5*x^4*y^8+3*x^4*y^ 7+x^3*y^4-x^2*y^4+x^2*y^3-2*x*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, 6 5 3 2 (a + a - 2 a + 3) -------------------------- 3 2 3 (8 a - 3 a + 3) (a + 1) 4 3 where a is the root of the polynomial, 2 x - x + 3 x - 1, and in decimals this is, 1.902027576 BTW the ratio for words with, 500, letters is, 1.898796400 ------------------------------------------------ "Theorem Number 7" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 1, 2, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 16 8 15 8 14 8 13 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 14 8 12 7 13 7 12 7 11 6 12 6 11 - 2 x y + x y + 6 x y - 6 x y + 2 x y + x y - 2 x y 6 10 4 8 4 7 4 6 3 6 3 5 / + x y - 2 x y + 4 x y - 2 x y + 2 x y - 2 x y + 1) / ( / 5 10 5 8 4 8 4 7 4 6 3 6 3 5 2 x y - x y - x y - 2 x y + 2 x y - 2 x y + 2 x y + 3 x y - 1 ) and in Maple notation -(x^8*y^16-4*x^8*y^15+6*x^8*y^14-4*x^8*y^13-2*x^7*y^14+x^8*y^12+6*x^7*y^13-6*x^ 7*y^12+2*x^7*y^11+x^6*y^12-2*x^6*y^11+x^6*y^10-2*x^4*y^8+4*x^4*y^7-2*x^4*y^6+2* x^3*y^6-2*x^3*y^5+1)/(x^5*y^10-x^5*y^8-x^4*y^8-2*x^4*y^7+2*x^4*y^6-2*x^3*y^6+2* x^3*y^5+3*x*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, 6 4 3 2 (a + a - 8 a + 3) - ---------------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.901954325 BTW the ratio for words with, 500, letters is, 1.898752405 ------------------------------------------------ "Theorem Number 8" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 2, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 18 9 17 9 16 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 15 8 16 8 15 8 14 8 13 7 14 8 12 - 2 x y + x y - 4 x y + 6 x y - 4 x y + x y + x y 7 13 7 12 7 11 6 12 7 10 6 11 - 4 x y + 6 x y - 4 x y - 5 x y + x y + 10 x y 6 10 6 9 5 10 5 9 5 8 4 8 4 7 - 7 x y + 2 x y - 2 x y + 4 x y - 2 x y - 2 x y + 4 x y 4 6 3 6 3 5 3 4 / 10 19 10 18 - 2 x y + 4 x y - 2 x y - x y + 1) / (4 x y - 12 x y / 10 17 10 16 9 17 9 16 9 15 8 16 + 12 x y - 4 x y - 2 x y + 6 x y - 6 x y + x y 9 14 8 15 8 14 8 13 7 14 8 12 7 13 + 2 x y - 4 x y + 6 x y - 4 x y - 3 x y + x y + 4 x y 7 12 7 11 6 12 7 10 6 11 6 10 6 9 - 4 x y + 4 x y + x y - x y - 2 x y + 3 x y - 2 x y 5 10 5 9 5 8 4 8 4 7 4 6 3 6 - 2 x y + 4 x y - 2 x y + 5 x y - 2 x y - x y - 4 x y 3 5 3 4 2 + 2 x y + x y + 3 x y - 1) and in Maple notation -(2*x^9*y^18-6*x^9*y^17+6*x^9*y^16-2*x^9*y^15+x^8*y^16-4*x^8*y^15+6*x^8*y^14-4* x^8*y^13+x^7*y^14+x^8*y^12-4*x^7*y^13+6*x^7*y^12-4*x^7*y^11-5*x^6*y^12+x^7*y^10 +10*x^6*y^11-7*x^6*y^10+2*x^6*y^9-2*x^5*y^10+4*x^5*y^9-2*x^5*y^8-2*x^4*y^8+4*x^ 4*y^7-2*x^4*y^6+4*x^3*y^6-2*x^3*y^5-x^3*y^4+1)/(4*x^10*y^19-12*x^10*y^18+12*x^ 10*y^17-4*x^10*y^16-2*x^9*y^17+6*x^9*y^16-6*x^9*y^15+x^8*y^16+2*x^9*y^14-4*x^8* y^15+6*x^8*y^14-4*x^8*y^13-3*x^7*y^14+x^8*y^12+4*x^7*y^13-4*x^7*y^12+4*x^7*y^11 +x^6*y^12-x^7*y^10-2*x^6*y^11+3*x^6*y^10-2*x^6*y^9-2*x^5*y^10+4*x^5*y^9-2*x^5*y ^8+5*x^4*y^8-2*x^4*y^7-x^4*y^6-4*x^3*y^6+2*x^3*y^5+x^3*y^4+3*x*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, 6 3 2 (a - 2 a + 3) -------------------------- 3 2 3 (8 a - 3 a + 3) (a + 1) 4 3 where a is the root of the polynomial, 2 x - x + 3 x - 1, and in decimals this is, 1.899184047 BTW the ratio for words with, 500, letters is, 1.895986467 ------------------------------------------------ "Theorem Number 9" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 3, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = - ( / | / | ----- | ----- | m = 0 \ n = 0 / 6 12 6 11 5 10 5 9 5 8 3 6 3 5 2 x y - 2 x y + 2 x y - 4 x y + 2 x y - 5 x y + 4 x y - 1) / 7 13 7 12 6 12 6 11 6 10 5 10 / (4 x y - 4 x y + 2 x y - 6 x y + 4 x y - 2 x y / 5 9 5 8 4 8 4 7 3 6 3 5 2 + 4 x y - 2 x y - 6 x y + 4 x y + 5 x y - 4 x y - 3 x y + 1) and in Maple notation -(2*x^6*y^12-2*x^6*y^11+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8-5*x^3*y^6+4*x^3*y^5-1)/( 4*x^7*y^13-4*x^7*y^12+2*x^6*y^12-6*x^6*y^11+4*x^6*y^10-2*x^5*y^10+4*x^5*y^9-2*x ^5*y^8-6*x^4*y^8+4*x^4*y^7+5*x^3*y^6-4*x^3*y^5-3*x*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, 6 3 2 (a - 2 a + 3) -------------------------- 3 2 3 (8 a - 3 a + 3) (a + 1) 4 3 where a is the root of the polynomial, 2 x - x + 3 x - 1, and in decimals this is, 1.899184047 BTW the ratio for words with, 500, letters is, 1.895986467 ------------------------------------------------ "Theorem Number 10" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 1, 3]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 5 10 ) | ) C(m, n) x y | = (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 9 5 8 4 8 4 7 4 6 3 6 3 5 - 4 x y + 2 x y + x y - 2 x y + x y + 2 x y - 2 x y 2 4 2 3 / 6 12 6 11 6 10 5 10 + 2 x y - 2 x y + 1) / (2 x y - 4 x y + 2 x y - x y / 5 9 5 8 4 8 4 7 4 6 3 6 3 5 + 2 x y - x y + 4 x y - 4 x y + x y - 4 x y + 4 x y 2 4 2 3 2 + 2 x y - 2 x y - 3 x y + 1) and in Maple notation (x^6*y^12-2*x^6*y^11+x^6*y^10+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8+x^4*y^8-2*x^4*y^7+ x^4*y^6+2*x^3*y^6-2*x^3*y^5+2*x^2*y^4-2*x^2*y^3+1)/(2*x^6*y^12-4*x^6*y^11+2*x^6 *y^10-x^5*y^10+2*x^5*y^9-x^5*y^8+4*x^4*y^8-4*x^4*y^7+x^4*y^6-4*x^3*y^6+4*x^3*y^ 5+2*x^2*y^4-2*x^2*y^3-3*x*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, 6 5 3 2 (a + a - 8 a + 3) - ---------------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.895903353 BTW the ratio for words with, 500, letters is, 1.892757625 ------------------------------------------------ "Theorem Number 11" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 3, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 5 10 ) | ) C(m, n) x y | = (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 9 5 8 4 8 4 7 4 6 3 6 3 5 - 4 x y + 2 x y + x y - 2 x y + x y + 2 x y - 2 x y 2 4 2 3 / 6 12 6 11 6 10 5 10 + 2 x y - 2 x y + 1) / (2 x y - 4 x y + 2 x y - x y / 5 9 5 8 4 8 4 7 4 6 3 6 3 5 + 2 x y - x y + 4 x y - 4 x y + x y - 4 x y + 4 x y 2 4 2 3 2 + 2 x y - 2 x y - 3 x y + 1) and in Maple notation (x^6*y^12-2*x^6*y^11+x^6*y^10+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8+x^4*y^8-2*x^4*y^7+ x^4*y^6+2*x^3*y^6-2*x^3*y^5+2*x^2*y^4-2*x^2*y^3+1)/(2*x^6*y^12-4*x^6*y^11+2*x^6 *y^10-x^5*y^10+2*x^5*y^9-x^5*y^8+4*x^4*y^8-4*x^4*y^7+x^4*y^6-4*x^3*y^6+4*x^3*y^ 5+2*x^2*y^4-2*x^2*y^3-3*x*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, 6 5 3 2 (a + a - 8 a + 3) - ---------------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.895903353 BTW the ratio for words with, 500, letters is, 1.892757625 ------------------------------------------------ "Theorem Number 12" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 1, 2, 3]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = - ( / | / | ----- | ----- | m = 0 \ n = 0 / 6 12 6 11 6 10 3 6 3 5 / 6 12 6 11 x y - 2 x y + x y + 2 x y - 2 x y + 1) / (x y - 2 x y / 6 10 5 10 5 9 5 8 4 8 4 7 3 6 + x y + 2 x y - 4 x y + 2 x y - 3 x y + 2 x y - 2 x y 3 5 2 + 2 x y + 3 x y - 1) and in Maple notation -(x^6*y^12-2*x^6*y^11+x^6*y^10+2*x^3*y^6-2*x^3*y^5+1)/(x^6*y^12-2*x^6*y^11+x^6* y^10+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8-3*x^4*y^8+2*x^4*y^7-2*x^3*y^6+2*x^3*y^5+3*x *y^2-1) As the length of the word goes to infinity, the average number of good neigh\ 6 3 2 (a - 8 a + 3) bors of a random word of length n tends to n times, - ----------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.892818483 BTW the ratio for words with, 500, letters is, 1.889710699 ------------------------------------------------ "Theorem Number 13" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 2, 3]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = - ( / | / | ----- | ----- | m = 0 \ n = 0 / 6 12 6 11 6 10 3 6 3 5 / 6 12 6 11 x y - 2 x y + x y + 2 x y - 2 x y + 1) / (x y - 2 x y / 6 10 5 10 5 9 5 8 4 8 4 7 3 6 + x y + 2 x y - 4 x y + 2 x y - 3 x y + 2 x y - 2 x y 3 5 2 + 2 x y + 3 x y - 1) and in Maple notation -(x^6*y^12-2*x^6*y^11+x^6*y^10+2*x^3*y^6-2*x^3*y^5+1)/(x^6*y^12-2*x^6*y^11+x^6* y^10+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8-3*x^4*y^8+2*x^4*y^7-2*x^3*y^6+2*x^3*y^5+3*x *y^2-1) As the length of the word goes to infinity, the average number of good neigh\ 6 3 2 (a - 8 a + 3) bors of a random word of length n tends to n times, - ----------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.892818483 BTW the ratio for words with, 500, letters is, 1.889710699 ------------------------------------------------ "Theorem Number 14" Let C(m,n) be the number of words of length n in the alphabet, {1, 2, 3}, avoiding consecutive substrings in the set, {[1, 2, 3, 3]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = - ( / | / | ----- | ----- | m = 0 \ n = 0 / 6 12 6 11 6 10 3 6 3 5 / 6 12 6 11 x y - 2 x y + x y + 2 x y - 2 x y + 1) / (x y - 2 x y / 6 10 5 10 5 9 5 8 4 8 4 7 3 6 + x y + 2 x y - 4 x y + 2 x y - 3 x y + 2 x y - 2 x y 3 5 2 + 2 x y + 3 x y - 1) and in Maple notation -(x^6*y^12-2*x^6*y^11+x^6*y^10+2*x^3*y^6-2*x^3*y^5+1)/(x^6*y^12-2*x^6*y^11+x^6* y^10+2*x^5*y^10-4*x^5*y^9+2*x^5*y^8-3*x^4*y^8+2*x^4*y^7-2*x^3*y^6+2*x^3*y^5+3*x *y^2-1) As the length of the word goes to infinity, the average number of good neigh\ 6 3 2 (a - 8 a + 3) bors of a random word of length n tends to n times, - ----------------- 3 4 a - 3 4 where a is the root of the polynomial, x - 3 x + 1, and in decimals this is, 1.892818483 BTW the ratio for words with, 500, letters is, 1.889710699 ---------------------------------- This ends this paper that took, 28081.154, seconds to produce ------------------------------ this took, 28081.155, seconds.