Counting Words in the Alphabet, {1, 2, 3}, That Avoid A Certain set of , 2, 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], [1, 2]}, and having \ m neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | 2 4 2 3 2 \ | \ n m| x y - x y - 2 x y + x y - 1 ) | ) C(m, n) x y | = - -------------------------------------- / | / | 3 4 3 3 2 2 2 ----- | ----- | x y - x y - x y - x y - x y + 1 m = 0 \ n = 0 / and in Maple notation -(x^2*y^4-x^2*y^3-2*x*y^2+x*y-1)/(x^3*y^4-x^3*y^3-x^2*y^2-x*y^2-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ a - 3 bors of a random word of length n tends to n times, - -------- 2 (a + 1) 2 where a is the root of the polynomial, x + 2 x - 1, and in decimals this is, 1.292893219 BTW the ratio for words with, 500, letters is, 1.291481828 ------------------------------------------------ "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, 1], [2, 1]}, and having \ m neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | 2 4 2 3 2 \ | \ n m| x y - x y - 2 x y + x y - 1 ) | ) C(m, n) x y | = - -------------------------------------- / | / | 3 4 3 3 2 2 2 ----- | ----- | x y - x y - x y - x y - x y + 1 m = 0 \ n = 0 / and in Maple notation -(x^2*y^4-x^2*y^3-2*x*y^2+x*y-1)/(x^3*y^4-x^3*y^3-x^2*y^2-x*y^2-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ a - 3 bors of a random word of length n tends to n times, - -------- 2 (a + 1) 2 where a is the root of the polynomial, x + 2 x - 1, and in decimals this is, 1.292893219 BTW the ratio for words with, 500, letters is, 1.291481828 ------------------------------------------------ "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, 2]}, and having \ m neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 3 5 3 4 3 3 2 4 ) | ) C(m, n) x y | = - (2 x y - 5 x y + 4 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 2 2 3 2 2 2 / - x y + x y + x y + 2 x y - x y + 1) / ( / 4 6 4 5 4 4 3 5 3 2 2 2 2 x y - 2 x y + x y - x y + x y + x y + x y + x y - 1) and in Maple notation -(2*x^3*y^5-5*x^3*y^4+4*x^3*y^3-2*x^2*y^4-x^3*y^2+x^2*y^3+x^2*y^2+2*x*y^2-x*y+1 )/(x^4*y^6-2*x^4*y^5+x^4*y^4-x^3*y^5+x^3*y^2+x^2*y^2+x*y^2+x*y-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 (a + 1) 2 where a is the root of the polynomial, x + 2 x - 1, and in decimals this is, 1.171572875 BTW the ratio for words with, 500, letters is, 1.170403641 ------------------------------------------------ "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, 1]}, and having \ m neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 3 5 3 4 3 3 2 4 ) | ) C(m, n) x y | = - (2 x y - 5 x y + 4 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 2 2 3 2 2 2 / - x y + x y + x y + 2 x y - x y + 1) / ( / 4 6 4 5 4 4 3 5 3 2 2 2 2 x y - 2 x y + x y - x y + x y + x y + x y + x y - 1) and in Maple notation -(2*x^3*y^5-5*x^3*y^4+4*x^3*y^3-2*x^2*y^4-x^3*y^2+x^2*y^3+x^2*y^2+2*x*y^2-x*y+1 )/(x^4*y^6-2*x^4*y^5+x^4*y^4-x^3*y^5+x^3*y^2+x^2*y^2+x*y^2+x*y-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 (a + 1) 2 where a is the root of the polynomial, x + 2 x - 1, and in decimals this is, 1.171572875 BTW the ratio for words with, 500, letters is, 1.170403641 ------------------------------------------------ "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], [2, 3]}, and having \ m neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 8 5 7 5 6 5 5 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 6 5 4 4 5 4 4 4 3 3 4 3 3 + 2 x y + x y - 6 x y + 6 x y - 2 x y + 2 x y - 4 x y 2 4 3 2 2 3 2 2 2 2 - x y + 2 x y + 2 x y + x y - 2 x y - 3 x y + 2 x y + x - 1) / 4 6 4 5 4 4 3 4 3 3 3 2 2 / (x y - 2 x y + x y + 2 x y - 2 x y + x y - 2 x y + 2 x y / + x - 1) and in Maple notation (x^5*y^8-4*x^5*y^7+6*x^5*y^6-4*x^5*y^5+2*x^4*y^6+x^5*y^4-6*x^4*y^5+6*x^4*y^4-2* x^4*y^3+2*x^3*y^4-4*x^3*y^3-x^2*y^4+2*x^3*y^2+2*x^2*y^3+x^2*y^2-2*x^2*y-3*x*y^2 +2*x*y+x-1)/(x^4*y^6-2*x^4*y^5+x^4*y^4+2*x^3*y^4-2*x^3*y^3+x^3*y^2-2*x^2*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, 4 3 2 2 (a - 4 a + 9 a - 8 a + 3) ------------------------------ 2 3 a - 4 a + 3 3 2 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.024724693 BTW the ratio for words with, 500, letters is, 1.024019915 ------------------------------------------------ "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, 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 / 2 4 2 3 2 2 2 2 2 x y - 2 x y - 2 x y + 2 x y + 3 x y - 2 x y - x + 1 ------------------------------------------------------------- 2 2 x y - 2 x y - x + 1 and in Maple notation (2*x^2*y^4-2*x^2*y^3-2*x^2*y^2+2*x^2*y+3*x*y^2-2*x*y-x+1)/(2*x^2*y-2*x*y-x+1) Error, (in AvNei) numeric exception: division by zero |Marko.txt:844| ------------------------------ Counting Words in the Alphabet, {1, 2, 3}, That Avoid A Certain set of , 2, 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], [1, 1, 2]}, and h\ aving m neighbors (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 | = - (x y - 2 x y + x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 7 4 6 3 6 3 5 3 4 2 4 2 3 2 + 2 x y - x y + 3 x y - 4 x y + x y - 3 x y + 2 x y - x y / 6 10 6 9 6 8 4 6 4 5 3 6 3 5 - 1) / (x y - 2 x y + x y + 2 x y - 2 x y - x y + 2 x y / 3 4 2 3 2 - 2 x y - 2 x y - 2 x y + 1) and in Maple notation -(x^5*y^10-2*x^5*y^9+x^5*y^8-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^4-3*x^2*y^4+2*x^2*y^3-x*y^2-1)/(x^6*y^10-2*x^6*y^9+x^6*y^8+2*x^4*y^6-2*x^4*y^ 5-x^3*y^6+2*x^3*y^5-2*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\ bors of a random word of length n tends to n times, 3 2 2 (a + 2 a - 6 a - 3) - ---------------------------- 4 3 2 3 a + 7 a + 9 a + 6 a + 2 3 2 where a is the root of the polynomial, x + 2 x + 2 x - 1, and in decimals this is, 1.724310870 BTW the ratio for words with, 500, letters is, 1.721772122 ------------------------------------------------ "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, 1, 1], [2, 1, 1]}, and h\ aving m neighbors (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 | = - (x y - 2 x y + x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 7 4 6 3 6 3 5 3 4 2 4 2 3 2 + 2 x y - x y + 3 x y - 4 x y + x y - 3 x y + 2 x y - x y / 6 10 6 9 6 8 4 6 4 5 3 6 3 5 - 1) / (x y - 2 x y + x y + 2 x y - 2 x y - x y + 2 x y / 3 4 2 3 2 - 2 x y - 2 x y - 2 x y + 1) and in Maple notation -(x^5*y^10-2*x^5*y^9+x^5*y^8-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^4-3*x^2*y^4+2*x^2*y^3-x*y^2-1)/(x^6*y^10-2*x^6*y^9+x^6*y^8+2*x^4*y^6-2*x^4*y^ 5-x^3*y^6+2*x^3*y^5-2*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\ bors of a random word of length n tends to n times, 3 2 2 (a + 2 a - 6 a - 3) - ---------------------------- 4 3 2 3 a + 7 a + 9 a + 6 a + 2 3 2 where a is the root of the polynomial, x + 2 x + 2 x - 1, and in decimals this is, 1.724310870 BTW the ratio for words with, 500, letters is, 1.721772122 ------------------------------------------------ "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], [1, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 12 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 11 7 12 7 11 7 10 6 11 7 9 6 10 - 2 x y - 2 x y + 2 x y + 2 x y - 2 x y - 2 x y + x y 6 9 5 10 6 8 5 9 5 8 5 7 4 8 + 4 x y - x y - 3 x y + 2 x y - 3 x y + 2 x y - 2 x y 4 7 4 6 3 6 3 5 3 4 2 4 2 2 2 - 2 x y + 4 x y - 3 x y + 6 x y - 2 x y + 3 x y - x y + x y / 9 15 9 14 9 13 8 14 9 12 + 1) / (4 x y - 12 x y + 12 x y - 2 x y - 4 x y / 8 13 8 12 8 11 8 10 7 11 7 10 + 2 x y - 2 x y + 6 x y - 4 x y - 6 x y + 12 x y 7 9 6 10 6 9 6 8 5 9 5 8 5 7 - 6 x y - x y - 2 x y + 3 x y - 4 x y - 2 x y + 6 x y 4 7 3 6 3 5 2 2 2 + 2 x y + x y + 2 x y + x y + 2 x y - 1) and in Maple notation -(2*x^8*y^14-6*x^8*y^13+6*x^8*y^12-2*x^8*y^11-2*x^7*y^12+2*x^7*y^11+2*x^7*y^10-\ 2*x^6*y^11-2*x^7*y^9+x^6*y^10+4*x^6*y^9-x^5*y^10-3*x^6*y^8+2*x^5*y^9-3*x^5*y^8+ 2*x^5*y^7-2*x^4*y^8-2*x^4*y^7+4*x^4*y^6-3*x^3*y^6+6*x^3*y^5-2*x^3*y^4+3*x^2*y^4 -x^2*y^2+x*y^2+1)/(4*x^9*y^15-12*x^9*y^14+12*x^9*y^13-2*x^8*y^14-4*x^9*y^12+2*x ^8*y^13-2*x^8*y^12+6*x^8*y^11-4*x^8*y^10-6*x^7*y^11+12*x^7*y^10-6*x^7*y^9-x^6*y ^10-2*x^6*y^9+3*x^6*y^8-4*x^5*y^9-2*x^5*y^8+6*x^5*y^7+2*x^4*y^7+x^3*y^6+2*x^3*y ^5+x^2*y^2+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 2 (a + 4 a + 5 a + 4 a + 4 a + 6 a + 3) -------------------------------------------- 3 2 3 2 (8 a + 9 a + 2 a + 2) (a + 2 a + a + 1) 4 3 2 where a is the root of the polynomial, 2 x + 3 x + x + 2 x - 1, and in decimals this is, 1.706934314 BTW the ratio for words with, 500, letters is, 1.704440601 ------------------------------------------------ "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, 1, 1], [2, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 6 9 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 10 5 9 5 8 5 7 4 8 5 6 4 7 + 4 x y - 8 x y + 5 x y - 2 x y + 2 x y + x y - 8 x y 4 6 4 5 3 6 3 5 3 4 3 3 2 4 + 10 x y - 4 x y + x y - 2 x y + 2 x y - x y - 5 x y 2 3 2 2 2 / 7 13 7 12 7 11 + 3 x y + x y - 2 x y + x y - 1) / (2 x y - 6 x y + 6 x y / 7 10 6 11 6 10 6 9 6 8 5 9 5 8 - 2 x y + 4 x y - 7 x y + 2 x y + x y + 2 x y - 5 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 5 + 4 x y - x y - x y + 4 x y - 4 x y + x y - 2 x y - x y 3 4 3 3 2 4 2 2 2 + x y + x y - x y - x y - x y - x y + 1) and in Maple notation -(2*x^6*y^12-6*x^6*y^11+6*x^6*y^10-2*x^6*y^9+4*x^5*y^10-8*x^5*y^9+5*x^5*y^8-2*x ^5*y^7+2*x^4*y^8+x^5*y^6-8*x^4*y^7+10*x^4*y^6-4*x^4*y^5+x^3*y^6-2*x^3*y^5+2*x^3 *y^4-x^3*y^3-5*x^2*y^4+3*x^2*y^3+x^2*y^2-2*x*y^2+x*y-1)/(2*x^7*y^13-6*x^7*y^12+ 6*x^7*y^11-2*x^7*y^10+4*x^6*y^11-7*x^6*y^10+2*x^6*y^9+x^6*y^8+2*x^5*y^9-5*x^5*y ^8+4*x^5*y^7-x^4*y^8-x^5*y^6+4*x^4*y^7-4*x^4*y^6+x^4*y^5-2*x^3*y^6-x^3*y^5+x^3* y^4+x^3*y^3-x^2*y^4-x^2*y^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 3 2 2 (a - 2 a - 3 a + 6 a + 3) ------------------------------ 4 3 2 3 a + 7 a + 9 a + 6 a + 2 3 2 where a is the root of the polynomial, x + 2 x + 2 x - 1, and in decimals this is, 1.669548605 BTW the ratio for words with, 500, letters is, 1.667284618 ------------------------------------------------ "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, 1], [2, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 6 9 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 10 5 9 5 8 5 7 4 8 5 6 4 7 + 4 x y - 8 x y + 5 x y - 2 x y + 2 x y + x y - 8 x y 4 6 4 5 3 6 3 5 3 4 3 3 2 4 + 10 x y - 4 x y + x y - 2 x y + 2 x y - x y - 5 x y 2 3 2 2 2 / 7 13 7 12 7 11 + 3 x y + x y - 2 x y + x y - 1) / (2 x y - 6 x y + 6 x y / 7 10 6 11 6 10 6 9 6 8 5 9 5 8 - 2 x y + 4 x y - 7 x y + 2 x y + x y + 2 x y - 5 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 5 + 4 x y - x y - x y + 4 x y - 4 x y + x y - 2 x y - x y 3 4 3 3 2 4 2 2 2 + x y + x y - x y - x y - x y - x y + 1) and in Maple notation -(2*x^6*y^12-6*x^6*y^11+6*x^6*y^10-2*x^6*y^9+4*x^5*y^10-8*x^5*y^9+5*x^5*y^8-2*x ^5*y^7+2*x^4*y^8+x^5*y^6-8*x^4*y^7+10*x^4*y^6-4*x^4*y^5+x^3*y^6-2*x^3*y^5+2*x^3 *y^4-x^3*y^3-5*x^2*y^4+3*x^2*y^3+x^2*y^2-2*x*y^2+x*y-1)/(2*x^7*y^13-6*x^7*y^12+ 6*x^7*y^11-2*x^7*y^10+4*x^6*y^11-7*x^6*y^10+2*x^6*y^9+x^6*y^8+2*x^5*y^9-5*x^5*y ^8+4*x^5*y^7-x^4*y^8-x^5*y^6+4*x^4*y^7-4*x^4*y^6+x^4*y^5-2*x^3*y^6-x^3*y^5+x^3* y^4+x^3*y^3-x^2*y^4-x^2*y^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 3 2 2 (a - 2 a - 3 a + 6 a + 3) ------------------------------ 4 3 2 3 a + 7 a + 9 a + 6 a + 2 3 2 where a is the root of the polynomial, x + 2 x + 2 x - 1, and in decimals this is, 1.669548605 BTW the ratio for words with, 500, letters is, 1.667284618 ------------------------------------------------ "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, 1, 2], [2, 1, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 16 27 16 26 15 27 ) | ) C(m, n) x y | = - (2 x y - 17 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 16 25 15 26 16 24 15 25 14 26 + 64 x y - 17 x y - 140 x y + 66 x y - 2 x y 16 23 15 24 14 25 16 22 15 23 + 196 x y - 155 x y + 17 x y - 182 x y + 245 x y 14 24 16 21 15 22 14 23 13 24 - 66 x y + 112 x y - 273 x y + 155 x y - 4 x y 16 20 15 21 14 22 13 23 16 19 - 44 x y + 217 x y - 246 x y + 28 x y + 10 x y 15 20 14 21 13 22 12 23 16 18 - 121 x y + 280 x y - 88 x y + 2 x y - x y 15 19 14 20 13 21 12 22 15 18 + 45 x y - 238 x y + 166 x y - 16 x y - 10 x y 14 19 13 20 12 21 15 17 14 18 + 156 x y - 211 x y + 56 x y + x y - 80 x y 13 19 12 20 11 21 14 17 13 18 + 189 x y - 111 x y + 3 x y + 31 x y - 118 x y 12 19 11 20 14 16 13 17 12 18 + 140 x y - 18 x y - 8 x y + 48 x y - 132 x y 11 19 14 15 13 16 12 17 11 18 10 19 + 49 x y + x y - 11 x y + 120 x y - 87 x y + 2 x y 13 15 12 16 11 17 10 18 12 15 + x y - 107 x y + 109 x y - 16 x y + 70 x y 11 16 10 17 9 18 12 14 11 15 10 16 - 84 x y + 41 x y - x y - 26 x y + 29 x y - 48 x y 9 17 12 13 11 14 10 15 9 16 11 13 + 7 x y + 4 x y - 3 x y + 31 x y - 14 x y + 8 x y 10 14 9 15 8 16 11 12 10 13 9 14 - 8 x y + 11 x y - x y - 8 x y - 21 x y + 9 x y 8 15 11 11 10 12 9 13 8 14 10 11 + 10 x y + 2 x y + 36 x y - 34 x y - 17 x y - 21 x y 9 12 8 13 7 14 10 10 9 11 8 12 + 28 x y + 3 x y - x y + 4 x y - 2 x y + 12 x y 7 13 9 10 8 11 7 12 9 9 8 10 + x y - 3 x y - 18 x y + 4 x y - 2 x y + 22 x y 7 11 6 12 9 8 8 9 7 10 6 11 8 8 - 15 x y - x y + x y - 13 x y + 19 x y - 9 x y + 2 x y 7 9 6 10 7 8 6 9 5 10 7 7 6 8 - 9 x y + 25 x y + 3 x y - 22 x y - 3 x y - 3 x y + 7 x y 5 9 7 6 6 7 5 8 6 6 5 7 4 8 5 6 - 2 x y + x y + x y + 8 x y - x y - 4 x y + 6 x y + 2 x y 4 7 5 5 4 6 3 6 4 4 3 5 3 4 - 7 x y - x y + 3 x y + 5 x y - x y - 2 x y - 2 x y 2 4 2 3 2 / 17 28 17 27 16 28 - 2 x y + 2 x y - x y + x y - 1) / (x y - 8 x y + x y / 17 26 16 27 17 25 16 26 15 27 17 24 + 28 x y - 8 x y - 56 x y + 29 x y - x y + 70 x y 16 25 15 26 17 23 16 24 15 25 - 63 x y + 8 x y - 56 x y + 91 x y - 29 x y 17 22 16 23 15 24 14 25 17 21 + 28 x y - 91 x y + 63 x y - 2 x y - 8 x y 16 22 15 23 14 24 17 20 16 21 15 22 + 63 x y - 92 x y + 13 x y + x y - 29 x y + 98 x y 14 23 13 24 16 20 15 21 14 22 13 23 - 38 x y + x y + 8 x y - 84 x y + 68 x y - 8 x y 16 19 15 20 14 21 13 22 15 19 14 20 - x y + 64 x y - 86 x y + 28 x y - 43 x y + 83 x y 13 21 12 22 15 18 14 19 13 20 - 56 x y + 2 x y + 22 x y - 63 x y + 74 x y 12 21 15 17 14 18 13 19 12 20 15 16 - 12 x y - 7 x y + 38 x y - 78 x y + 33 x y + x y 14 17 13 18 12 19 11 20 14 16 - 18 x y + 77 x y - 59 x y + 2 x y + 6 x y 13 17 12 18 11 19 14 15 13 16 12 17 - 64 x y + 73 x y - 14 x y - x y + 35 x y - 56 x y 11 18 10 19 13 15 12 16 11 17 10 18 + 34 x y - x y - 10 x y + 23 x y - 40 x y + 7 x y 13 14 12 15 11 16 10 17 12 14 11 15 + x y - 7 x y + 29 x y - 16 x y + 5 x y - 16 x y 10 16 9 17 12 13 11 14 10 15 9 16 + 19 x y - x y - 2 x y + 4 x y - 11 x y + 8 x y 10 14 9 15 11 12 10 13 9 14 8 15 + 2 x y - 15 x y + 3 x y - 7 x y + 13 x y - x y 11 11 10 12 9 13 8 14 10 11 9 12 - 2 x y + 10 x y - 11 x y + 4 x y - x y + 6 x y 8 13 10 10 9 11 8 12 7 13 8 10 - 6 x y - 2 x y + x y + 2 x y - 2 x y + 2 x y 7 11 9 8 8 9 7 10 6 11 8 8 7 9 + 5 x y - x y + x y - 6 x y - 5 x y - 2 x y + 4 x y 6 10 7 8 6 9 5 10 7 7 6 8 5 9 7 6 + 4 x y - x y + 2 x y + x y + x y - 3 x y + x y - x y 6 7 5 8 6 6 5 7 4 8 5 6 4 7 5 5 + 4 x y - 2 x y - 2 x y + 3 x y + x y - x y + 3 x y - x y 4 6 3 6 2 4 2 3 2 - 3 x y + x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(2*x^16*y^27-17*x^16*y^26+2*x^15*y^27+64*x^16*y^25-17*x^15*y^26-140*x^16*y^24+ 66*x^15*y^25-2*x^14*y^26+196*x^16*y^23-155*x^15*y^24+17*x^14*y^25-182*x^16*y^22 +245*x^15*y^23-66*x^14*y^24+112*x^16*y^21-273*x^15*y^22+155*x^14*y^23-4*x^13*y^ 24-44*x^16*y^20+217*x^15*y^21-246*x^14*y^22+28*x^13*y^23+10*x^16*y^19-121*x^15* y^20+280*x^14*y^21-88*x^13*y^22+2*x^12*y^23-x^16*y^18+45*x^15*y^19-238*x^14*y^ 20+166*x^13*y^21-16*x^12*y^22-10*x^15*y^18+156*x^14*y^19-211*x^13*y^20+56*x^12* y^21+x^15*y^17-80*x^14*y^18+189*x^13*y^19-111*x^12*y^20+3*x^11*y^21+31*x^14*y^ 17-118*x^13*y^18+140*x^12*y^19-18*x^11*y^20-8*x^14*y^16+48*x^13*y^17-132*x^12*y ^18+49*x^11*y^19+x^14*y^15-11*x^13*y^16+120*x^12*y^17-87*x^11*y^18+2*x^10*y^19+ x^13*y^15-107*x^12*y^16+109*x^11*y^17-16*x^10*y^18+70*x^12*y^15-84*x^11*y^16+41 *x^10*y^17-x^9*y^18-26*x^12*y^14+29*x^11*y^15-48*x^10*y^16+7*x^9*y^17+4*x^12*y^ 13-3*x^11*y^14+31*x^10*y^15-14*x^9*y^16+8*x^11*y^13-8*x^10*y^14+11*x^9*y^15-x^8 *y^16-8*x^11*y^12-21*x^10*y^13+9*x^9*y^14+10*x^8*y^15+2*x^11*y^11+36*x^10*y^12-\ 34*x^9*y^13-17*x^8*y^14-21*x^10*y^11+28*x^9*y^12+3*x^8*y^13-x^7*y^14+4*x^10*y^ 10-2*x^9*y^11+12*x^8*y^12+x^7*y^13-3*x^9*y^10-18*x^8*y^11+4*x^7*y^12-2*x^9*y^9+ 22*x^8*y^10-15*x^7*y^11-x^6*y^12+x^9*y^8-13*x^8*y^9+19*x^7*y^10-9*x^6*y^11+2*x^ 8*y^8-9*x^7*y^9+25*x^6*y^10+3*x^7*y^8-22*x^6*y^9-3*x^5*y^10-3*x^7*y^7+7*x^6*y^8 -2*x^5*y^9+x^7*y^6+x^6*y^7+8*x^5*y^8-x^6*y^6-4*x^5*y^7+6*x^4*y^8+2*x^5*y^6-7*x^ 4*y^7-x^5*y^5+3*x^4*y^6+5*x^3*y^6-x^4*y^4-2*x^3*y^5-2*x^3*y^4-2*x^2*y^4+2*x^2*y ^3-x*y^2+x*y-1)/(x^17*y^28-8*x^17*y^27+x^16*y^28+28*x^17*y^26-8*x^16*y^27-56*x^ 17*y^25+29*x^16*y^26-x^15*y^27+70*x^17*y^24-63*x^16*y^25+8*x^15*y^26-56*x^17*y^ 23+91*x^16*y^24-29*x^15*y^25+28*x^17*y^22-91*x^16*y^23+63*x^15*y^24-2*x^14*y^25 -8*x^17*y^21+63*x^16*y^22-92*x^15*y^23+13*x^14*y^24+x^17*y^20-29*x^16*y^21+98*x ^15*y^22-38*x^14*y^23+x^13*y^24+8*x^16*y^20-84*x^15*y^21+68*x^14*y^22-8*x^13*y^ 23-x^16*y^19+64*x^15*y^20-86*x^14*y^21+28*x^13*y^22-43*x^15*y^19+83*x^14*y^20-\ 56*x^13*y^21+2*x^12*y^22+22*x^15*y^18-63*x^14*y^19+74*x^13*y^20-12*x^12*y^21-7* x^15*y^17+38*x^14*y^18-78*x^13*y^19+33*x^12*y^20+x^15*y^16-18*x^14*y^17+77*x^13 *y^18-59*x^12*y^19+2*x^11*y^20+6*x^14*y^16-64*x^13*y^17+73*x^12*y^18-14*x^11*y^ 19-x^14*y^15+35*x^13*y^16-56*x^12*y^17+34*x^11*y^18-x^10*y^19-10*x^13*y^15+23*x ^12*y^16-40*x^11*y^17+7*x^10*y^18+x^13*y^14-7*x^12*y^15+29*x^11*y^16-16*x^10*y^ 17+5*x^12*y^14-16*x^11*y^15+19*x^10*y^16-x^9*y^17-2*x^12*y^13+4*x^11*y^14-11*x^ 10*y^15+8*x^9*y^16+2*x^10*y^14-15*x^9*y^15+3*x^11*y^12-7*x^10*y^13+13*x^9*y^14- x^8*y^15-2*x^11*y^11+10*x^10*y^12-11*x^9*y^13+4*x^8*y^14-x^10*y^11+6*x^9*y^12-6 *x^8*y^13-2*x^10*y^10+x^9*y^11+2*x^8*y^12-2*x^7*y^13+2*x^8*y^10+5*x^7*y^11-x^9* y^8+x^8*y^9-6*x^7*y^10-5*x^6*y^11-2*x^8*y^8+4*x^7*y^9+4*x^6*y^10-x^7*y^8+2*x^6* y^9+x^5*y^10+x^7*y^7-3*x^6*y^8+x^5*y^9-x^7*y^6+4*x^6*y^7-2*x^5*y^8-2*x^6*y^6+3* x^5*y^7+x^4*y^8-x^5*y^6+3*x^4*y^7-x^5*y^5-3*x^4*y^6+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, 8 7 6 5 4 3 2 2 (a + 2 a - 5 a - 6 a + 14 a + 4 a - 12 a + 3) ------------------------------------------------------ 4 3 2 4 3 (5 a + 4 a + 3 a - 3) (a + a - 1) 5 4 3 where a is the root of the polynomial, x + x + x - 3 x + 1, and in decimals this is, 1.663390809 BTW the ratio for words with, 500, letters is, 1.661070640 ------------------------------------------------ "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], [3, 1, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 14 26 14 25 14 24 ) | ) C(m, n) x y | = (x y - 8 x y + 28 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 23 13 24 14 22 13 23 14 21 - 56 x y - 2 x y + 70 x y + 14 x y - 56 x y 13 22 14 20 13 21 14 19 13 20 14 18 - 42 x y + 28 x y + 70 x y - 8 x y - 70 x y + x y 13 19 13 18 11 20 13 17 11 19 + 42 x y - 14 x y - 2 x y + 2 x y + 14 x y 11 18 11 17 10 18 11 16 10 17 9 18 - 40 x y + 60 x y + 5 x y - 50 x y - 26 x y - x y 11 15 10 16 9 17 11 14 10 15 9 16 + 22 x y + 54 x y + 8 x y - 4 x y - 56 x y - 25 x y 10 14 9 15 8 16 10 13 9 14 8 15 + 29 x y + 40 x y + 2 x y - 6 x y - 35 x y - 12 x y 9 13 8 14 9 12 8 13 7 14 8 12 + 16 x y + 30 x y - 3 x y - 42 x y - x y + 36 x y 7 13 8 11 7 12 8 10 7 11 6 12 + 6 x y - 18 x y - 17 x y + 4 x y + 26 x y + x y 7 10 6 11 7 9 6 10 7 8 6 9 - 21 x y - 10 x y + 8 x y + 24 x y - x y - 24 x y 5 10 6 8 5 9 6 7 5 8 5 7 4 8 - 3 x y + 11 x y + 10 x y - 2 x y - 12 x y + 6 x y - x y 5 6 4 6 4 5 3 6 4 4 3 5 3 4 - x y + 3 x y - 4 x y - 3 x y + 2 x y + 6 x y - 5 x y 3 3 2 4 2 3 2 2 2 / 12 22 + 2 x y + 3 x y - 2 x y - x y + x y - x + 1) / (x y / 12 21 12 20 12 19 11 20 12 18 - 6 x y + 15 x y - 20 x y - 2 x y + 15 x y 11 19 12 17 11 18 12 16 11 17 11 16 + 10 x y - 6 x y - 20 x y + x y + 20 x y - 10 x y 11 15 10 16 10 15 9 16 10 14 9 15 + 2 x y + x y - 4 x y - 2 x y + 6 x y + 10 x y 10 13 9 14 10 12 9 13 8 14 9 12 - 4 x y - 20 x y + x y + 20 x y + 5 x y - 10 x y 8 13 9 11 8 12 7 13 8 11 7 12 - 16 x y + 2 x y + 18 x y + 2 x y - 8 x y - 6 x y 8 10 7 11 7 10 6 11 7 9 6 10 7 8 + x y + 6 x y - 5 x y - 2 x y + 6 x y + 8 x y - 3 x y 6 9 5 10 6 8 5 9 6 7 5 8 5 7 - 12 x y - x y + 10 x y + 2 x y - 4 x y - 7 x y + 8 x y 5 6 4 7 4 6 4 5 4 4 3 5 3 4 - 3 x y - 4 x y + 4 x y - 4 x y + 3 x y + 4 x y - 4 x y 3 3 2 3 2 2 2 + 2 x y - 2 x y + 2 x y - 2 x y - x + 1) and in Maple notation (x^14*y^26-8*x^14*y^25+28*x^14*y^24-56*x^14*y^23-2*x^13*y^24+70*x^14*y^22+14*x^ 13*y^23-56*x^14*y^21-42*x^13*y^22+28*x^14*y^20+70*x^13*y^21-8*x^14*y^19-70*x^13 *y^20+x^14*y^18+42*x^13*y^19-14*x^13*y^18-2*x^11*y^20+2*x^13*y^17+14*x^11*y^19-\ 40*x^11*y^18+60*x^11*y^17+5*x^10*y^18-50*x^11*y^16-26*x^10*y^17-x^9*y^18+22*x^ 11*y^15+54*x^10*y^16+8*x^9*y^17-4*x^11*y^14-56*x^10*y^15-25*x^9*y^16+29*x^10*y^ 14+40*x^9*y^15+2*x^8*y^16-6*x^10*y^13-35*x^9*y^14-12*x^8*y^15+16*x^9*y^13+30*x^ 8*y^14-3*x^9*y^12-42*x^8*y^13-x^7*y^14+36*x^8*y^12+6*x^7*y^13-18*x^8*y^11-17*x^ 7*y^12+4*x^8*y^10+26*x^7*y^11+x^6*y^12-21*x^7*y^10-10*x^6*y^11+8*x^7*y^9+24*x^6 *y^10-x^7*y^8-24*x^6*y^9-3*x^5*y^10+11*x^6*y^8+10*x^5*y^9-2*x^6*y^7-12*x^5*y^8+ 6*x^5*y^7-x^4*y^8-x^5*y^6+3*x^4*y^6-4*x^4*y^5-3*x^3*y^6+2*x^4*y^4+6*x^3*y^5-5*x ^3*y^4+2*x^3*y^3+3*x^2*y^4-2*x^2*y^3-x^2*y^2+x*y^2-x+1)/(x^12*y^22-6*x^12*y^21+ 15*x^12*y^20-20*x^12*y^19-2*x^11*y^20+15*x^12*y^18+10*x^11*y^19-6*x^12*y^17-20* x^11*y^18+x^12*y^16+20*x^11*y^17-10*x^11*y^16+2*x^11*y^15+x^10*y^16-4*x^10*y^15 -2*x^9*y^16+6*x^10*y^14+10*x^9*y^15-4*x^10*y^13-20*x^9*y^14+x^10*y^12+20*x^9*y^ 13+5*x^8*y^14-10*x^9*y^12-16*x^8*y^13+2*x^9*y^11+18*x^8*y^12+2*x^7*y^13-8*x^8*y ^11-6*x^7*y^12+x^8*y^10+6*x^7*y^11-5*x^7*y^10-2*x^6*y^11+6*x^7*y^9+8*x^6*y^10-3 *x^7*y^8-12*x^6*y^9-x^5*y^10+10*x^6*y^8+2*x^5*y^9-4*x^6*y^7-7*x^5*y^8+8*x^5*y^7 -3*x^5*y^6-4*x^4*y^7+4*x^4*y^6-4*x^4*y^5+3*x^4*y^4+4*x^3*y^5-4*x^3*y^4+2*x^3*y^ 3-2*x^2*y^3+2*x^2*y^2-2*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, 3 2 5 4 3 2 2 (a - 2 a + a + 1) (a + 4 a + 4 a - 3 a - 3 a + 3) --------------------------------------------------------- 4 3 2 5 a + 4 a - 6 a + 3 5 4 3 where a is the root of the polynomial, x + x - 2 x + 3 x - 1, and in decimals this is, 1.654156238 BTW the ratio for words with, 500, letters is, 1.651865368 ------------------------------------------------ "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, 1, 2], [1, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 12 7 11 7 10 7 9 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 10 7 8 6 9 6 8 5 9 6 7 5 8 + 2 x y + x y - 6 x y + 6 x y - 2 x y - 2 x y + 8 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 - 10 x y + x y + 4 x y - 6 x y + 10 x y - 6 x y - x y 4 4 3 5 3 4 3 3 2 4 2 3 2 + x y + 2 x y + x y - 2 x y - 4 x y + 4 x y - 2 x y + 2 x y / 5 8 5 7 5 6 4 6 4 5 4 4 3 5 - 1) / (x y - 2 x y + x y + 2 x y - 2 x y + x y - 2 x y / 3 4 3 3 2 4 2 3 2 + 2 x y - 2 x y + 2 x y - 2 x y + x y + 2 x y - 1) and in Maple notation (x^7*y^12-4*x^7*y^11+6*x^7*y^10-4*x^7*y^9+2*x^6*y^10+x^7*y^8-6*x^6*y^9+6*x^6*y^ 8-2*x^5*y^9-2*x^6*y^7+8*x^5*y^8-10*x^5*y^7+x^4*y^8+4*x^5*y^6-6*x^4*y^7+10*x^4*y ^6-6*x^4*y^5-x^3*y^6+x^4*y^4+2*x^3*y^5+x^3*y^4-2*x^3*y^3-4*x^2*y^4+4*x^2*y^3-2* x*y^2+2*x*y-1)/(x^5*y^8-2*x^5*y^7+x^5*y^6+2*x^4*y^6-2*x^4*y^5+x^4*y^4-2*x^3*y^5 +2*x^3*y^4-2*x^3*y^3+2*x^2*y^4-2*x^2*y^3+x*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, 4 3 2 2 2 (a - 2 a - 2 a + 6 a + 3) (a - 1) --------------------------------------- 3 2 4 a - 6 a + 3 4 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.643252858 BTW the ratio for words with, 500, letters is, 1.641323560 ------------------------------------------------ "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, 1, 2], [1, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 11 7 12 ) | ) C(m, n) x y | = - (x y - 2 x y + 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 10 7 11 6 12 7 10 6 11 7 9 6 10 - x y - 3 x y - x y + 3 x y - x y - x y + 3 x y 6 9 5 10 6 8 5 9 6 7 5 8 5 7 4 8 + 2 x y + x y - 4 x y - 3 x y + x y + x y + x y + 3 x y 4 7 4 6 3 6 3 5 3 4 2 4 2 3 2 - 5 x y + 2 x y + 4 x y - x y - 2 x y - 3 x y + 2 x y - x y / 9 15 9 14 9 13 9 12 8 13 9 11 + x y - 1) / (x y - x y - 2 x y + 2 x y + x y + x y / 8 12 7 13 9 10 7 12 8 10 7 11 8 9 - 2 x y - x y - x y - 2 x y + 2 x y + 3 x y - x y 7 10 6 11 7 9 6 10 7 8 6 9 6 8 6 7 + 3 x y + x y - 2 x y - x y - x y - 3 x y + x y + 2 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 5 + 2 x y + x y - 2 x y + 4 x y - 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^8*y^14-2*x^8*y^13+2*x^8*y^11+x^7*y^12-x^8*y^10-3*x^7*y^11-x^6*y^12+3*x^7*y^ 10-x^6*y^11-x^7*y^9+3*x^6*y^10+2*x^6*y^9+x^5*y^10-4*x^6*y^8-3*x^5*y^9+x^6*y^7+x ^5*y^8+x^5*y^7+3*x^4*y^8-5*x^4*y^7+2*x^4*y^6+4*x^3*y^6-x^3*y^5-2*x^3*y^4-3*x^2* y^4+2*x^2*y^3-x*y^2+x*y-1)/(x^9*y^15-x^9*y^14-2*x^9*y^13+2*x^9*y^12+x^8*y^13+x^ 9*y^11-2*x^8*y^12-x^7*y^13-x^9*y^10-2*x^7*y^12+2*x^8*y^10+3*x^7*y^11-x^8*y^9+3* x^7*y^10+x^6*y^11-2*x^7*y^9-x^6*y^10-x^7*y^8-3*x^6*y^9+x^6*y^8+2*x^6*y^7+2*x^5* y^7+x^4*y^8-2*x^5*y^6+4*x^4*y^7-x^4*y^6-2*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 3 2 2 (2 a - 4 a + 6 a - 3) - ------------------------------------- 3 2 3 2 (8 a - 6 a + 2 a - 3) (a - a - 1) 4 3 2 where a is the root of the polynomial, 2 x - 2 x + x - 3 x + 1, and in decimals this is, 1.642482036 BTW the ratio for words with, 500, letters is, 1.640391876 ------------------------------------------------ "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], [2, 1, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 11 7 12 ) | ) C(m, n) x y | = - (x y - 2 x y + 2 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 10 7 11 6 12 7 10 6 11 7 9 6 10 - x y - 3 x y - x y + 3 x y - x y - x y + 3 x y 6 9 5 10 6 8 5 9 6 7 5 8 5 7 4 8 + 2 x y + x y - 4 x y - 3 x y + x y + x y + x y + 3 x y 4 7 4 6 3 6 3 5 3 4 2 4 2 3 2 - 5 x y + 2 x y + 4 x y - x y - 2 x y - 3 x y + 2 x y - x y / 9 15 9 14 9 13 9 12 8 13 9 11 + x y - 1) / (x y - x y - 2 x y + 2 x y + x y + x y / 8 12 7 13 9 10 7 12 8 10 7 11 8 9 - 2 x y - x y - x y - 2 x y + 2 x y + 3 x y - x y 7 10 6 11 7 9 6 10 7 8 6 9 6 8 6 7 + 3 x y + x y - 2 x y - x y - x y - 3 x y + x y + 2 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 5 + 2 x y + x y - 2 x y + 4 x y - 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^8*y^14-2*x^8*y^13+2*x^8*y^11+x^7*y^12-x^8*y^10-3*x^7*y^11-x^6*y^12+3*x^7*y^ 10-x^6*y^11-x^7*y^9+3*x^6*y^10+2*x^6*y^9+x^5*y^10-4*x^6*y^8-3*x^5*y^9+x^6*y^7+x ^5*y^8+x^5*y^7+3*x^4*y^8-5*x^4*y^7+2*x^4*y^6+4*x^3*y^6-x^3*y^5-2*x^3*y^4-3*x^2* y^4+2*x^2*y^3-x*y^2+x*y-1)/(x^9*y^15-x^9*y^14-2*x^9*y^13+2*x^9*y^12+x^8*y^13+x^ 9*y^11-2*x^8*y^12-x^7*y^13-x^9*y^10-2*x^7*y^12+2*x^8*y^10+3*x^7*y^11-x^8*y^9+3* x^7*y^10+x^6*y^11-2*x^7*y^9-x^6*y^10-x^7*y^8-3*x^6*y^9+x^6*y^8+2*x^6*y^7+2*x^5* y^7+x^4*y^8-2*x^5*y^6+4*x^4*y^7-x^4*y^6-2*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 3 2 2 (2 a - 4 a + 6 a - 3) - ------------------------------------- 3 2 3 2 (8 a - 6 a + 2 a - 3) (a - a - 1) 4 3 2 where a is the root of the polynomial, 2 x - 2 x + x - 3 x + 1, and in decimals this is, 1.642482036 BTW the ratio for words with, 500, letters is, 1.640391876 ------------------------------------------------ "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, 1], [1, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 ) | ) C(m, n) x y | = - (8 x y - 8 x y - 8 x y + 8 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 5 3 6 4 4 3 5 3 4 + 12 x y - 16 x y + 8 x y + 2 x y - 4 x y - 8 x y + 6 x y 2 4 2 2 2 / 6 11 6 10 6 9 - 3 x y + x y - x y + x - 1) / (16 x y - 16 x y - 16 x y / 6 8 5 9 5 8 5 7 5 6 4 7 4 6 + 16 x y + 8 x y - 8 x y + 8 x y - 8 x y - 8 x y + 4 x y 4 4 3 4 2 2 2 + 4 x y - 4 x y + 2 x y - 2 x y - x + 1) and in Maple notation -(8*x^5*y^10-8*x^5*y^9-8*x^5*y^8+8*x^5*y^7+12*x^4*y^8-16*x^4*y^7+8*x^4*y^5+2*x^ 3*y^6-4*x^4*y^4-8*x^3*y^5+6*x^3*y^4-3*x^2*y^4+x^2*y^2-x*y^2+x-1)/(16*x^6*y^11-\ 16*x^6*y^10-16*x^6*y^9+16*x^6*y^8+8*x^5*y^9-8*x^5*y^8+8*x^5*y^7-8*x^5*y^6-8*x^4 *y^7+4*x^4*y^6+4*x^4*y^4-4*x^3*y^4+2*x^2*y^2-2*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, 4 2 2 (4 a + a + 3) ---------------------------- 2 2 (12 a - 4 a + 3) (2 a + 1) 3 2 where a is the root of the polynomial, 4 x - 2 x + 3 x - 1, and in decimals this is, 1.638365856 BTW the ratio for words with, 500, letters is, 1.636292224 ------------------------------------------------ "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], [1, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (4 x y - 8 x y + 8 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 7 4 6 4 5 3 6 4 4 3 5 - 4 x y - 6 x y + 2 x y + 6 x y + 2 x y - 4 x y - 6 x y 3 4 3 3 2 4 2 3 2 2 2 / + 6 x y - 2 x y - 3 x y + 2 x y + x y - x y + x - 1) / ( / 4 5 4 4 3 5 3 4 3 3 2 3 2 2 4 x y - 4 x y - 4 x y + 4 x y - 2 x y + 2 x y - 2 x y 2 + 2 x y + x - 1) and in Maple notation (4*x^5*y^10-8*x^5*y^9+8*x^5*y^7+2*x^4*y^8-4*x^5*y^6-6*x^4*y^7+2*x^4*y^6+6*x^4*y ^5+2*x^3*y^6-4*x^4*y^4-6*x^3*y^5+6*x^3*y^4-2*x^3*y^3-3*x^2*y^4+2*x^2*y^3+x^2*y^ 2-x*y^2+x-1)/(4*x^4*y^5-4*x^4*y^4-4*x^3*y^5+4*x^3*y^4-2*x^3*y^3+2*x^2*y^3-2*x^2 *y^2+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, approximaley to, 1.632091611 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "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, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (4 x y - 8 x y + 8 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 7 4 6 4 5 3 6 4 4 3 5 - 4 x y - 6 x y + 2 x y + 6 x y + 2 x y - 4 x y - 6 x y 3 4 3 3 2 4 2 3 2 2 2 / + 6 x y - 2 x y - 3 x y + 2 x y + x y - x y + x - 1) / ( / 3 5 3 4 2 3 2 (4 x y - 4 x y - 2 x y - 2 x y + 1) (x - 1)) and in Maple notation (4*x^5*y^10-8*x^5*y^9+8*x^5*y^7+2*x^4*y^8-4*x^5*y^6-6*x^4*y^7+2*x^4*y^6+6*x^4*y ^5+2*x^3*y^6-4*x^4*y^4-6*x^3*y^5+6*x^3*y^4-2*x^3*y^3-3*x^2*y^4+2*x^2*y^3+x^2*y^ 2-x*y^2+x-1)/(4*x^3*y^5-4*x^3*y^4-2*x^2*y^3-2*x*y^2+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, 1.632091611 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "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, 1, 2], [2, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 4 8 4 7 3 6 3 5 ) | ) C(m, n) x y | = - (x y - x y - 3 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 4 2 4 2 3 2 2 2 / 5 8 - 3 x y - 5 x y + 5 x y - x y - 2 x y + 2 x y - 1) / (x y / 5 7 4 8 4 7 4 6 4 5 3 6 3 5 3 4 - x y - x y + 2 x y - 2 x y + x y - x y + x y - x y 2 4 2 3 2 2 2 - x y + x y + x y - x y - 2 x y + 1) and in Maple notation -(x^4*y^8-x^4*y^7-3*x^3*y^6+6*x^3*y^5-3*x^3*y^4-5*x^2*y^4+5*x^2*y^3-x^2*y^2-2*x *y^2+2*x*y-1)/(x^5*y^8-x^5*y^7-x^4*y^8+2*x^4*y^7-2*x^4*y^6+x^4*y^5-x^3*y^6+x^3* y^5-x^3*y^4-x^2*y^4+x^2*y^3+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\ bors of a random word of length n tends to n times, 3 2 2 (2 a - 5 a + 3) ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.618954467 BTW the ratio for words with, 500, letters is, 1.617147402 ------------------------------------------------ "Theorem Number 15" 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, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 4 8 4 7 3 6 3 5 ) | ) C(m, n) x y | = - (x y - x y - 3 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 3 4 2 4 2 3 2 2 2 / 5 8 - 3 x y - 5 x y + 5 x y - x y - 2 x y + 2 x y - 1) / (x y / 5 7 4 8 4 7 4 6 4 5 3 6 3 5 3 4 - x y - x y + 2 x y - 2 x y + x y - x y + x y - x y 2 4 2 3 2 2 2 - x y + x y + x y - x y - 2 x y + 1) and in Maple notation -(x^4*y^8-x^4*y^7-3*x^3*y^6+6*x^3*y^5-3*x^3*y^4-5*x^2*y^4+5*x^2*y^3-x^2*y^2-2*x *y^2+2*x*y-1)/(x^5*y^8-x^5*y^7-x^4*y^8+2*x^4*y^7-2*x^4*y^6+x^4*y^5-x^3*y^6+x^3* y^5-x^3*y^4-x^2*y^4+x^2*y^3+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\ bors of a random word of length n tends to n times, 3 2 2 (2 a - 5 a + 3) ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.618954467 BTW the ratio for words with, 500, letters is, 1.617147402 ------------------------------------------------ "Theorem Number 16" 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, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 6 12 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 7 9 6 10 7 8 6 9 5 10 7 7 - 3 x y + 3 x y - 6 x y - 3 x y + 10 x y + x y + x y 6 8 5 9 6 7 5 8 6 6 5 7 4 8 5 6 - 8 x y + x y + 4 x y - 6 x y - x y + 4 x y - x y + x y 4 7 5 5 4 6 4 5 3 6 3 5 3 4 2 4 + x y - x y + x y - x y - 2 x y + 3 x y - x y - 5 x y 2 3 2 2 2 / 7 13 7 12 7 10 + 3 x y + x y - 2 x y + x y - 1) / (x y - 2 x y + x y / 6 11 7 8 6 9 7 7 6 8 5 9 6 7 5 8 + x y + x y - 2 x y - x y + x y + x y - x y - 2 x y 6 6 5 7 4 8 5 6 4 7 5 5 4 6 4 5 3 6 + x y + x y - x y - x y + 2 x y + x y + x y - x y - x y 3 5 3 4 2 4 2 2 2 - 2 x y + 2 x y - x y - x y - x y - x y + 1) and in Maple notation -(2*x^7*y^13-6*x^7*y^12+6*x^7*y^11+x^6*y^12-3*x^7*y^10+3*x^7*y^9-6*x^6*y^10-3*x ^7*y^8+10*x^6*y^9+x^5*y^10+x^7*y^7-8*x^6*y^8+x^5*y^9+4*x^6*y^7-6*x^5*y^8-x^6*y^ 6+4*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^7-x^5*y^5+x^4*y^6-x^4*y^5-2*x^3*y^6+3*x^3*y^5 -x^3*y^4-5*x^2*y^4+3*x^2*y^3+x^2*y^2-2*x*y^2+x*y-1)/(x^7*y^13-2*x^7*y^12+x^7*y^ 10+x^6*y^11+x^7*y^8-2*x^6*y^9-x^7*y^7+x^6*y^8+x^5*y^9-x^6*y^7-2*x^5*y^8+x^6*y^6 +x^5*y^7-x^4*y^8-x^5*y^6+2*x^4*y^7+x^5*y^5+x^4*y^6-x^4*y^5-x^3*y^6-2*x^3*y^5+2* x^3*y^4-x^2*y^4-x^2*y^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 2 (a + a - 3 a - 8 a - 3 a + 6 a + 3) - ------------------------------------------ 5 4 3 2 4 a + a - 3 a - 9 a - 6 a - 2 4 3 2 where a is the root of the polynomial, x - x - 2 x - 2 x + 1, and in decimals this is, 1.615125426 BTW the ratio for words with, 500, letters is, 1.613271274 ------------------------------------------------ "Theorem Number 17" 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, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 6 12 ) | ) C(m, n) x y | = - (2 x y - 6 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 7 9 6 10 7 8 6 9 5 10 7 7 - 3 x y + 3 x y - 6 x y - 3 x y + 10 x y + x y + x y 6 8 5 9 6 7 5 8 6 6 5 7 4 8 5 6 - 8 x y + x y + 4 x y - 6 x y - x y + 4 x y - x y + x y 4 7 5 5 4 6 4 5 3 6 3 5 3 4 2 4 + x y - x y + x y - x y - 2 x y + 3 x y - x y - 5 x y 2 3 2 2 2 / 7 13 7 12 7 10 + 3 x y + x y - 2 x y + x y - 1) / (x y - 2 x y + x y / 6 11 7 8 6 9 7 7 6 8 5 9 6 7 5 8 + x y + x y - 2 x y - x y + x y + x y - x y - 2 x y 6 6 5 7 4 8 5 6 4 7 5 5 4 6 4 5 3 6 + x y + x y - x y - x y + 2 x y + x y + x y - x y - x y 3 5 3 4 2 4 2 2 2 - 2 x y + 2 x y - x y - x y - x y - x y + 1) and in Maple notation -(2*x^7*y^13-6*x^7*y^12+6*x^7*y^11+x^6*y^12-3*x^7*y^10+3*x^7*y^9-6*x^6*y^10-3*x ^7*y^8+10*x^6*y^9+x^5*y^10+x^7*y^7-8*x^6*y^8+x^5*y^9+4*x^6*y^7-6*x^5*y^8-x^6*y^ 6+4*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^7-x^5*y^5+x^4*y^6-x^4*y^5-2*x^3*y^6+3*x^3*y^5 -x^3*y^4-5*x^2*y^4+3*x^2*y^3+x^2*y^2-2*x*y^2+x*y-1)/(x^7*y^13-2*x^7*y^12+x^7*y^ 10+x^6*y^11+x^7*y^8-2*x^6*y^9-x^7*y^7+x^6*y^8+x^5*y^9-x^6*y^7-2*x^5*y^8+x^6*y^6 +x^5*y^7-x^4*y^8-x^5*y^6+2*x^4*y^7+x^5*y^5+x^4*y^6-x^4*y^5-x^3*y^6-2*x^3*y^5+2* x^3*y^4-x^2*y^4-x^2*y^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 2 (a + a - 3 a - 8 a - 3 a + 6 a + 3) - ------------------------------------------ 5 4 3 2 4 a + a - 3 a - 9 a - 6 a - 2 4 3 2 where a is the root of the polynomial, x - x - 2 x - 2 x + 1, and in decimals this is, 1.615125426 BTW the ratio for words with, 500, letters is, 1.613271274 ------------------------------------------------ "Theorem Number 18" 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, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (x y - 6 x y + 16 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 - 27 x y + 2 x y + 37 x y - 10 x y - 47 x y 11 19 12 17 11 18 10 19 12 16 11 17 + 22 x y + 51 x y - 31 x y + x y - 41 x y + 37 x y 10 18 12 15 11 16 10 17 9 18 12 14 - 7 x y + 22 x y - 42 x y + 18 x y - x y - 7 x y 11 15 10 16 9 17 12 13 11 14 10 15 + 40 x y - 23 x y + 7 x y + x y - 27 x y + 18 x y 9 16 11 13 10 14 9 15 8 16 11 12 - 23 x y + 11 x y - 13 x y + 41 x y - x y - 2 x y 10 13 9 14 8 15 10 12 9 13 8 14 + 10 x y - 45 x y + 4 x y - 5 x y + 37 x y - 8 x y 10 11 9 12 8 13 7 14 9 11 8 12 + x y - 25 x y + 13 x y + x y + 11 x y - 19 x y 9 10 8 11 7 12 8 10 7 11 6 12 - 2 x y + 20 x y - 5 x y - 12 x y + 14 x y + x y 8 9 7 10 6 11 7 9 6 10 7 8 + 3 x y - 24 x y + 3 x y + 20 x y - 10 x y - 6 x y 6 9 6 8 5 9 6 7 5 8 6 6 5 7 + 12 x y - 11 x y + 6 x y + 6 x y - 12 x y - x y + 10 x y 4 8 5 6 4 7 5 5 4 6 4 5 3 6 - 6 x y - 6 x y + 10 x y + 2 x y - 9 x y + 5 x y - 4 x y 4 4 3 5 3 4 3 3 2 4 2 3 2 - x y + 6 x y - 4 x y + x y - 6 x y + 4 x y - 2 x y + x y - 1 / 13 23 13 22 13 21 13 20 12 21 ) / (x y - 6 x y + 16 x y - 26 x y + 2 x y / 13 19 12 20 13 18 12 19 13 17 + 30 x y - 10 x y - 26 x y + 22 x y + 16 x y 12 18 13 16 12 17 11 18 13 15 12 16 - 29 x y - 6 x y + 25 x y - x y + x y - 12 x y 11 17 10 18 11 16 10 17 12 14 11 15 + 2 x y - x y + 3 x y + 4 x y + 3 x y - 12 x y 10 16 9 17 12 13 11 14 10 15 9 16 - 7 x y + x y - x y + 13 x y + 4 x y - 6 x y 11 13 10 14 9 15 11 12 10 13 9 14 - 6 x y + 5 x y + 13 x y + x y - 8 x y - 13 x y 8 15 10 12 9 13 8 14 9 12 8 13 + 2 x y + 3 x y + 5 x y - 6 x y + 2 x y + 9 x y 9 11 8 12 7 13 9 10 8 11 7 12 7 11 - 3 x y - 8 x y + 2 x y + x y + 3 x y - 3 x y + x y 6 12 7 10 6 11 7 9 6 10 7 8 6 9 - x y + 3 x y + 5 x y - 6 x y - 4 x y + 3 x y - x y 5 10 6 8 5 9 6 7 5 8 6 6 5 7 - 2 x y + 2 x y + 3 x y - 2 x y - 2 x y + x y - 2 x y 4 8 5 6 4 7 5 5 4 6 4 5 3 6 - 2 x y + 3 x y + 2 x y - x y + x y - 2 x y - 2 x y 3 5 3 4 3 3 2 3 2 - 2 x y + 2 x y - x y - x y - x y - x y + 1) and in Maple notation -(x^12*y^23-6*x^12*y^22+16*x^12*y^21-27*x^12*y^20+2*x^11*y^21+37*x^12*y^19-10*x ^11*y^20-47*x^12*y^18+22*x^11*y^19+51*x^12*y^17-31*x^11*y^18+x^10*y^19-41*x^12* y^16+37*x^11*y^17-7*x^10*y^18+22*x^12*y^15-42*x^11*y^16+18*x^10*y^17-x^9*y^18-7 *x^12*y^14+40*x^11*y^15-23*x^10*y^16+7*x^9*y^17+x^12*y^13-27*x^11*y^14+18*x^10* y^15-23*x^9*y^16+11*x^11*y^13-13*x^10*y^14+41*x^9*y^15-x^8*y^16-2*x^11*y^12+10* x^10*y^13-45*x^9*y^14+4*x^8*y^15-5*x^10*y^12+37*x^9*y^13-8*x^8*y^14+x^10*y^11-\ 25*x^9*y^12+13*x^8*y^13+x^7*y^14+11*x^9*y^11-19*x^8*y^12-2*x^9*y^10+20*x^8*y^11 -5*x^7*y^12-12*x^8*y^10+14*x^7*y^11+x^6*y^12+3*x^8*y^9-24*x^7*y^10+3*x^6*y^11+ 20*x^7*y^9-10*x^6*y^10-6*x^7*y^8+12*x^6*y^9-11*x^6*y^8+6*x^5*y^9+6*x^6*y^7-12*x ^5*y^8-x^6*y^6+10*x^5*y^7-6*x^4*y^8-6*x^5*y^6+10*x^4*y^7+2*x^5*y^5-9*x^4*y^6+5* x^4*y^5-4*x^3*y^6-x^4*y^4+6*x^3*y^5-4*x^3*y^4+x^3*y^3-6*x^2*y^4+4*x^2*y^3-2*x*y ^2+x*y-1)/(x^13*y^23-6*x^13*y^22+16*x^13*y^21-26*x^13*y^20+2*x^12*y^21+30*x^13* y^19-10*x^12*y^20-26*x^13*y^18+22*x^12*y^19+16*x^13*y^17-29*x^12*y^18-6*x^13*y^ 16+25*x^12*y^17-x^11*y^18+x^13*y^15-12*x^12*y^16+2*x^11*y^17-x^10*y^18+3*x^11*y ^16+4*x^10*y^17+3*x^12*y^14-12*x^11*y^15-7*x^10*y^16+x^9*y^17-x^12*y^13+13*x^11 *y^14+4*x^10*y^15-6*x^9*y^16-6*x^11*y^13+5*x^10*y^14+13*x^9*y^15+x^11*y^12-8*x^ 10*y^13-13*x^9*y^14+2*x^8*y^15+3*x^10*y^12+5*x^9*y^13-6*x^8*y^14+2*x^9*y^12+9*x ^8*y^13-3*x^9*y^11-8*x^8*y^12+2*x^7*y^13+x^9*y^10+3*x^8*y^11-3*x^7*y^12+x^7*y^ 11-x^6*y^12+3*x^7*y^10+5*x^6*y^11-6*x^7*y^9-4*x^6*y^10+3*x^7*y^8-x^6*y^9-2*x^5* y^10+2*x^6*y^8+3*x^5*y^9-2*x^6*y^7-2*x^5*y^8+x^6*y^6-2*x^5*y^7-2*x^4*y^8+3*x^5* y^6+2*x^4*y^7-x^5*y^5+x^4*y^6-2*x^4*y^5-2*x^3*y^6-2*x^3*y^5+2*x^3*y^4-x^3*y^3-x ^2*y^3-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 2 (4 a + 2 a + 7 a - 4 a - 3 a - 6 a - 3) - ------------------------------------------------------- 4 3 2 4 3 2 (5 a + 4 a + 9 a + 2 a + 2) (a + a + 2 a + a + 1) 5 4 3 2 where a is the root of the polynomial, x + x + 3 x + x + 2 x - 1, and in decimals this is, 1.614691227 BTW the ratio for words with, 500, letters is, 1.612828031 ------------------------------------------------ "Theorem Number 19" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 - 70 x y - 2 x y + 70 x y + 13 x y - 42 x y 11 19 12 17 11 18 10 19 12 16 - 36 x y + 14 x y + 55 x y - 5 x y - 2 x y 11 17 10 18 11 16 10 17 9 18 11 15 - 50 x y + 25 x y + 27 x y - 50 x y + x y - 8 x y 10 16 9 17 11 14 10 15 9 16 10 14 + 50 x y - 4 x y + x y - 25 x y + 3 x y + 5 x y 9 15 9 14 8 15 9 13 8 14 9 12 + 8 x y - 17 x y + 4 x y + 12 x y - 14 x y - 3 x y 8 13 7 14 8 12 7 13 8 11 7 12 + 16 x y - 3 x y - 4 x y + 12 x y - 4 x y - 17 x y 8 10 7 11 6 12 6 11 7 9 6 10 6 9 + 2 x y + 9 x y - 2 x y + 5 x y - x y - 5 x y + 2 x y 5 10 6 8 5 9 6 7 5 8 5 7 5 6 + 4 x y + x y - 10 x y - x y + 11 x y - 8 x y + 3 x y 4 6 4 5 3 6 4 4 3 5 3 3 2 4 2 3 + x y - 2 x y - 3 x y + x y + 4 x y - x y + 3 x y - 2 x y 2 / 9 16 9 15 9 14 8 15 + x y - x y + 1) / (3 x y - 12 x y + 18 x y - x y / 9 13 8 14 9 12 8 13 8 12 7 12 7 11 - 12 x y + x y + 3 x y + x y - x y - 6 x y + 12 x y 7 10 6 11 7 9 6 10 6 9 6 8 5 9 - 8 x y + 3 x y + 2 x y - x y - 5 x y + 4 x y + 2 x y 6 7 5 8 5 7 4 8 4 7 4 5 3 6 4 4 - x y + 2 x y - 3 x y - x y - 2 x y + x y + 2 x y + x y 3 5 3 3 2 3 2 - 2 x y - x y + x y - 2 x y - x y + 1) and in Maple notation (2*x^12*y^23-14*x^12*y^22+42*x^12*y^21-70*x^12*y^20-2*x^11*y^21+70*x^12*y^19+13 *x^11*y^20-42*x^12*y^18-36*x^11*y^19+14*x^12*y^17+55*x^11*y^18-5*x^10*y^19-2*x^ 12*y^16-50*x^11*y^17+25*x^10*y^18+27*x^11*y^16-50*x^10*y^17+x^9*y^18-8*x^11*y^ 15+50*x^10*y^16-4*x^9*y^17+x^11*y^14-25*x^10*y^15+3*x^9*y^16+5*x^10*y^14+8*x^9* y^15-17*x^9*y^14+4*x^8*y^15+12*x^9*y^13-14*x^8*y^14-3*x^9*y^12+16*x^8*y^13-3*x^ 7*y^14-4*x^8*y^12+12*x^7*y^13-4*x^8*y^11-17*x^7*y^12+2*x^8*y^10+9*x^7*y^11-2*x^ 6*y^12+5*x^6*y^11-x^7*y^9-5*x^6*y^10+2*x^6*y^9+4*x^5*y^10+x^6*y^8-10*x^5*y^9-x^ 6*y^7+11*x^5*y^8-8*x^5*y^7+3*x^5*y^6+x^4*y^6-2*x^4*y^5-3*x^3*y^6+x^4*y^4+4*x^3* y^5-x^3*y^3+3*x^2*y^4-2*x^2*y^3+x*y^2-x*y+1)/(3*x^9*y^16-12*x^9*y^15+18*x^9*y^ 14-x^8*y^15-12*x^9*y^13+x^8*y^14+3*x^9*y^12+x^8*y^13-x^8*y^12-6*x^7*y^12+12*x^7 *y^11-8*x^7*y^10+3*x^6*y^11+2*x^7*y^9-x^6*y^10-5*x^6*y^9+4*x^6*y^8+2*x^5*y^9-x^ 6*y^7+2*x^5*y^8-3*x^5*y^7-x^4*y^8-2*x^4*y^7+x^4*y^5+2*x^3*y^6+x^4*y^4-2*x^3*y^5 -x^3*y^3+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, 8 7 6 5 4 3 2 2 (a - 2 a - 2 a + 3 a - 10 a + 9 a - 6 a + 3) - ----------------------------------------------------- 4 3 2 2 (5 a - 4 a - 3 a + 2 a - 3) (a + 1) 5 4 3 2 where a is the root of the polynomial, x - x - x + x - 3 x + 1, and in decimals this is, 1.597371424 BTW the ratio for words with, 500, letters is, 1.595644252 ------------------------------------------------ "Theorem Number 20" 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, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 - 70 x y - 2 x y + 70 x y + 13 x y - 42 x y 11 19 12 17 11 18 10 19 12 16 - 36 x y + 14 x y + 55 x y - 5 x y - 2 x y 11 17 10 18 11 16 10 17 9 18 11 15 - 50 x y + 25 x y + 27 x y - 50 x y + x y - 8 x y 10 16 9 17 11 14 10 15 9 16 10 14 + 50 x y - 4 x y + x y - 25 x y + 3 x y + 5 x y 9 15 9 14 8 15 9 13 8 14 9 12 + 8 x y - 17 x y + 4 x y + 12 x y - 14 x y - 3 x y 8 13 7 14 8 12 7 13 8 11 7 12 + 16 x y - 3 x y - 4 x y + 12 x y - 4 x y - 17 x y 8 10 7 11 6 12 6 11 7 9 6 10 6 9 + 2 x y + 9 x y - 2 x y + 5 x y - x y - 5 x y + 2 x y 5 10 6 8 5 9 6 7 5 8 5 7 5 6 + 4 x y + x y - 10 x y - x y + 11 x y - 8 x y + 3 x y 4 6 4 5 3 6 4 4 3 5 3 3 2 4 2 3 + x y - 2 x y - 3 x y + x y + 4 x y - x y + 3 x y - 2 x y 2 / 9 16 9 15 9 14 8 15 + x y - x y + 1) / (3 x y - 12 x y + 18 x y - x y / 9 13 8 14 9 12 8 13 8 12 7 12 7 11 - 12 x y + x y + 3 x y + x y - x y - 6 x y + 12 x y 7 10 6 11 7 9 6 10 6 9 6 8 5 9 - 8 x y + 3 x y + 2 x y - x y - 5 x y + 4 x y + 2 x y 6 7 5 8 5 7 4 8 4 7 4 5 3 6 4 4 - x y + 2 x y - 3 x y - x y - 2 x y + x y + 2 x y + x y 3 5 3 3 2 3 2 - 2 x y - x y + x y - 2 x y - x y + 1) and in Maple notation (2*x^12*y^23-14*x^12*y^22+42*x^12*y^21-70*x^12*y^20-2*x^11*y^21+70*x^12*y^19+13 *x^11*y^20-42*x^12*y^18-36*x^11*y^19+14*x^12*y^17+55*x^11*y^18-5*x^10*y^19-2*x^ 12*y^16-50*x^11*y^17+25*x^10*y^18+27*x^11*y^16-50*x^10*y^17+x^9*y^18-8*x^11*y^ 15+50*x^10*y^16-4*x^9*y^17+x^11*y^14-25*x^10*y^15+3*x^9*y^16+5*x^10*y^14+8*x^9* y^15-17*x^9*y^14+4*x^8*y^15+12*x^9*y^13-14*x^8*y^14-3*x^9*y^12+16*x^8*y^13-3*x^ 7*y^14-4*x^8*y^12+12*x^7*y^13-4*x^8*y^11-17*x^7*y^12+2*x^8*y^10+9*x^7*y^11-2*x^ 6*y^12+5*x^6*y^11-x^7*y^9-5*x^6*y^10+2*x^6*y^9+4*x^5*y^10+x^6*y^8-10*x^5*y^9-x^ 6*y^7+11*x^5*y^8-8*x^5*y^7+3*x^5*y^6+x^4*y^6-2*x^4*y^5-3*x^3*y^6+x^4*y^4+4*x^3* y^5-x^3*y^3+3*x^2*y^4-2*x^2*y^3+x*y^2-x*y+1)/(3*x^9*y^16-12*x^9*y^15+18*x^9*y^ 14-x^8*y^15-12*x^9*y^13+x^8*y^14+3*x^9*y^12+x^8*y^13-x^8*y^12-6*x^7*y^12+12*x^7 *y^11-8*x^7*y^10+3*x^6*y^11+2*x^7*y^9-x^6*y^10-5*x^6*y^9+4*x^6*y^8+2*x^5*y^9-x^ 6*y^7+2*x^5*y^8-3*x^5*y^7-x^4*y^8-2*x^4*y^7+x^4*y^5+2*x^3*y^6+x^4*y^4-2*x^3*y^5 -x^3*y^3+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, 8 7 6 5 4 3 2 2 (a - 2 a - 2 a + 3 a - 10 a + 9 a - 6 a + 3) - ----------------------------------------------------- 4 3 2 2 (5 a - 4 a - 3 a + 2 a - 3) (a + 1) 5 4 3 2 where a is the root of the polynomial, x - x - x + x - 3 x + 1, and in decimals this is, 1.597371424 BTW the ratio for words with, 500, letters is, 1.595644252 ------------------------------------------------ "Theorem Number 21" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 6 12 7 10 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 11 7 9 6 10 6 9 5 10 6 8 5 9 - 2 x y - x y - 3 x y + 10 x y - x y - 7 x y + 2 x y 5 8 6 6 5 7 4 8 5 6 4 7 4 6 - 2 x y + x y + 2 x y + 3 x y - x y - 6 x y + 2 x y 4 5 3 6 4 4 3 5 3 4 2 4 2 3 2 2 + 2 x y - x y - x y + 2 x y - x y + 4 x y - 2 x y - x y 2 / 7 12 7 11 7 10 7 9 6 10 + x y + 1) / (2 x y - 2 x y - 2 x y + 2 x y - 2 x y / 6 8 6 7 5 8 6 6 5 7 4 8 4 7 4 6 + 5 x y - 2 x y + 2 x y - x y - 2 x y - x y + x y - 3 x y 4 5 3 6 4 4 2 4 2 3 2 2 2 + x y + x y + x y - x y + 2 x y + x y + 2 x y - 1) and in Maple notation -(x^7*y^13-2*x^7*y^12+x^6*y^12+2*x^7*y^10-2*x^6*y^11-x^7*y^9-3*x^6*y^10+10*x^6* y^9-x^5*y^10-7*x^6*y^8+2*x^5*y^9-2*x^5*y^8+x^6*y^6+2*x^5*y^7+3*x^4*y^8-x^5*y^6-\ 6*x^4*y^7+2*x^4*y^6+2*x^4*y^5-x^3*y^6-x^4*y^4+2*x^3*y^5-x^3*y^4+4*x^2*y^4-2*x^2 *y^3-x^2*y^2+x*y^2+1)/(2*x^7*y^12-2*x^7*y^11-2*x^7*y^10+2*x^7*y^9-2*x^6*y^10+5* x^6*y^8-2*x^6*y^7+2*x^5*y^8-x^6*y^6-2*x^5*y^7-x^4*y^8+x^4*y^7-3*x^4*y^6+x^4*y^5 +x^3*y^6+x^4*y^4-x^2*y^4+2*x^2*y^3+x^2*y^2+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 4 3 2 2 (a - 4 a - 9 a - 3 a + 6 a + 3) - ------------------------------------- 5 4 3 2 4 a + a - 3 a - 9 a - 6 a - 2 4 3 2 where a is the root of the polynomial, x - x - 2 x - 2 x + 1, and in decimals this is, 1.589954475 BTW the ratio for words with, 500, letters is, 1.588299214 ------------------------------------------------ "Theorem Number 22" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 6 12 7 10 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 11 7 9 6 10 6 9 5 10 6 8 5 9 - 2 x y - x y - 3 x y + 10 x y - x y - 7 x y + 2 x y 5 8 6 6 5 7 4 8 5 6 4 7 4 6 - 2 x y + x y + 2 x y + 3 x y - x y - 6 x y + 2 x y 4 5 3 6 4 4 3 5 3 4 2 4 2 3 2 2 + 2 x y - x y - x y + 2 x y - x y + 4 x y - 2 x y - x y 2 / 7 12 7 11 7 10 7 9 6 10 + x y + 1) / (2 x y - 2 x y - 2 x y + 2 x y - 2 x y / 6 8 6 7 5 8 6 6 5 7 4 8 4 7 4 6 + 5 x y - 2 x y + 2 x y - x y - 2 x y - x y + x y - 3 x y 4 5 3 6 4 4 2 4 2 3 2 2 2 + x y + x y + x y - x y + 2 x y + x y + 2 x y - 1) and in Maple notation -(x^7*y^13-2*x^7*y^12+x^6*y^12+2*x^7*y^10-2*x^6*y^11-x^7*y^9-3*x^6*y^10+10*x^6* y^9-x^5*y^10-7*x^6*y^8+2*x^5*y^9-2*x^5*y^8+x^6*y^6+2*x^5*y^7+3*x^4*y^8-x^5*y^6-\ 6*x^4*y^7+2*x^4*y^6+2*x^4*y^5-x^3*y^6-x^4*y^4+2*x^3*y^5-x^3*y^4+4*x^2*y^4-2*x^2 *y^3-x^2*y^2+x*y^2+1)/(2*x^7*y^12-2*x^7*y^11-2*x^7*y^10+2*x^7*y^9-2*x^6*y^10+5* x^6*y^8-2*x^6*y^7+2*x^5*y^8-x^6*y^6-2*x^5*y^7-x^4*y^8+x^4*y^7-3*x^4*y^6+x^4*y^5 +x^3*y^6+x^4*y^4-x^2*y^4+2*x^2*y^3+x^2*y^2+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 4 3 2 2 (a - 4 a - 9 a - 3 a + 6 a + 3) - ------------------------------------- 5 4 3 2 4 a + a - 3 a - 9 a - 6 a - 2 4 3 2 where a is the root of the polynomial, x - x - 2 x - 2 x + 1, and in decimals this is, 1.589954475 BTW the ratio for words with, 500, letters is, 1.588299214 ------------------------------------------------ "Theorem Number 23" 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, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 ) | ) C(m, n) x y | = - (2 x y - 8 x y + 13 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 9 5 10 6 8 5 9 6 7 5 8 4 7 - 11 x y + 2 x y + 5 x y - 4 x y - x y + 2 x y - 2 x y 4 6 4 5 3 5 3 4 2 4 2 3 2 2 2 + 3 x y - x y - 2 x y + 2 x y + 5 x y - 5 x y + x y + 2 x y / 7 12 7 11 7 10 7 9 6 10 6 9 - 2 x y + 1) / (x y - 3 x y + 3 x y - x y + x y - x y / 6 8 6 7 5 8 5 7 4 6 4 5 3 6 3 5 3 4 - x y + x y - x y + x y - x y + x y + x y - x y + x y 2 4 2 3 2 2 2 + x y - x y - x y + x y + 2 x y - 1) and in Maple notation -(2*x^6*y^12-8*x^6*y^11+13*x^6*y^10-11*x^6*y^9+2*x^5*y^10+5*x^6*y^8-4*x^5*y^9-x ^6*y^7+2*x^5*y^8-2*x^4*y^7+3*x^4*y^6-x^4*y^5-2*x^3*y^5+2*x^3*y^4+5*x^2*y^4-5*x^ 2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(x^7*y^12-3*x^7*y^11+3*x^7*y^10-x^7*y^9+x^6*y^10 -x^6*y^9-x^6*y^8+x^6*y^7-x^5*y^8+x^5*y^7-x^4*y^6+x^4*y^5+x^3*y^6-x^3*y^5+x^3*y^ 4+x^2*y^4-x^2*y^3-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\ bors of a random word of length n tends to n times, 4 3 2 2 (a - 4 a + 6 a - 3) - ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.583670432 BTW the ratio for words with, 500, letters is, 1.581993451 ------------------------------------------------ "Theorem Number 24" 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, 1, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 ) | ) C(m, n) x y | = - (2 x y - 8 x y + 13 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 9 5 10 6 8 5 9 6 7 5 8 4 7 - 11 x y + 2 x y + 5 x y - 4 x y - x y + 2 x y - 2 x y 4 6 4 5 3 5 3 4 2 4 2 3 2 2 2 + 3 x y - x y - 2 x y + 2 x y + 5 x y - 5 x y + x y + 2 x y / 7 12 7 11 7 10 7 9 6 10 6 9 - 2 x y + 1) / (x y - 3 x y + 3 x y - x y + x y - x y / 6 8 6 7 5 8 5 7 4 6 4 5 3 6 3 5 3 4 - x y + x y - x y + x y - x y + x y + x y - x y + x y 2 4 2 3 2 2 2 + x y - x y - x y + x y + 2 x y - 1) and in Maple notation -(2*x^6*y^12-8*x^6*y^11+13*x^6*y^10-11*x^6*y^9+2*x^5*y^10+5*x^6*y^8-4*x^5*y^9-x ^6*y^7+2*x^5*y^8-2*x^4*y^7+3*x^4*y^6-x^4*y^5-2*x^3*y^5+2*x^3*y^4+5*x^2*y^4-5*x^ 2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(x^7*y^12-3*x^7*y^11+3*x^7*y^10-x^7*y^9+x^6*y^10 -x^6*y^9-x^6*y^8+x^6*y^7-x^5*y^8+x^5*y^7-x^4*y^6+x^4*y^5+x^3*y^6-x^3*y^5+x^3*y^ 4+x^2*y^4-x^2*y^3-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\ bors of a random word of length n tends to n times, 4 3 2 2 (a - 4 a + 6 a - 3) - ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.583670432 BTW the ratio for words with, 500, letters is, 1.581993451 ------------------------------------------------ "Theorem Number 25" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (4 x y - 24 x y + 60 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 - 80 x y + 8 x y + 60 x y - 45 x y - 24 x y 11 19 12 17 11 18 11 17 10 18 + 108 x y + 4 x y - 145 x y + 120 x y - 12 x y 11 16 10 17 9 18 11 15 10 16 9 17 - 63 x y + 55 x y - x y + 20 x y - 101 x y - 6 x y 11 14 10 15 9 16 10 14 9 15 8 16 - 3 x y + 94 x y + 27 x y - 46 x y - 33 x y - 2 x y 10 13 9 14 10 12 9 13 8 14 9 12 + 11 x y + 9 x y - x y + 12 x y + 21 x y - 11 x y 8 13 7 14 9 11 8 12 7 13 8 11 - 44 x y - 2 x y + 3 x y + 35 x y + 14 x y - 9 x y 7 12 8 10 7 11 6 12 8 9 7 10 - 20 x y - 2 x y - 3 x y + 2 x y + x y + 21 x y 6 11 7 9 6 10 7 8 6 9 5 10 6 8 + 12 x y - 11 x y - 33 x y + x y + 22 x y + 4 x y - x y 5 9 6 7 5 8 5 7 5 6 4 7 4 6 + 3 x y - 2 x y - 16 x y + 10 x y - x y - 7 x y + 8 x y 4 5 3 6 3 5 3 4 3 3 2 4 2 3 2 2 - 2 x y - 3 x y + x y + 2 x y - x y - 6 x y + 3 x y + x y 2 / 13 24 13 23 13 22 13 21 - 2 x y + x y - 1) / (4 x y - 24 x y + 60 x y - 80 x y / 12 22 13 20 12 21 13 19 12 20 + 4 x y + 60 x y - 18 x y - 24 x y + 30 x y 13 18 12 19 11 20 11 19 12 17 11 18 + 4 x y - 20 x y - 4 x y + 12 x y + 6 x y - 6 x y 10 19 12 16 11 17 10 18 11 16 10 17 - 2 x y - 2 x y - 16 x y + 6 x y + 24 x y - 14 x y 11 15 10 16 11 14 10 15 9 16 10 14 - 12 x y + 29 x y + 2 x y - 34 x y - 5 x y + 20 x y 9 15 10 13 9 14 8 15 10 12 9 13 + 20 x y - 6 x y - 29 x y + 2 x y + x y + 17 x y 8 14 9 12 9 11 8 12 7 13 8 11 7 12 - 3 x y - 2 x y - x y + x y + 2 x y - x y + 5 x y 8 10 7 11 6 12 8 9 7 10 6 11 7 9 + 2 x y - 13 x y - x y - x y + 4 x y + 6 x y + 3 x y 6 10 7 8 6 9 6 8 5 9 6 7 5 8 5 7 - 3 x y - x y - 3 x y - x y - 2 x y + 2 x y + x y - x y 5 6 4 5 3 6 3 5 3 4 3 3 2 2 2 + x y - x y - x y - 4 x y + x y + x y - x y - x y - x y + 1 ) and in Maple notation -(4*x^12*y^23-24*x^12*y^22+60*x^12*y^21-80*x^12*y^20+8*x^11*y^21+60*x^12*y^19-\ 45*x^11*y^20-24*x^12*y^18+108*x^11*y^19+4*x^12*y^17-145*x^11*y^18+120*x^11*y^17 -12*x^10*y^18-63*x^11*y^16+55*x^10*y^17-x^9*y^18+20*x^11*y^15-101*x^10*y^16-6*x ^9*y^17-3*x^11*y^14+94*x^10*y^15+27*x^9*y^16-46*x^10*y^14-33*x^9*y^15-2*x^8*y^ 16+11*x^10*y^13+9*x^9*y^14-x^10*y^12+12*x^9*y^13+21*x^8*y^14-11*x^9*y^12-44*x^8 *y^13-2*x^7*y^14+3*x^9*y^11+35*x^8*y^12+14*x^7*y^13-9*x^8*y^11-20*x^7*y^12-2*x^ 8*y^10-3*x^7*y^11+2*x^6*y^12+x^8*y^9+21*x^7*y^10+12*x^6*y^11-11*x^7*y^9-33*x^6* y^10+x^7*y^8+22*x^6*y^9+4*x^5*y^10-x^6*y^8+3*x^5*y^9-2*x^6*y^7-16*x^5*y^8+10*x^ 5*y^7-x^5*y^6-7*x^4*y^7+8*x^4*y^6-2*x^4*y^5-3*x^3*y^6+x^3*y^5+2*x^3*y^4-x^3*y^3 -6*x^2*y^4+3*x^2*y^3+x^2*y^2-2*x*y^2+x*y-1)/(4*x^13*y^24-24*x^13*y^23+60*x^13*y ^22-80*x^13*y^21+4*x^12*y^22+60*x^13*y^20-18*x^12*y^21-24*x^13*y^19+30*x^12*y^ 20+4*x^13*y^18-20*x^12*y^19-4*x^11*y^20+12*x^11*y^19+6*x^12*y^17-6*x^11*y^18-2* x^10*y^19-2*x^12*y^16-16*x^11*y^17+6*x^10*y^18+24*x^11*y^16-14*x^10*y^17-12*x^ 11*y^15+29*x^10*y^16+2*x^11*y^14-34*x^10*y^15-5*x^9*y^16+20*x^10*y^14+20*x^9*y^ 15-6*x^10*y^13-29*x^9*y^14+2*x^8*y^15+x^10*y^12+17*x^9*y^13-3*x^8*y^14-2*x^9*y^ 12-x^9*y^11+x^8*y^12+2*x^7*y^13-x^8*y^11+5*x^7*y^12+2*x^8*y^10-13*x^7*y^11-x^6* y^12-x^8*y^9+4*x^7*y^10+6*x^6*y^11+3*x^7*y^9-3*x^6*y^10-x^7*y^8-3*x^6*y^9-x^6*y ^8-2*x^5*y^9+2*x^6*y^7+x^5*y^8-x^5*y^7+x^5*y^6-x^4*y^5-x^3*y^6-4*x^3*y^5+x^3*y^ 4+x^3*y^3-x^2*y^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, 8 6 5 4 3 2 2 (a - 3 a - 4 a - 8 a + 2 a + 3 a + 6 a + 3) ------------------------------------------------------- 4 3 2 4 3 2 (5 a + 4 a + 9 a + 2 a + 2) (a + a + 2 a + a + 1) 5 4 3 2 where a is the root of the polynomial, x + x + 3 x + x + 2 x - 1, and in decimals this is, 1.580933418 BTW the ratio for words with, 500, letters is, 1.579294827 ------------------------------------------------ "Theorem Number 26" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 12 7 13 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 11 7 12 8 10 7 11 6 12 8 9 - 10 x y - 10 x y + 5 x y + 20 x y + 2 x y - x y 7 10 6 11 7 9 6 10 7 8 6 9 5 10 - 20 x y - 8 x y + 10 x y + 11 x y - 2 x y - 4 x y + x y 6 8 5 9 6 7 5 8 6 6 5 7 4 8 - 4 x y - 5 x y + 4 x y + 9 x y - x y - 6 x y - 4 x y 4 7 5 5 4 6 4 5 3 6 4 4 3 5 + 5 x y + x y + 4 x y - 9 x y - 3 x y + 4 x y + 8 x y 3 4 2 4 3 2 2 2 2 / 8 13 - 6 x y + 3 x y + x y - 2 x y + x y - x + 1) / (2 x y / 8 12 8 11 7 12 8 10 7 11 8 9 - 8 x y + 12 x y + 2 x y - 8 x y - 6 x y + 2 x y 7 10 7 9 6 10 6 9 6 8 5 9 6 7 + 6 x y - 2 x y - 2 x y + 2 x y + 4 x y - 2 x y - 6 x y 5 8 6 6 5 7 5 6 4 7 5 5 4 6 + 2 x y + 2 x y - x y + 2 x y + 3 x y - x y - 4 x y 4 5 4 4 3 4 3 2 2 2 2 + 3 x y - 2 x y + 2 x y - x y - x y + 2 x y + x - 1) and in Maple notation -(x^8*y^14-5*x^8*y^13+10*x^8*y^12+2*x^7*y^13-10*x^8*y^11-10*x^7*y^12+5*x^8*y^10 +20*x^7*y^11+2*x^6*y^12-x^8*y^9-20*x^7*y^10-8*x^6*y^11+10*x^7*y^9+11*x^6*y^10-2 *x^7*y^8-4*x^6*y^9+x^5*y^10-4*x^6*y^8-5*x^5*y^9+4*x^6*y^7+9*x^5*y^8-x^6*y^6-6*x ^5*y^7-4*x^4*y^8+5*x^4*y^7+x^5*y^5+4*x^4*y^6-9*x^4*y^5-3*x^3*y^6+4*x^4*y^4+8*x^ 3*y^5-6*x^3*y^4+3*x^2*y^4+x^3*y^2-2*x^2*y^2+x*y^2-x+1)/(2*x^8*y^13-8*x^8*y^12+ 12*x^8*y^11+2*x^7*y^12-8*x^8*y^10-6*x^7*y^11+2*x^8*y^9+6*x^7*y^10-2*x^7*y^9-2*x ^6*y^10+2*x^6*y^9+4*x^6*y^8-2*x^5*y^9-6*x^6*y^7+2*x^5*y^8+2*x^6*y^6-x^5*y^7+2*x ^5*y^6+3*x^4*y^7-x^5*y^5-4*x^4*y^6+3*x^4*y^5-2*x^4*y^4+2*x^3*y^4-x^3*y^2-x^2*y^ 2+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, 5 4 3 2 2 (a + a + 3 a - 6 a + 3) ----------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.579069555 BTW the ratio for words with, 500, letters is, 1.577367857 ------------------------------------------------ "Theorem Number 27" 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, 1, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 12 7 13 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 11 7 12 8 10 7 11 6 12 8 9 - 10 x y - 10 x y + 5 x y + 20 x y + 2 x y - x y 7 10 6 11 7 9 6 10 7 8 6 9 5 10 - 20 x y - 8 x y + 10 x y + 11 x y - 2 x y - 4 x y + x y 6 8 5 9 6 7 5 8 6 6 5 7 4 8 - 4 x y - 5 x y + 4 x y + 9 x y - x y - 6 x y - 4 x y 4 7 5 5 4 6 4 5 3 6 4 4 3 5 + 5 x y + x y + 4 x y - 9 x y - 3 x y + 4 x y + 8 x y 3 4 2 4 3 2 2 2 2 / 8 13 - 6 x y + 3 x y + x y - 2 x y + x y - x + 1) / (2 x y / 8 12 8 11 7 12 8 10 7 11 8 9 - 8 x y + 12 x y + 2 x y - 8 x y - 6 x y + 2 x y 7 10 7 9 6 10 6 9 6 8 5 9 6 7 + 6 x y - 2 x y - 2 x y + 2 x y + 4 x y - 2 x y - 6 x y 5 8 6 6 5 7 5 6 4 7 5 5 4 6 + 2 x y + 2 x y - x y + 2 x y + 3 x y - x y - 4 x y 4 5 4 4 3 4 3 2 2 2 2 + 3 x y - 2 x y + 2 x y - x y - x y + 2 x y + x - 1) and in Maple notation -(x^8*y^14-5*x^8*y^13+10*x^8*y^12+2*x^7*y^13-10*x^8*y^11-10*x^7*y^12+5*x^8*y^10 +20*x^7*y^11+2*x^6*y^12-x^8*y^9-20*x^7*y^10-8*x^6*y^11+10*x^7*y^9+11*x^6*y^10-2 *x^7*y^8-4*x^6*y^9+x^5*y^10-4*x^6*y^8-5*x^5*y^9+4*x^6*y^7+9*x^5*y^8-x^6*y^6-6*x ^5*y^7-4*x^4*y^8+5*x^4*y^7+x^5*y^5+4*x^4*y^6-9*x^4*y^5-3*x^3*y^6+4*x^4*y^4+8*x^ 3*y^5-6*x^3*y^4+3*x^2*y^4+x^3*y^2-2*x^2*y^2+x*y^2-x+1)/(2*x^8*y^13-8*x^8*y^12+ 12*x^8*y^11+2*x^7*y^12-8*x^8*y^10-6*x^7*y^11+2*x^8*y^9+6*x^7*y^10-2*x^7*y^9-2*x ^6*y^10+2*x^6*y^9+4*x^6*y^8-2*x^5*y^9-6*x^6*y^7+2*x^5*y^8+2*x^6*y^6-x^5*y^7+2*x ^5*y^6+3*x^4*y^7-x^5*y^5-4*x^4*y^6+3*x^4*y^5-2*x^4*y^4+2*x^3*y^4-x^3*y^2-x^2*y^ 2+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, 5 4 3 2 2 (a + a + 3 a - 6 a + 3) ----------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.579069555 BTW the ratio for words with, 500, letters is, 1.577367857 ------------------------------------------------ "Theorem Number 28" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 12 7 13 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 11 7 12 8 10 7 11 6 12 7 10 6 11 - 4 x y + 3 x y + x y - 2 x y + x y - 2 x y - 2 x y 7 9 7 8 6 9 5 10 6 8 5 9 5 8 5 7 + 3 x y - x y + 2 x y - 2 x y - x y + 7 x y - 7 x y + x y 4 8 5 6 4 6 3 6 4 4 3 4 2 4 2 3 - x y + x y + 2 x y + x y - x y - x y - 3 x y + 2 x y 2 2 2 / 6 10 6 9 6 8 5 9 5 8 + x y - x y + x y - 1) / (x y - 2 x y + x y - x y + x y / 4 8 4 7 4 6 4 5 3 6 4 4 3 5 3 4 + x y + x y - x y + x y - 2 x y - x y + 2 x y - 2 x y 2 3 2 2 2 - x y + x y + 2 x y + x y - 1) and in Maple notation (x^8*y^14-4*x^8*y^13+6*x^8*y^12-x^7*y^13-4*x^8*y^11+3*x^7*y^12+x^8*y^10-2*x^7*y ^11+x^6*y^12-2*x^7*y^10-2*x^6*y^11+3*x^7*y^9-x^7*y^8+2*x^6*y^9-2*x^5*y^10-x^6*y ^8+7*x^5*y^9-7*x^5*y^8+x^5*y^7-x^4*y^8+x^5*y^6+2*x^4*y^6+x^3*y^6-x^4*y^4-x^3*y^ 4-3*x^2*y^4+2*x^2*y^3+x^2*y^2-x*y^2+x*y-1)/(x^6*y^10-2*x^6*y^9+x^6*y^8-x^5*y^9+ x^5*y^8+x^4*y^8+x^4*y^7-x^4*y^6+x^4*y^5-2*x^3*y^6-x^4*y^4+2*x^3*y^5-2*x^3*y^4-x ^2*y^3+x^2*y^2+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, 5 4 2 2 (a - 1) (a - 3 a + 9 a - 3 a - 3) -------------------------------------- 3 2 4 a - 6 a + 3 4 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.574902525 BTW the ratio for words with, 500, letters is, 1.573362199 ------------------------------------------------ "Theorem Number 29" 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, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 8 14 8 13 8 12 7 13 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 11 7 12 8 10 7 11 6 12 7 10 6 11 - 4 x y + 3 x y + x y - 2 x y + x y - 2 x y - 2 x y 7 9 7 8 6 9 5 10 6 8 5 9 5 8 5 7 + 3 x y - x y + 2 x y - 2 x y - x y + 7 x y - 7 x y + x y 4 8 5 6 4 6 3 6 4 4 3 4 2 4 2 3 - x y + x y + 2 x y + x y - x y - x y - 3 x y + 2 x y 2 2 2 / 6 10 6 9 6 8 5 9 5 8 + x y - x y + x y - 1) / (x y - 2 x y + x y - x y + x y / 4 8 4 7 4 6 4 5 3 6 4 4 3 5 3 4 + x y + x y - x y + x y - 2 x y - x y + 2 x y - 2 x y 2 3 2 2 2 - x y + x y + 2 x y + x y - 1) and in Maple notation (x^8*y^14-4*x^8*y^13+6*x^8*y^12-x^7*y^13-4*x^8*y^11+3*x^7*y^12+x^8*y^10-2*x^7*y ^11+x^6*y^12-2*x^7*y^10-2*x^6*y^11+3*x^7*y^9-x^7*y^8+2*x^6*y^9-2*x^5*y^10-x^6*y ^8+7*x^5*y^9-7*x^5*y^8+x^5*y^7-x^4*y^8+x^5*y^6+2*x^4*y^6+x^3*y^6-x^4*y^4-x^3*y^ 4-3*x^2*y^4+2*x^2*y^3+x^2*y^2-x*y^2+x*y-1)/(x^6*y^10-2*x^6*y^9+x^6*y^8-x^5*y^9+ x^5*y^8+x^4*y^8+x^4*y^7-x^4*y^6+x^4*y^5-2*x^3*y^6-x^4*y^4+2*x^3*y^5-2*x^3*y^4-x ^2*y^3+x^2*y^2+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, 5 4 2 2 (a - 1) (a - 3 a + 9 a - 3 a - 3) -------------------------------------- 3 2 4 a - 6 a + 3 4 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.574902525 BTW the ratio for words with, 500, letters is, 1.573362199 ------------------------------------------------ "Theorem Number 30" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 22 12 21 12 20 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 19 11 20 12 18 11 19 12 17 - 70 x y - 5 x y + 70 x y + 30 x y - 42 x y 11 18 12 16 11 17 10 18 12 15 11 16 - 75 x y + 14 x y + 100 x y + x y - 2 x y - 75 x y 10 17 9 18 11 15 10 16 9 17 11 14 - 4 x y - x y + 30 x y + 5 x y + 4 x y - 5 x y 9 16 10 14 9 15 8 16 10 13 9 14 - 4 x y - 5 x y - 8 x y + 2 x y + 4 x y + 26 x y 8 15 10 12 9 13 8 14 9 12 8 13 - 8 x y - x y - 28 x y + 14 x y + 12 x y - 16 x y 7 14 8 12 7 13 9 10 8 11 7 12 8 10 - x y + 14 x y + 6 x y - x y - 8 x y - 22 x y + 2 x y 7 11 6 12 7 10 7 9 6 10 7 8 6 9 + 38 x y + x y - 27 x y + 4 x y - 5 x y + 2 x y + 6 x y 5 10 6 8 5 9 5 8 5 7 4 8 5 6 + x y - 2 x y - 8 x y + 12 x y - 4 x y + 3 x y - x y 4 7 4 6 3 6 4 4 3 5 3 4 2 4 - 2 x y - 2 x y - 2 x y + x y + 4 x y - 2 x y + 3 x y 2 2 2 / 11 18 11 17 11 16 - 2 x y + x y + 1) / (2 x y - 10 x y + 20 x y / 11 15 10 16 11 14 10 15 11 13 - 20 x y - 5 x y + 10 x y + 20 x y - 2 x y 10 14 9 15 10 13 9 14 10 12 9 13 - 30 x y + 4 x y + 20 x y - 14 x y - 5 x y + 18 x y 8 14 9 12 8 13 9 11 8 12 8 11 + x y - 10 x y - 12 x y + 2 x y + 29 x y - 28 x y 8 10 6 11 8 8 6 10 6 9 5 10 6 8 + 11 x y - 2 x y - x y + 3 x y + 4 x y - x y - 9 x y 5 9 6 7 5 8 6 6 5 7 4 8 5 6 + 6 x y + 2 x y - 7 x y + 2 x y + 4 x y + x y - x y 4 6 4 5 3 6 3 5 3 4 2 2 2 + 2 x y - 2 x y + x y - 4 x y + 2 x y - 2 x y - 2 x y + 1) and in Maple notation (2*x^12*y^22-14*x^12*y^21+42*x^12*y^20-70*x^12*y^19-5*x^11*y^20+70*x^12*y^18+30 *x^11*y^19-42*x^12*y^17-75*x^11*y^18+14*x^12*y^16+100*x^11*y^17+x^10*y^18-2*x^ 12*y^15-75*x^11*y^16-4*x^10*y^17-x^9*y^18+30*x^11*y^15+5*x^10*y^16+4*x^9*y^17-5 *x^11*y^14-4*x^9*y^16-5*x^10*y^14-8*x^9*y^15+2*x^8*y^16+4*x^10*y^13+26*x^9*y^14 -8*x^8*y^15-x^10*y^12-28*x^9*y^13+14*x^8*y^14+12*x^9*y^12-16*x^8*y^13-x^7*y^14+ 14*x^8*y^12+6*x^7*y^13-x^9*y^10-8*x^8*y^11-22*x^7*y^12+2*x^8*y^10+38*x^7*y^11+x ^6*y^12-27*x^7*y^10+4*x^7*y^9-5*x^6*y^10+2*x^7*y^8+6*x^6*y^9+x^5*y^10-2*x^6*y^8 -8*x^5*y^9+12*x^5*y^8-4*x^5*y^7+3*x^4*y^8-x^5*y^6-2*x^4*y^7-2*x^4*y^6-2*x^3*y^6 +x^4*y^4+4*x^3*y^5-2*x^3*y^4+3*x^2*y^4-2*x^2*y^2+x*y^2+1)/(2*x^11*y^18-10*x^11* y^17+20*x^11*y^16-20*x^11*y^15-5*x^10*y^16+10*x^11*y^14+20*x^10*y^15-2*x^11*y^ 13-30*x^10*y^14+4*x^9*y^15+20*x^10*y^13-14*x^9*y^14-5*x^10*y^12+18*x^9*y^13+x^8 *y^14-10*x^9*y^12-12*x^8*y^13+2*x^9*y^11+29*x^8*y^12-28*x^8*y^11+11*x^8*y^10-2* x^6*y^11-x^8*y^8+3*x^6*y^10+4*x^6*y^9-x^5*y^10-9*x^6*y^8+6*x^5*y^9+2*x^6*y^7-7* x^5*y^8+2*x^6*y^6+4*x^5*y^7+x^4*y^8-x^5*y^6+2*x^4*y^6-2*x^4*y^5+x^3*y^6-4*x^3*y ^5+2*x^3*y^4-2*x^2*y^2-2*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, 8 7 6 5 4 3 2 2 (a + 2 a - a - 6 a - 12 a - 8 a - 3 a + 6 a + 3) - --------------------------------------------------------- 6 5 4 3 2 5 a + 9 a + 6 a - 3 a - 9 a - 6 a - 2 5 4 3 2 where a is the root of the polynomial, x + x - x - 2 x - 2 x + 1, and in decimals this is, 1.573198359 BTW the ratio for words with, 500, letters is, 1.571760554 ------------------------------------------------ "Theorem Number 31" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 12 19 11 20 12 18 11 19 12 17 - 70 x y + 70 x y - x y - 42 x y + 6 x y + 14 x y 11 18 10 19 12 16 11 17 10 18 - 15 x y - 2 x y - 2 x y + 20 x y + 10 x y 11 16 10 17 9 18 11 15 10 16 9 17 - 15 x y - 20 x y - x y + 6 x y + 20 x y + 2 x y 11 14 10 15 9 16 10 14 9 15 8 16 - x y - 10 x y + 4 x y + 2 x y - 16 x y + x y 9 14 8 15 9 13 8 14 9 12 7 14 + 19 x y - 4 x y - 10 x y + 5 x y + 2 x y - x y 8 12 7 13 8 11 7 12 8 10 7 11 6 12 - 5 x y + 2 x y + 4 x y + 2 x y - x y - 8 x y + 3 x y 7 10 6 11 7 9 6 10 6 9 6 8 5 9 + 7 x y + 2 x y - 2 x y - 18 x y + 18 x y - 5 x y - 6 x y 5 8 5 7 4 8 5 6 4 7 4 6 4 5 3 6 + 11 x y - 4 x y + x y - x y - 2 x y - x y + 2 x y - 7 x y 3 5 2 4 2 3 2 2 / 12 21 12 20 + 6 x y + 3 x y - 2 x y - x y + 1) / (2 x y - 10 x y / 12 19 11 20 12 18 11 19 12 17 + 20 x y - 4 x y - 20 x y + 16 x y + 10 x y 11 18 12 16 11 17 10 18 11 16 10 17 - 24 x y - 2 x y + 16 x y + 8 x y - 4 x y - 30 x y 10 16 9 17 10 15 9 16 10 14 9 15 + 42 x y - 2 x y - 26 x y - 3 x y + 6 x y + 24 x y 9 14 8 15 9 13 8 14 9 12 8 13 - 34 x y + 8 x y + 18 x y - 8 x y - 3 x y - 16 x y 8 12 7 13 8 11 7 12 7 11 6 12 + 24 x y - 10 x y - 8 x y + 13 x y + 4 x y + x y 7 10 6 11 6 10 6 9 5 10 6 8 5 9 - 7 x y + 6 x y - 16 x y + 10 x y - 3 x y - x y - 8 x y 5 8 5 7 4 8 5 6 4 7 4 6 4 5 + 19 x y - 8 x y + 3 x y - x y + 8 x y - 11 x y + 2 x y 3 6 3 4 2 4 2 3 2 2 2 - 2 x y + 3 x y + 3 x y - 2 x y - x y - 3 x y + 1) and in Maple notation (2*x^12*y^23-14*x^12*y^22+42*x^12*y^21-70*x^12*y^20+70*x^12*y^19-x^11*y^20-42*x ^12*y^18+6*x^11*y^19+14*x^12*y^17-15*x^11*y^18-2*x^10*y^19-2*x^12*y^16+20*x^11* y^17+10*x^10*y^18-15*x^11*y^16-20*x^10*y^17-x^9*y^18+6*x^11*y^15+20*x^10*y^16+2 *x^9*y^17-x^11*y^14-10*x^10*y^15+4*x^9*y^16+2*x^10*y^14-16*x^9*y^15+x^8*y^16+19 *x^9*y^14-4*x^8*y^15-10*x^9*y^13+5*x^8*y^14+2*x^9*y^12-x^7*y^14-5*x^8*y^12+2*x^ 7*y^13+4*x^8*y^11+2*x^7*y^12-x^8*y^10-8*x^7*y^11+3*x^6*y^12+7*x^7*y^10+2*x^6*y^ 11-2*x^7*y^9-18*x^6*y^10+18*x^6*y^9-5*x^6*y^8-6*x^5*y^9+11*x^5*y^8-4*x^5*y^7+x^ 4*y^8-x^5*y^6-2*x^4*y^7-x^4*y^6+2*x^4*y^5-7*x^3*y^6+6*x^3*y^5+3*x^2*y^4-2*x^2*y ^3-x^2*y^2+1)/(2*x^12*y^21-10*x^12*y^20+20*x^12*y^19-4*x^11*y^20-20*x^12*y^18+ 16*x^11*y^19+10*x^12*y^17-24*x^11*y^18-2*x^12*y^16+16*x^11*y^17+8*x^10*y^18-4*x ^11*y^16-30*x^10*y^17+42*x^10*y^16-2*x^9*y^17-26*x^10*y^15-3*x^9*y^16+6*x^10*y^ 14+24*x^9*y^15-34*x^9*y^14+8*x^8*y^15+18*x^9*y^13-8*x^8*y^14-3*x^9*y^12-16*x^8* y^13+24*x^8*y^12-10*x^7*y^13-8*x^8*y^11+13*x^7*y^12+4*x^7*y^11+x^6*y^12-7*x^7*y ^10+6*x^6*y^11-16*x^6*y^10+10*x^6*y^9-3*x^5*y^10-x^6*y^8-8*x^5*y^9+19*x^5*y^8-8 *x^5*y^7+3*x^4*y^8-x^5*y^6+8*x^4*y^7-11*x^4*y^6+2*x^4*y^5-2*x^3*y^6+3*x^3*y^4+3 *x^2*y^4-2*x^2*y^3-x^2*y^2-3*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 4 3 2 2 (a - 4 a - 10 a - 3 a + 6 a + 3) (5 a - 8 a - 3 a + 3) - --------------------------------------------------------------- 3 2 2 3 (4 a - 3 a - 4 a - 2) (a - 1) 5 4 3 where a is the root of the polynomial, x - 2 x - x + 3 x - 1, and in decimals this is, 1.573008002 BTW the ratio for words with, 500, letters is, 1.571472684 ------------------------------------------------ "Theorem Number 32" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 12 19 11 20 12 18 11 19 - 70 x y + 70 x y - 3 x y - 42 x y + 18 x y 12 17 11 18 10 19 12 16 11 17 + 14 x y - 45 x y + 3 x y - 2 x y + 60 x y 10 18 11 16 10 17 9 18 11 15 10 16 - 14 x y - 45 x y + 26 x y - x y + 18 x y - 25 x y 9 17 11 14 10 15 9 16 10 14 9 15 + 6 x y - 3 x y + 15 x y - 19 x y - 8 x y + 36 x y 10 13 9 14 8 15 10 12 9 13 8 14 + 4 x y - 39 x y - x y - x y + 22 x y + 3 x y 9 12 8 13 7 14 8 12 7 13 8 11 - 5 x y - 2 x y - 2 x y - 2 x y + 2 x y + 3 x y 7 12 8 10 7 11 6 12 7 10 6 11 7 9 + 2 x y - x y + 2 x y + x y - 8 x y - 7 x y + 4 x y 6 10 6 9 5 10 6 8 5 9 6 7 5 8 + 16 x y - 14 x y - x y + 3 x y - 4 x y + x y + 12 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 - 8 x y + 5 x y + x y - 10 x y + 7 x y - 2 x y + 3 x y 3 5 3 4 2 4 2 3 2 / 11 19 - 5 x y + 2 x y + 6 x y - 5 x y + 2 x y - x y + 1) / (2 x y / 11 18 11 17 10 18 11 16 10 17 - 10 x y + 20 x y - 2 x y - 20 x y + 8 x y 11 15 10 16 11 14 10 15 10 14 9 15 + 10 x y - 13 x y - 2 x y + 12 x y - 8 x y + 5 x y 10 13 9 14 10 12 9 13 8 14 9 12 + 4 x y - 15 x y - x y + 15 x y - 7 x y - 5 x y 8 13 8 12 7 13 8 11 7 12 8 10 7 11 + 18 x y - 16 x y + x y + 6 x y - 5 x y - x y + 9 x y 7 10 6 11 7 9 6 10 6 8 5 9 6 7 - 7 x y + 2 x y + 2 x y - 4 x y + x y + 3 x y + x y 5 8 5 6 4 7 4 5 3 6 3 5 3 4 - 3 x y + x y + 3 x y - 2 x y - x y - 2 x y + 2 x y 2 3 2 - 2 x y - x y - x y + 1) and in Maple notation (2*x^12*y^23-14*x^12*y^22+42*x^12*y^21-70*x^12*y^20+70*x^12*y^19-3*x^11*y^20-42 *x^12*y^18+18*x^11*y^19+14*x^12*y^17-45*x^11*y^18+3*x^10*y^19-2*x^12*y^16+60*x^ 11*y^17-14*x^10*y^18-45*x^11*y^16+26*x^10*y^17-x^9*y^18+18*x^11*y^15-25*x^10*y^ 16+6*x^9*y^17-3*x^11*y^14+15*x^10*y^15-19*x^9*y^16-8*x^10*y^14+36*x^9*y^15+4*x^ 10*y^13-39*x^9*y^14-x^8*y^15-x^10*y^12+22*x^9*y^13+3*x^8*y^14-5*x^9*y^12-2*x^8* y^13-2*x^7*y^14-2*x^8*y^12+2*x^7*y^13+3*x^8*y^11+2*x^7*y^12-x^8*y^10+2*x^7*y^11 +x^6*y^12-8*x^7*y^10-7*x^6*y^11+4*x^7*y^9+16*x^6*y^10-14*x^6*y^9-x^5*y^10+3*x^6 *y^8-4*x^5*y^9+x^6*y^7+12*x^5*y^8-8*x^5*y^7+5*x^4*y^8+x^5*y^6-10*x^4*y^7+7*x^4* y^6-2*x^4*y^5+3*x^3*y^6-5*x^3*y^5+2*x^3*y^4+6*x^2*y^4-5*x^2*y^3+2*x*y^2-x*y+1)/ (2*x^11*y^19-10*x^11*y^18+20*x^11*y^17-2*x^10*y^18-20*x^11*y^16+8*x^10*y^17+10* x^11*y^15-13*x^10*y^16-2*x^11*y^14+12*x^10*y^15-8*x^10*y^14+5*x^9*y^15+4*x^10*y ^13-15*x^9*y^14-x^10*y^12+15*x^9*y^13-7*x^8*y^14-5*x^9*y^12+18*x^8*y^13-16*x^8* y^12+x^7*y^13+6*x^8*y^11-5*x^7*y^12-x^8*y^10+9*x^7*y^11-7*x^7*y^10+2*x^6*y^11+2 *x^7*y^9-4*x^6*y^10+x^6*y^8+3*x^5*y^9+x^6*y^7-3*x^5*y^8+x^5*y^6+3*x^4*y^7-2*x^4 *y^5-x^3*y^6-2*x^3*y^5+2*x^3*y^4-2*x^2*y^3-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, 8 7 6 5 4 3 2 2 (a + 3 a + a - 5 a - 12 a - 9 a - 3 a + 6 a + 3) - --------------------------------------------------------- 6 5 4 3 2 5 a + 9 a + 6 a - 3 a - 9 a - 6 a - 2 5 4 3 2 where a is the root of the polynomial, x + x - x - 2 x - 2 x + 1, and in decimals this is, 1.559858032 BTW the ratio for words with, 500, letters is, 1.558495775 ------------------------------------------------ "Theorem Number 33" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (2 x y - 14 x y + 42 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 12 19 11 20 12 18 11 19 - 70 x y + 70 x y - 3 x y - 42 x y + 18 x y 12 17 11 18 10 19 12 16 11 17 + 14 x y - 45 x y + 3 x y - 2 x y + 60 x y 10 18 11 16 10 17 9 18 11 15 10 16 - 14 x y - 45 x y + 26 x y - x y + 18 x y - 25 x y 9 17 11 14 10 15 9 16 10 14 9 15 + 6 x y - 3 x y + 15 x y - 19 x y - 8 x y + 36 x y 10 13 9 14 8 15 10 12 9 13 8 14 + 4 x y - 39 x y - x y - x y + 22 x y + 3 x y 9 12 8 13 7 14 8 12 7 13 8 11 - 5 x y - 2 x y - 2 x y - 2 x y + 2 x y + 3 x y 7 12 8 10 7 11 6 12 7 10 6 11 7 9 + 2 x y - x y + 2 x y + x y - 8 x y - 7 x y + 4 x y 6 10 6 9 5 10 6 8 5 9 6 7 5 8 + 16 x y - 14 x y - x y + 3 x y - 4 x y + x y + 12 x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 - 8 x y + 5 x y + x y - 10 x y + 7 x y - 2 x y + 3 x y 3 5 3 4 2 4 2 3 2 / 11 19 - 5 x y + 2 x y + 6 x y - 5 x y + 2 x y - x y + 1) / (2 x y / 11 18 11 17 10 18 11 16 10 17 - 10 x y + 20 x y - 2 x y - 20 x y + 8 x y 11 15 10 16 11 14 10 15 10 14 9 15 + 10 x y - 13 x y - 2 x y + 12 x y - 8 x y + 5 x y 10 13 9 14 10 12 9 13 8 14 9 12 + 4 x y - 15 x y - x y + 15 x y - 7 x y - 5 x y 8 13 8 12 7 13 8 11 7 12 8 10 7 11 + 18 x y - 16 x y + x y + 6 x y - 5 x y - x y + 9 x y 7 10 6 11 7 9 6 10 6 8 5 9 6 7 - 7 x y + 2 x y + 2 x y - 4 x y + x y + 3 x y + x y 5 8 5 6 4 7 4 5 3 6 3 5 3 4 - 3 x y + x y + 3 x y - 2 x y - x y - 2 x y + 2 x y 2 3 2 - 2 x y - x y - x y + 1) and in Maple notation (2*x^12*y^23-14*x^12*y^22+42*x^12*y^21-70*x^12*y^20+70*x^12*y^19-3*x^11*y^20-42 *x^12*y^18+18*x^11*y^19+14*x^12*y^17-45*x^11*y^18+3*x^10*y^19-2*x^12*y^16+60*x^ 11*y^17-14*x^10*y^18-45*x^11*y^16+26*x^10*y^17-x^9*y^18+18*x^11*y^15-25*x^10*y^ 16+6*x^9*y^17-3*x^11*y^14+15*x^10*y^15-19*x^9*y^16-8*x^10*y^14+36*x^9*y^15+4*x^ 10*y^13-39*x^9*y^14-x^8*y^15-x^10*y^12+22*x^9*y^13+3*x^8*y^14-5*x^9*y^12-2*x^8* y^13-2*x^7*y^14-2*x^8*y^12+2*x^7*y^13+3*x^8*y^11+2*x^7*y^12-x^8*y^10+2*x^7*y^11 +x^6*y^12-8*x^7*y^10-7*x^6*y^11+4*x^7*y^9+16*x^6*y^10-14*x^6*y^9-x^5*y^10+3*x^6 *y^8-4*x^5*y^9+x^6*y^7+12*x^5*y^8-8*x^5*y^7+5*x^4*y^8+x^5*y^6-10*x^4*y^7+7*x^4* y^6-2*x^4*y^5+3*x^3*y^6-5*x^3*y^5+2*x^3*y^4+6*x^2*y^4-5*x^2*y^3+2*x*y^2-x*y+1)/ (2*x^11*y^19-10*x^11*y^18+20*x^11*y^17-2*x^10*y^18-20*x^11*y^16+8*x^10*y^17+10* x^11*y^15-13*x^10*y^16-2*x^11*y^14+12*x^10*y^15-8*x^10*y^14+5*x^9*y^15+4*x^10*y ^13-15*x^9*y^14-x^10*y^12+15*x^9*y^13-7*x^8*y^14-5*x^9*y^12+18*x^8*y^13-16*x^8* y^12+x^7*y^13+6*x^8*y^11-5*x^7*y^12-x^8*y^10+9*x^7*y^11-7*x^7*y^10+2*x^6*y^11+2 *x^7*y^9-4*x^6*y^10+x^6*y^8+3*x^5*y^9+x^6*y^7-3*x^5*y^8+x^5*y^6+3*x^4*y^7-2*x^4 *y^5-x^3*y^6-2*x^3*y^5+2*x^3*y^4-2*x^2*y^3-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, 8 7 6 5 4 3 2 2 (a + 3 a + a - 5 a - 12 a - 9 a - 3 a + 6 a + 3) - --------------------------------------------------------- 6 5 4 3 2 5 a + 9 a + 6 a - 3 a - 9 a - 6 a - 2 5 4 3 2 where a is the root of the polynomial, x + x - x - 2 x - 2 x + 1, and in decimals this is, 1.559858032 BTW the ratio for words with, 500, letters is, 1.558495775 ------------------------------------------------ "Theorem Number 34" 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, 2, 3]}, and h\ aving m neighbors (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 | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 7 4 5 3 6 4 4 3 5 3 4 3 3 2 4 - 3 x y + 2 x y - x y - x y + 2 x y - 2 x y + x y - 4 x y 2 3 2 / 6 10 6 9 6 8 5 7 5 6 + 3 x y - x y + x y - 1) / (x y - 2 x y + x y + x y - x y / 4 6 4 4 3 5 3 4 3 3 2 4 2 - x y + x y - x y + x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(x^5*y^10-2*x^5*y^9+x^5*y^8+2*x^4*y^8-3*x^4*y^7+2*x^4*y^5-x^3*y^6-x^4*y^4+2*x^ 3*y^5-2*x^3*y^4+x^3*y^3-4*x^2*y^4+3*x^2*y^3-x*y^2+x*y-1)/(x^6*y^10-2*x^6*y^9+x^ 6*y^8+x^5*y^7-x^5*y^6-x^4*y^6+x^4*y^4-x^3*y^5+x^3*y^4-x^3*y^3+x^2*y^4-2*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, 2 2 (5 a - 3) - ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.556526695 BTW the ratio for words with, 500, letters is, 1.555215246 ------------------------------------------------ "Theorem Number 35" 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, 2, 1]}, and h\ aving m neighbors (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 | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 7 4 5 3 6 4 4 3 5 3 4 3 3 2 4 - 3 x y + 2 x y - x y - x y + 2 x y - 2 x y + x y - 4 x y 2 3 2 / 6 10 6 9 6 8 5 7 5 6 + 3 x y - x y + x y - 1) / (x y - 2 x y + x y + x y - x y / 4 6 4 4 3 5 3 4 3 3 2 4 2 - x y + x y - x y + x y - x y + x y - 2 x y - x y + 1) and in Maple notation -(x^5*y^10-2*x^5*y^9+x^5*y^8+2*x^4*y^8-3*x^4*y^7+2*x^4*y^5-x^3*y^6-x^4*y^4+2*x^ 3*y^5-2*x^3*y^4+x^3*y^3-4*x^2*y^4+3*x^2*y^3-x*y^2+x*y-1)/(x^6*y^10-2*x^6*y^9+x^ 6*y^8+x^5*y^7-x^5*y^6-x^4*y^6+x^4*y^4-x^3*y^5+x^3*y^4-x^3*y^3+x^2*y^4-2*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, 2 2 (5 a - 3) - ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.556526695 BTW the ratio for words with, 500, letters is, 1.555215246 ------------------------------------------------ "Theorem Number 36" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (2 x y - 4 x y + 4 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 7 4 6 4 5 3 6 4 4 3 5 - 2 x y + 6 x y - 4 x y + 2 x y - x y - x y + 3 x y 3 4 3 3 2 4 2 3 2 / - 3 x y + x y - 4 x y + 4 x y - x y + x y - 1) / ( / 4 6 4 4 3 5 3 4 3 3 2 4 2 3 2 x y - x y - x y - 2 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+4*x^5*y^7-3*x^4*y^8-2*x^5*y^6+6*x^4*y^7-4*x^4*y^6+2*x^4*y ^5-x^3*y^6-x^4*y^4+3*x^3*y^5-3*x^3*y^4+x^3*y^3-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1) /(x^4*y^6-x^4*y^4-x^3*y^5-2*x^3*y^4+x^3*y^3-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, 4 3 2 2 (4 a + 2 a - 11 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.544658198 BTW the ratio for words with, 500, letters is, 1.543618826 ------------------------------------------------ "Theorem Number 37" 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, 1, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (2 x y - 4 x y + 4 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 6 4 4 2 4 2 3 2 2 2 - 2 x y + 2 x y - x y - 3 x y + 2 x y + x y - x y + x y - 1) / 4 8 4 7 4 6 3 6 4 4 3 5 3 4 / (x y - 2 x y + 2 x y - 2 x y - x y + 2 x y - 2 x y / 2 3 2 2 2 - x y + x y + 2 x y + x y - 1) and in Maple notation (2*x^5*y^10-4*x^5*y^9+4*x^5*y^7-x^4*y^8-2*x^5*y^6+2*x^4*y^6-x^4*y^4-3*x^2*y^4+2 *x^2*y^3+x^2*y^2-x*y^2+x*y-1)/(x^4*y^8-2*x^4*y^7+2*x^4*y^6-2*x^3*y^6-x^4*y^4+2* x^3*y^5-2*x^3*y^4-x^2*y^3+x^2*y^2+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 3 2 2 (4 a + 2 a - 11 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.544658198 BTW the ratio for words with, 500, letters is, 1.543618826 ------------------------------------------------ "Theorem Number 38" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (2 x y - 4 x y + 4 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 6 4 4 2 4 2 3 2 2 2 - 2 x y + 2 x y - x y - 3 x y + 2 x y + x y - x y + x y - 1) / 4 8 4 7 4 6 3 6 4 4 3 5 3 4 / (x y - 2 x y + 2 x y - 2 x y - x y + 2 x y - 2 x y / 2 3 2 2 2 - x y + x y + 2 x y + x y - 1) and in Maple notation (2*x^5*y^10-4*x^5*y^9+4*x^5*y^7-x^4*y^8-2*x^5*y^6+2*x^4*y^6-x^4*y^4-3*x^2*y^4+2 *x^2*y^3+x^2*y^2-x*y^2+x*y-1)/(x^4*y^8-2*x^4*y^7+2*x^4*y^6-2*x^3*y^6-x^4*y^4+2* x^3*y^5-2*x^3*y^4-x^2*y^3+x^2*y^2+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 3 2 2 (4 a + 2 a - 11 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.544658198 BTW the ratio for words with, 500, letters is, 1.543618826 ------------------------------------------------ "Theorem Number 39" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 7 4 8 ) | ) C(m, n) x y | = (2 x y - 4 x y + 4 x y - 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 6 4 7 4 6 4 5 3 6 4 4 3 5 - 2 x y + 6 x y - 4 x y + 2 x y - x y - x y + 3 x y 3 4 3 3 2 4 2 3 2 / - 3 x y + x y - 4 x y + 4 x y - x y + x y - 1) / ( / 4 6 4 4 3 5 3 4 3 3 2 4 2 3 2 x y - x y - x y - 2 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+4*x^5*y^7-3*x^4*y^8-2*x^5*y^6+6*x^4*y^7-4*x^4*y^6+2*x^4*y ^5-x^3*y^6-x^4*y^4+3*x^3*y^5-3*x^3*y^4+x^3*y^3-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1) /(x^4*y^6-x^4*y^4-x^3*y^5-2*x^3*y^4+x^3*y^3-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, 4 3 2 2 (4 a + 2 a - 11 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.544658198 BTW the ratio for words with, 500, letters is, 1.543618826 ------------------------------------------------ "Theorem Number 40" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (2 x y - 11 x y + 24 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 11 19 - 24 x y + 2 x y + 4 x y - 9 x y + 18 x y + 15 x y 12 17 11 18 12 16 11 17 10 18 12 15 - 24 x y - 9 x y + 16 x y - 5 x y + 2 x y - 6 x y 11 16 10 17 12 14 11 15 10 16 9 17 + 13 x y - 12 x y + x y - 11 x y + 30 x y + 3 x y 11 14 10 15 9 16 11 13 10 14 9 15 + 5 x y - 40 x y - 8 x y - x y + 30 x y + 2 x y 8 16 10 13 9 14 8 15 10 12 9 13 - 2 x y - 12 x y + 17 x y + 6 x y + 2 x y - 29 x y 8 14 9 12 8 13 7 14 9 11 8 12 - 15 x y + 22 x y + 29 x y - x y - 8 x y - 28 x y 7 13 9 10 8 11 7 12 8 10 8 9 7 10 + 3 x y + x y + 8 x y - 3 x y + 5 x y - 3 x y + 4 x y 7 9 6 10 7 8 6 9 5 10 7 7 6 8 - 6 x y + 4 x y + 4 x y - 7 x y - 4 x y - x y + 3 x y 5 9 6 7 5 8 6 6 5 7 4 8 5 6 + 10 x y - x y - 10 x y + x y + 5 x y + 3 x y - x y 4 7 4 6 4 5 3 6 4 4 3 5 3 4 3 3 + 4 x y - 7 x y + 2 x y + 2 x y - x y - x y - 2 x y + x y 2 4 2 3 2 / 13 24 13 23 13 22 - 4 x y + 4 x y - x y + x y - 1) / (x y - 4 x y + 4 x y / 13 21 13 20 12 21 13 19 12 20 13 18 + 4 x y - 10 x y + x y + 4 x y - 3 x y + 4 x y 12 19 13 17 12 18 13 16 12 17 12 16 + x y - 4 x y + 5 x y + x y - 5 x y - x y 10 18 12 15 10 17 12 14 10 16 9 17 + 2 x y + 3 x y - 5 x y - x y + 3 x y - 2 x y 10 15 9 16 10 14 9 15 10 13 9 14 + 2 x y + 4 x y - 4 x y - 3 x y + 3 x y - x y 8 15 10 12 9 13 8 14 9 12 8 13 9 11 + x y - x y + 8 x y - 2 x y - 10 x y + 3 x y + 5 x y 8 12 9 10 7 12 8 10 7 11 7 10 6 11 - 3 x y - x y - x y + x y + x y + 2 x y - 3 x y 7 9 6 10 7 8 6 9 5 10 7 7 6 8 5 9 - 2 x y + 5 x y - x y - 3 x y + x y + x y + x y + 3 x y 6 7 5 8 6 6 5 7 4 8 5 6 4 7 4 6 + x y + x y - x y - 2 x y - x y - 2 x y - x y - x y 4 5 4 4 3 5 3 4 3 3 2 4 2 3 2 + x y + x y + x y + 2 x y - x y + x y - x y - 2 x y - x y + 1) and in Maple notation -(2*x^12*y^23-11*x^12*y^22+24*x^12*y^21-24*x^12*y^20+2*x^11*y^21+4*x^12*y^19-9* x^11*y^20+18*x^12*y^18+15*x^11*y^19-24*x^12*y^17-9*x^11*y^18+16*x^12*y^16-5*x^ 11*y^17+2*x^10*y^18-6*x^12*y^15+13*x^11*y^16-12*x^10*y^17+x^12*y^14-11*x^11*y^ 15+30*x^10*y^16+3*x^9*y^17+5*x^11*y^14-40*x^10*y^15-8*x^9*y^16-x^11*y^13+30*x^ 10*y^14+2*x^9*y^15-2*x^8*y^16-12*x^10*y^13+17*x^9*y^14+6*x^8*y^15+2*x^10*y^12-\ 29*x^9*y^13-15*x^8*y^14+22*x^9*y^12+29*x^8*y^13-x^7*y^14-8*x^9*y^11-28*x^8*y^12 +3*x^7*y^13+x^9*y^10+8*x^8*y^11-3*x^7*y^12+5*x^8*y^10-3*x^8*y^9+4*x^7*y^10-6*x^ 7*y^9+4*x^6*y^10+4*x^7*y^8-7*x^6*y^9-4*x^5*y^10-x^7*y^7+3*x^6*y^8+10*x^5*y^9-x^ 6*y^7-10*x^5*y^8+x^6*y^6+5*x^5*y^7+3*x^4*y^8-x^5*y^6+4*x^4*y^7-7*x^4*y^6+2*x^4* y^5+2*x^3*y^6-x^4*y^4-x^3*y^5-2*x^3*y^4+x^3*y^3-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1 )/(x^13*y^24-4*x^13*y^23+4*x^13*y^22+4*x^13*y^21-10*x^13*y^20+x^12*y^21+4*x^13* y^19-3*x^12*y^20+4*x^13*y^18+x^12*y^19-4*x^13*y^17+5*x^12*y^18+x^13*y^16-5*x^12 *y^17-x^12*y^16+2*x^10*y^18+3*x^12*y^15-5*x^10*y^17-x^12*y^14+3*x^10*y^16-2*x^9 *y^17+2*x^10*y^15+4*x^9*y^16-4*x^10*y^14-3*x^9*y^15+3*x^10*y^13-x^9*y^14+x^8*y^ 15-x^10*y^12+8*x^9*y^13-2*x^8*y^14-10*x^9*y^12+3*x^8*y^13+5*x^9*y^11-3*x^8*y^12 -x^9*y^10-x^7*y^12+x^8*y^10+x^7*y^11+2*x^7*y^10-3*x^6*y^11-2*x^7*y^9+5*x^6*y^10 -x^7*y^8-3*x^6*y^9+x^5*y^10+x^7*y^7+x^6*y^8+3*x^5*y^9+x^6*y^7+x^5*y^8-x^6*y^6-2 *x^5*y^7-x^4*y^8-2*x^5*y^6-x^4*y^7-x^4*y^6+x^4*y^5+x^4*y^4+x^3*y^5+2*x^3*y^4-x^ 3*y^3+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, 4 3 2 3 2 2 (5 a - 4 a + 6 a - 3) (a + 3 a - 3 a - 3) ------------------------------------------------ 3 2 2 2 (a + a + a + 1) (3 a - 2 a + 3) 5 4 3 where a is the root of the polynomial, x - x + 2 x - 3 x + 1, and in decimals this is, 1.543785523 BTW the ratio for words with, 500, letters is, 1.542380192 ------------------------------------------------ "Theorem Number 41" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (2 x y - 11 x y + 24 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 11 19 - 24 x y + 2 x y + 4 x y - 9 x y + 18 x y + 15 x y 12 17 11 18 12 16 11 17 10 18 12 15 - 24 x y - 9 x y + 16 x y - 5 x y + 2 x y - 6 x y 11 16 10 17 12 14 11 15 10 16 9 17 + 13 x y - 12 x y + x y - 11 x y + 30 x y + 3 x y 11 14 10 15 9 16 11 13 10 14 9 15 + 5 x y - 40 x y - 8 x y - x y + 30 x y + 2 x y 8 16 10 13 9 14 8 15 10 12 9 13 - 2 x y - 12 x y + 17 x y + 6 x y + 2 x y - 29 x y 8 14 9 12 8 13 7 14 9 11 8 12 - 15 x y + 22 x y + 29 x y - x y - 8 x y - 28 x y 7 13 9 10 8 11 7 12 8 10 8 9 7 10 + 3 x y + x y + 8 x y - 3 x y + 5 x y - 3 x y + 4 x y 7 9 6 10 7 8 6 9 5 10 7 7 6 8 - 6 x y + 4 x y + 4 x y - 7 x y - 4 x y - x y + 3 x y 5 9 6 7 5 8 6 6 5 7 4 8 5 6 + 10 x y - x y - 10 x y + x y + 5 x y + 3 x y - x y 4 7 4 6 4 5 3 6 4 4 3 5 3 4 3 3 + 4 x y - 7 x y + 2 x y + 2 x y - x y - x y - 2 x y + x y 2 4 2 3 2 / 13 24 13 23 13 22 - 4 x y + 4 x y - x y + x y - 1) / (x y - 4 x y + 4 x y / 13 21 13 20 12 21 13 19 12 20 13 18 + 4 x y - 10 x y + x y + 4 x y - 3 x y + 4 x y 12 19 13 17 12 18 13 16 12 17 12 16 + x y - 4 x y + 5 x y + x y - 5 x y - x y 10 18 12 15 10 17 12 14 10 16 9 17 + 2 x y + 3 x y - 5 x y - x y + 3 x y - 2 x y 10 15 9 16 10 14 9 15 10 13 9 14 + 2 x y + 4 x y - 4 x y - 3 x y + 3 x y - x y 8 15 10 12 9 13 8 14 9 12 8 13 9 11 + x y - x y + 8 x y - 2 x y - 10 x y + 3 x y + 5 x y 8 12 9 10 7 12 8 10 7 11 7 10 6 11 - 3 x y - x y - x y + x y + x y + 2 x y - 3 x y 7 9 6 10 7 8 6 9 5 10 7 7 6 8 5 9 - 2 x y + 5 x y - x y - 3 x y + x y + x y + x y + 3 x y 6 7 5 8 6 6 5 7 4 8 5 6 4 7 4 6 + x y + x y - x y - 2 x y - x y - 2 x y - x y - x y 4 5 4 4 3 5 3 4 3 3 2 4 2 3 2 + x y + x y + x y + 2 x y - x y + x y - x y - 2 x y - x y + 1) and in Maple notation -(2*x^12*y^23-11*x^12*y^22+24*x^12*y^21-24*x^12*y^20+2*x^11*y^21+4*x^12*y^19-9* x^11*y^20+18*x^12*y^18+15*x^11*y^19-24*x^12*y^17-9*x^11*y^18+16*x^12*y^16-5*x^ 11*y^17+2*x^10*y^18-6*x^12*y^15+13*x^11*y^16-12*x^10*y^17+x^12*y^14-11*x^11*y^ 15+30*x^10*y^16+3*x^9*y^17+5*x^11*y^14-40*x^10*y^15-8*x^9*y^16-x^11*y^13+30*x^ 10*y^14+2*x^9*y^15-2*x^8*y^16-12*x^10*y^13+17*x^9*y^14+6*x^8*y^15+2*x^10*y^12-\ 29*x^9*y^13-15*x^8*y^14+22*x^9*y^12+29*x^8*y^13-x^7*y^14-8*x^9*y^11-28*x^8*y^12 +3*x^7*y^13+x^9*y^10+8*x^8*y^11-3*x^7*y^12+5*x^8*y^10-3*x^8*y^9+4*x^7*y^10-6*x^ 7*y^9+4*x^6*y^10+4*x^7*y^8-7*x^6*y^9-4*x^5*y^10-x^7*y^7+3*x^6*y^8+10*x^5*y^9-x^ 6*y^7-10*x^5*y^8+x^6*y^6+5*x^5*y^7+3*x^4*y^8-x^5*y^6+4*x^4*y^7-7*x^4*y^6+2*x^4* y^5+2*x^3*y^6-x^4*y^4-x^3*y^5-2*x^3*y^4+x^3*y^3-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1 )/(x^13*y^24-4*x^13*y^23+4*x^13*y^22+4*x^13*y^21-10*x^13*y^20+x^12*y^21+4*x^13* y^19-3*x^12*y^20+4*x^13*y^18+x^12*y^19-4*x^13*y^17+5*x^12*y^18+x^13*y^16-5*x^12 *y^17-x^12*y^16+2*x^10*y^18+3*x^12*y^15-5*x^10*y^17-x^12*y^14+3*x^10*y^16-2*x^9 *y^17+2*x^10*y^15+4*x^9*y^16-4*x^10*y^14-3*x^9*y^15+3*x^10*y^13-x^9*y^14+x^8*y^ 15-x^10*y^12+8*x^9*y^13-2*x^8*y^14-10*x^9*y^12+3*x^8*y^13+5*x^9*y^11-3*x^8*y^12 -x^9*y^10-x^7*y^12+x^8*y^10+x^7*y^11+2*x^7*y^10-3*x^6*y^11-2*x^7*y^9+5*x^6*y^10 -x^7*y^8-3*x^6*y^9+x^5*y^10+x^7*y^7+x^6*y^8+3*x^5*y^9+x^6*y^7+x^5*y^8-x^6*y^6-2 *x^5*y^7-x^4*y^8-2*x^5*y^6-x^4*y^7-x^4*y^6+x^4*y^5+x^4*y^4+x^3*y^5+2*x^3*y^4-x^ 3*y^3+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, 4 3 2 3 2 2 (5 a - 4 a + 6 a - 3) (a + 3 a - 3 a - 3) ------------------------------------------------ 3 2 2 2 (a + a + a + 1) (3 a - 2 a + 3) 5 4 3 where a is the root of the polynomial, x - x + 2 x - 3 x + 1, and in decimals this is, 1.543785523 BTW the ratio for words with, 500, letters is, 1.542380192 ------------------------------------------------ "Theorem Number 42" 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, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 11 20 12 18 11 19 12 17 - 14 x y - x y + 4 x y + 14 x y - 4 x y - 14 x y 11 18 10 19 12 16 11 17 10 18 12 15 - 4 x y - x y + 6 x y + 10 x y + 8 x y - x y 11 16 10 17 11 15 10 16 9 17 11 14 - 4 x y - 25 x y - 4 x y + 40 x y + 3 x y + 4 x y 10 15 9 16 11 13 10 14 9 15 8 16 - 35 x y - 16 x y - x y + 16 x y + 36 x y - x y 10 13 9 14 8 15 9 13 8 14 9 12 - 3 x y - 43 x y + x y + 27 x y + 11 x y - 6 x y 8 13 7 14 9 11 8 12 7 13 9 10 8 11 - 34 x y + x y - 2 x y + 39 x y - x y + x y - 17 x y 7 12 8 10 7 11 6 12 8 9 7 10 6 11 - 5 x y - x y + 8 x y - x y + 2 x y + x y + 2 x y 7 9 6 10 7 8 6 9 5 10 6 8 5 9 - 7 x y - 3 x y + 3 x y + 5 x y + 2 x y - 4 x y - 9 x y 6 7 5 8 5 7 4 8 5 6 4 6 4 5 + x y + 14 x y - 9 x y - x y + 2 x y + 2 x y - 2 x y 3 6 4 4 3 5 3 4 2 4 2 3 2 2 2 - 3 x y + x y + 4 x y - x y + 3 x y - 2 x y - x y + x y / 10 18 10 17 10 16 9 17 9 16 - x y + 1) / (x y - 4 x y + 5 x y - x y + x y / 10 14 9 15 10 13 9 14 8 15 10 12 - 5 x y + 4 x y + 4 x y - 7 x y + 2 x y - x y 9 13 8 14 9 12 9 11 8 12 7 13 9 10 + 3 x y - 4 x y - x y + 2 x y + 6 x y - x y - x y 8 11 7 12 8 10 7 11 8 9 7 10 6 11 - 8 x y + 3 x y + 6 x y - x y - 2 x y - 6 x y - x y 7 9 6 10 7 8 6 9 6 8 5 9 6 7 + 8 x y - 2 x y - 3 x y + 5 x y - x y + 2 x y - x y 5 8 5 7 4 8 5 6 4 7 4 5 3 6 4 4 + 2 x y - 4 x y + x y + x y - 2 x y + 2 x y - 2 x y - x y 3 5 3 4 2 3 2 2 2 + 2 x y - 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation -(x^12*y^23-6*x^12*y^22+14*x^12*y^21-14*x^12*y^20-x^11*y^21+4*x^11*y^20+14*x^12 *y^18-4*x^11*y^19-14*x^12*y^17-4*x^11*y^18-x^10*y^19+6*x^12*y^16+10*x^11*y^17+8 *x^10*y^18-x^12*y^15-4*x^11*y^16-25*x^10*y^17-4*x^11*y^15+40*x^10*y^16+3*x^9*y^ 17+4*x^11*y^14-35*x^10*y^15-16*x^9*y^16-x^11*y^13+16*x^10*y^14+36*x^9*y^15-x^8* y^16-3*x^10*y^13-43*x^9*y^14+x^8*y^15+27*x^9*y^13+11*x^8*y^14-6*x^9*y^12-34*x^8 *y^13+x^7*y^14-2*x^9*y^11+39*x^8*y^12-x^7*y^13+x^9*y^10-17*x^8*y^11-5*x^7*y^12- x^8*y^10+8*x^7*y^11-x^6*y^12+2*x^8*y^9+x^7*y^10+2*x^6*y^11-7*x^7*y^9-3*x^6*y^10 +3*x^7*y^8+5*x^6*y^9+2*x^5*y^10-4*x^6*y^8-9*x^5*y^9+x^6*y^7+14*x^5*y^8-9*x^5*y^ 7-x^4*y^8+2*x^5*y^6+2*x^4*y^6-2*x^4*y^5-3*x^3*y^6+x^4*y^4+4*x^3*y^5-x^3*y^4+3*x ^2*y^4-2*x^2*y^3-x^2*y^2+x*y^2-x*y+1)/(x^10*y^18-4*x^10*y^17+5*x^10*y^16-x^9*y^ 17+x^9*y^16-5*x^10*y^14+4*x^9*y^15+4*x^10*y^13-7*x^9*y^14+2*x^8*y^15-x^10*y^12+ 3*x^9*y^13-4*x^8*y^14-x^9*y^12+2*x^9*y^11+6*x^8*y^12-x^7*y^13-x^9*y^10-8*x^8*y^ 11+3*x^7*y^12+6*x^8*y^10-x^7*y^11-2*x^8*y^9-6*x^7*y^10-x^6*y^11+8*x^7*y^9-2*x^6 *y^10-3*x^7*y^8+5*x^6*y^9-x^6*y^8+2*x^5*y^9-x^6*y^7+2*x^5*y^8-4*x^5*y^7+x^4*y^8 +x^5*y^6-2*x^4*y^7+2*x^4*y^5-2*x^3*y^6-x^4*y^4+2*x^3*y^5-2*x^3*y^4-x^2*y^3+x^2* y^2+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, 7 5 4 3 2 2 (a + 1) (a - 4 a + a + 12 a - 9 a - 3 a + 3) --------------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.533503217 BTW the ratio for words with, 500, letters is, 1.532223901 ------------------------------------------------ "Theorem Number 43" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 11 20 12 18 11 19 12 17 - 14 x y - x y + 4 x y + 14 x y - 4 x y - 14 x y 11 18 10 19 12 16 11 17 10 18 12 15 - 4 x y - x y + 6 x y + 10 x y + 8 x y - x y 11 16 10 17 11 15 10 16 9 17 11 14 - 4 x y - 25 x y - 4 x y + 40 x y + 3 x y + 4 x y 10 15 9 16 11 13 10 14 9 15 8 16 - 35 x y - 16 x y - x y + 16 x y + 36 x y - x y 10 13 9 14 8 15 9 13 8 14 9 12 - 3 x y - 43 x y + x y + 27 x y + 11 x y - 6 x y 8 13 7 14 9 11 8 12 7 13 9 10 8 11 - 34 x y + x y - 2 x y + 39 x y - x y + x y - 17 x y 7 12 8 10 7 11 6 12 8 9 7 10 6 11 - 5 x y - x y + 8 x y - x y + 2 x y + x y + 2 x y 7 9 6 10 7 8 6 9 5 10 6 8 5 9 - 7 x y - 3 x y + 3 x y + 5 x y + 2 x y - 4 x y - 9 x y 6 7 5 8 5 7 4 8 5 6 4 6 4 5 + x y + 14 x y - 9 x y - x y + 2 x y + 2 x y - 2 x y 3 6 4 4 3 5 3 4 2 4 2 3 2 2 2 - 3 x y + x y + 4 x y - x y + 3 x y - 2 x y - x y + x y / 10 18 10 17 10 16 9 17 9 16 - x y + 1) / (x y - 4 x y + 5 x y - x y + x y / 10 14 9 15 10 13 9 14 8 15 10 12 - 5 x y + 4 x y + 4 x y - 7 x y + 2 x y - x y 9 13 8 14 9 12 9 11 8 12 7 13 9 10 + 3 x y - 4 x y - x y + 2 x y + 6 x y - x y - x y 8 11 7 12 8 10 7 11 8 9 7 10 6 11 - 8 x y + 3 x y + 6 x y - x y - 2 x y - 6 x y - x y 7 9 6 10 7 8 6 9 6 8 5 9 6 7 + 8 x y - 2 x y - 3 x y + 5 x y - x y + 2 x y - x y 5 8 5 7 4 8 5 6 4 7 4 5 3 6 4 4 + 2 x y - 4 x y + x y + x y - 2 x y + 2 x y - 2 x y - x y 3 5 3 4 2 3 2 2 2 + 2 x y - 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation -(x^12*y^23-6*x^12*y^22+14*x^12*y^21-14*x^12*y^20-x^11*y^21+4*x^11*y^20+14*x^12 *y^18-4*x^11*y^19-14*x^12*y^17-4*x^11*y^18-x^10*y^19+6*x^12*y^16+10*x^11*y^17+8 *x^10*y^18-x^12*y^15-4*x^11*y^16-25*x^10*y^17-4*x^11*y^15+40*x^10*y^16+3*x^9*y^ 17+4*x^11*y^14-35*x^10*y^15-16*x^9*y^16-x^11*y^13+16*x^10*y^14+36*x^9*y^15-x^8* y^16-3*x^10*y^13-43*x^9*y^14+x^8*y^15+27*x^9*y^13+11*x^8*y^14-6*x^9*y^12-34*x^8 *y^13+x^7*y^14-2*x^9*y^11+39*x^8*y^12-x^7*y^13+x^9*y^10-17*x^8*y^11-5*x^7*y^12- x^8*y^10+8*x^7*y^11-x^6*y^12+2*x^8*y^9+x^7*y^10+2*x^6*y^11-7*x^7*y^9-3*x^6*y^10 +3*x^7*y^8+5*x^6*y^9+2*x^5*y^10-4*x^6*y^8-9*x^5*y^9+x^6*y^7+14*x^5*y^8-9*x^5*y^ 7-x^4*y^8+2*x^5*y^6+2*x^4*y^6-2*x^4*y^5-3*x^3*y^6+x^4*y^4+4*x^3*y^5-x^3*y^4+3*x ^2*y^4-2*x^2*y^3-x^2*y^2+x*y^2-x*y+1)/(x^10*y^18-4*x^10*y^17+5*x^10*y^16-x^9*y^ 17+x^9*y^16-5*x^10*y^14+4*x^9*y^15+4*x^10*y^13-7*x^9*y^14+2*x^8*y^15-x^10*y^12+ 3*x^9*y^13-4*x^8*y^14-x^9*y^12+2*x^9*y^11+6*x^8*y^12-x^7*y^13-x^9*y^10-8*x^8*y^ 11+3*x^7*y^12+6*x^8*y^10-x^7*y^11-2*x^8*y^9-6*x^7*y^10-x^6*y^11+8*x^7*y^9-2*x^6 *y^10-3*x^7*y^8+5*x^6*y^9-x^6*y^8+2*x^5*y^9-x^6*y^7+2*x^5*y^8-4*x^5*y^7+x^4*y^8 +x^5*y^6-2*x^4*y^7+2*x^4*y^5-2*x^3*y^6-x^4*y^4+2*x^3*y^5-2*x^3*y^4-x^2*y^3+x^2* y^2+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, 7 5 4 3 2 2 (a + 1) (a - 4 a + a + 12 a - 9 a - 3 a + 3) --------------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.533503217 BTW the ratio for words with, 500, letters is, 1.532223901 ------------------------------------------------ "Theorem Number 44" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = - (2 x y - 13 x y + 37 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 12 19 11 20 12 18 - 61 x y + 4 x y + 65 x y - 18 x y - 47 x y 11 19 12 17 11 18 12 16 11 17 + 29 x y + 23 x y - 14 x y - 7 x y - 15 x y 10 18 12 15 11 16 10 17 11 15 10 16 + 5 x y + x y + 26 x y - 26 x y - 17 x y + 57 x y 9 17 11 14 10 15 9 16 11 13 10 14 - 2 x y + 6 x y - 68 x y + 5 x y - x y + 47 x y 9 15 8 16 10 13 9 14 8 15 10 12 - 10 x y + x y - 18 x y + 26 x y - 2 x y + 3 x y 9 13 8 14 9 12 8 13 7 14 9 11 8 12 - 38 x y - x y + 25 x y + x y + 3 x y - 6 x y + 3 x y 7 13 8 11 8 10 6 12 8 9 7 10 - 3 x y + 2 x y - 7 x y + 5 x y + 3 x y - 10 x y 6 11 7 9 6 10 7 8 6 9 5 10 7 7 - 17 x y + 18 x y + 31 x y - 9 x y - 24 x y + 6 x y + x y 6 8 5 9 6 7 5 8 6 6 5 7 4 8 - 2 x y - 27 x y + 10 x y + 36 x y - 3 x y - 14 x y + 2 x y 5 6 4 7 5 5 4 6 4 5 3 6 4 4 - 4 x y - 12 x y + 3 x y + 13 x y - 2 x y - 3 x y - 2 x y 3 5 3 4 3 3 2 4 2 3 2 2 2 + 3 x y - x y + x y - 6 x y + 5 x y - x y - 2 x y + 2 x y - 1 / 13 23 13 22 13 21 13 20 12 21 ) / (2 x y - 12 x y + 30 x y - 40 x y + 4 x y / 13 19 12 20 13 18 12 19 13 17 + 30 x y - 18 x y - 12 x y + 30 x y + 2 x y 12 18 11 19 11 18 12 16 11 17 12 15 - 20 x y - 2 x y + 11 x y + 6 x y - 23 x y - 2 x y 11 16 10 17 11 15 10 16 9 17 11 14 + 22 x y - 4 x y - 8 x y + 9 x y + 2 x y - x y 9 16 11 13 10 14 9 15 10 13 9 14 - 8 x y + x y - 14 x y + 15 x y + 12 x y - 19 x y 8 15 10 12 9 13 8 14 9 12 8 13 + 4 x y - 3 x y + 15 x y - 11 x y - 5 x y + 11 x y 8 12 7 13 8 11 7 12 8 10 7 11 + x y + 2 x y - 11 x y - 5 x y + 6 x y + 3 x y 7 10 6 11 7 9 6 10 6 9 7 7 6 8 + 2 x y + 2 x y - x y - 5 x y + 4 x y - x y - 3 x y 5 9 6 7 5 8 6 6 5 7 4 8 5 6 4 7 + x y - x y - 3 x y + 3 x y + 6 x y - x y - 2 x y + x y 5 5 4 6 4 5 3 6 4 4 3 4 3 3 2 3 - 3 x y - 6 x y + 5 x y - x y + 2 x y - 2 x y - x y + x y 2 2 2 + x y - x y - 2 x y + 1) and in Maple notation -(2*x^12*y^23-13*x^12*y^22+37*x^12*y^21-61*x^12*y^20+4*x^11*y^21+65*x^12*y^19-\ 18*x^11*y^20-47*x^12*y^18+29*x^11*y^19+23*x^12*y^17-14*x^11*y^18-7*x^12*y^16-15 *x^11*y^17+5*x^10*y^18+x^12*y^15+26*x^11*y^16-26*x^10*y^17-17*x^11*y^15+57*x^10 *y^16-2*x^9*y^17+6*x^11*y^14-68*x^10*y^15+5*x^9*y^16-x^11*y^13+47*x^10*y^14-10* x^9*y^15+x^8*y^16-18*x^10*y^13+26*x^9*y^14-2*x^8*y^15+3*x^10*y^12-38*x^9*y^13-x ^8*y^14+25*x^9*y^12+x^8*y^13+3*x^7*y^14-6*x^9*y^11+3*x^8*y^12-3*x^7*y^13+2*x^8* y^11-7*x^8*y^10+5*x^6*y^12+3*x^8*y^9-10*x^7*y^10-17*x^6*y^11+18*x^7*y^9+31*x^6* y^10-9*x^7*y^8-24*x^6*y^9+6*x^5*y^10+x^7*y^7-2*x^6*y^8-27*x^5*y^9+10*x^6*y^7+36 *x^5*y^8-3*x^6*y^6-14*x^5*y^7+2*x^4*y^8-4*x^5*y^6-12*x^4*y^7+3*x^5*y^5+13*x^4*y ^6-2*x^4*y^5-3*x^3*y^6-2*x^4*y^4+3*x^3*y^5-x^3*y^4+x^3*y^3-6*x^2*y^4+5*x^2*y^3- x^2*y^2-2*x*y^2+2*x*y-1)/(2*x^13*y^23-12*x^13*y^22+30*x^13*y^21-40*x^13*y^20+4* x^12*y^21+30*x^13*y^19-18*x^12*y^20-12*x^13*y^18+30*x^12*y^19+2*x^13*y^17-20*x^ 12*y^18-2*x^11*y^19+11*x^11*y^18+6*x^12*y^16-23*x^11*y^17-2*x^12*y^15+22*x^11*y ^16-4*x^10*y^17-8*x^11*y^15+9*x^10*y^16+2*x^9*y^17-x^11*y^14-8*x^9*y^16+x^11*y^ 13-14*x^10*y^14+15*x^9*y^15+12*x^10*y^13-19*x^9*y^14+4*x^8*y^15-3*x^10*y^12+15* x^9*y^13-11*x^8*y^14-5*x^9*y^12+11*x^8*y^13+x^8*y^12+2*x^7*y^13-11*x^8*y^11-5*x ^7*y^12+6*x^8*y^10+3*x^7*y^11+2*x^7*y^10+2*x^6*y^11-x^7*y^9-5*x^6*y^10+4*x^6*y^ 9-x^7*y^7-3*x^6*y^8+x^5*y^9-x^6*y^7-3*x^5*y^8+3*x^6*y^6+6*x^5*y^7-x^4*y^8-2*x^5 *y^6+x^4*y^7-3*x^5*y^5-6*x^4*y^6+5*x^4*y^5-x^3*y^6+2*x^4*y^4-2*x^3*y^4-x^3*y^3+ x^2*y^3+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\ bors of a random word of length n tends to n times, 3 2 4 3 2 2 (2 a - 6 a + 3) (5 a - 4 a + 12 a - 4 a + 3) --------------------------------------------------- 2 2 2 2 (3 a - 2 a + 3) (a + 1) 5 4 3 2 where a is the root of the polynomial, x - x + 4 x - 2 x + 3 x - 1, and in decimals this is, 1.532514094 BTW the ratio for words with, 500, letters is, 1.531220744 ------------------------------------------------ "Theorem Number 45" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 22 12 21 12 20 ) | ) C(m, n) x y | = - (x y - 7 x y + 21 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 19 11 20 12 18 11 19 12 17 11 18 - 35 x y - x y + 35 x y + 6 x y - 21 x y - 15 x y 12 16 11 17 10 18 12 15 11 16 10 17 + 7 x y + 20 x y + x y - x y - 15 x y - 7 x y 11 15 10 16 9 17 11 14 10 15 9 16 + 6 x y + 20 x y - x y - x y - 30 x y + 6 x y 10 14 9 15 8 16 10 13 9 14 8 15 + 25 x y - 14 x y + x y - 11 x y + 17 x y - 2 x y 10 12 9 13 8 14 9 12 8 13 7 14 + 2 x y - 13 x y - 7 x y + 8 x y + 27 x y + x y 9 11 8 12 7 13 9 10 8 11 7 12 - 4 x y - 34 x y - 3 x y + x y + 19 x y + 4 x y 8 10 7 11 6 12 7 10 6 11 7 9 - 4 x y - 2 x y - 3 x y - 3 x y + 11 x y + 5 x y 6 10 7 8 6 9 5 10 6 8 5 9 - 17 x y - 2 x y + 13 x y - 6 x y - 4 x y + 18 x y 5 8 5 7 4 8 5 6 4 7 4 6 3 6 - 16 x y + 2 x y - 2 x y + 2 x y + 4 x y - 3 x y + x y 4 4 3 5 3 4 3 3 2 4 2 3 2 2 2 + x y - 3 x y + 3 x y - x y + 5 x y - 6 x y + x y + 2 x y / 9 16 9 15 9 14 8 14 9 12 - 2 x y + 1) / (x y - 4 x y + 5 x y - x y - 5 x y / 8 13 9 11 8 12 9 10 8 11 7 12 8 10 + 4 x y + 4 x y - 6 x y - x y + 4 x y - x y - x y 7 11 7 10 7 9 6 10 7 8 6 8 5 9 + 2 x y + x y - 4 x y - x y + 2 x y + x y + 2 x y 5 7 5 6 4 7 4 6 4 5 4 4 3 5 3 3 - 2 x y + x y - x y + 5 x y - 3 x y - x y - 3 x y + x y 2 4 2 2 2 + x y - x y + x y + 2 x y - 1) and in Maple notation -(x^12*y^22-7*x^12*y^21+21*x^12*y^20-35*x^12*y^19-x^11*y^20+35*x^12*y^18+6*x^11 *y^19-21*x^12*y^17-15*x^11*y^18+7*x^12*y^16+20*x^11*y^17+x^10*y^18-x^12*y^15-15 *x^11*y^16-7*x^10*y^17+6*x^11*y^15+20*x^10*y^16-x^9*y^17-x^11*y^14-30*x^10*y^15 +6*x^9*y^16+25*x^10*y^14-14*x^9*y^15+x^8*y^16-11*x^10*y^13+17*x^9*y^14-2*x^8*y^ 15+2*x^10*y^12-13*x^9*y^13-7*x^8*y^14+8*x^9*y^12+27*x^8*y^13+x^7*y^14-4*x^9*y^ 11-34*x^8*y^12-3*x^7*y^13+x^9*y^10+19*x^8*y^11+4*x^7*y^12-4*x^8*y^10-2*x^7*y^11 -3*x^6*y^12-3*x^7*y^10+11*x^6*y^11+5*x^7*y^9-17*x^6*y^10-2*x^7*y^8+13*x^6*y^9-6 *x^5*y^10-4*x^6*y^8+18*x^5*y^9-16*x^5*y^8+2*x^5*y^7-2*x^4*y^8+2*x^5*y^6+4*x^4*y ^7-3*x^4*y^6+x^3*y^6+x^4*y^4-3*x^3*y^5+3*x^3*y^4-x^3*y^3+5*x^2*y^4-6*x^2*y^3+x^ 2*y^2+2*x*y^2-2*x*y+1)/(x^9*y^16-4*x^9*y^15+5*x^9*y^14-x^8*y^14-5*x^9*y^12+4*x^ 8*y^13+4*x^9*y^11-6*x^8*y^12-x^9*y^10+4*x^8*y^11-x^7*y^12-x^8*y^10+2*x^7*y^11+x ^7*y^10-4*x^7*y^9-x^6*y^10+2*x^7*y^8+x^6*y^8+2*x^5*y^9-2*x^5*y^7+x^5*y^6-x^4*y^ 7+5*x^4*y^6-3*x^4*y^5-x^4*y^4-3*x^3*y^5+x^3*y^3+x^2*y^4-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\ bors of a random word of length n tends to n times, 8 6 5 4 3 2 2 (a - 4 a - 2 a + 12 a + 3 a - 12 a + 3) ----------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.523127339 BTW the ratio for words with, 500, letters is, 1.521956075 ------------------------------------------------ "Theorem Number 46" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 22 12 21 12 20 ) | ) C(m, n) x y | = - (x y - 7 x y + 21 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 19 11 20 12 18 11 19 12 17 11 18 - 35 x y - x y + 35 x y + 6 x y - 21 x y - 15 x y 12 16 11 17 10 18 12 15 11 16 10 17 + 7 x y + 20 x y + x y - x y - 15 x y - 7 x y 11 15 10 16 9 17 11 14 10 15 9 16 + 6 x y + 20 x y - x y - x y - 30 x y + 6 x y 10 14 9 15 8 16 10 13 9 14 8 15 + 25 x y - 14 x y + x y - 11 x y + 17 x y - 2 x y 10 12 9 13 8 14 9 12 8 13 7 14 + 2 x y - 13 x y - 7 x y + 8 x y + 27 x y + x y 9 11 8 12 7 13 9 10 8 11 7 12 - 4 x y - 34 x y - 3 x y + x y + 19 x y + 4 x y 8 10 7 11 6 12 7 10 6 11 7 9 - 4 x y - 2 x y - 3 x y - 3 x y + 11 x y + 5 x y 6 10 7 8 6 9 5 10 6 8 5 9 - 17 x y - 2 x y + 13 x y - 6 x y - 4 x y + 18 x y 5 8 5 7 4 8 5 6 4 7 4 6 3 6 - 16 x y + 2 x y - 2 x y + 2 x y + 4 x y - 3 x y + x y 4 4 3 5 3 4 3 3 2 4 2 3 2 2 2 + x y - 3 x y + 3 x y - x y + 5 x y - 6 x y + x y + 2 x y / 9 16 9 15 9 14 8 14 9 12 - 2 x y + 1) / (x y - 4 x y + 5 x y - x y - 5 x y / 8 13 9 11 8 12 9 10 8 11 7 12 8 10 + 4 x y + 4 x y - 6 x y - x y + 4 x y - x y - x y 7 11 7 10 7 9 6 10 7 8 6 8 5 9 + 2 x y + x y - 4 x y - x y + 2 x y + x y + 2 x y 5 7 5 6 4 7 4 6 4 5 4 4 3 5 3 3 - 2 x y + x y - x y + 5 x y - 3 x y - x y - 3 x y + x y 2 4 2 2 2 + x y - x y + x y + 2 x y - 1) and in Maple notation -(x^12*y^22-7*x^12*y^21+21*x^12*y^20-35*x^12*y^19-x^11*y^20+35*x^12*y^18+6*x^11 *y^19-21*x^12*y^17-15*x^11*y^18+7*x^12*y^16+20*x^11*y^17+x^10*y^18-x^12*y^15-15 *x^11*y^16-7*x^10*y^17+6*x^11*y^15+20*x^10*y^16-x^9*y^17-x^11*y^14-30*x^10*y^15 +6*x^9*y^16+25*x^10*y^14-14*x^9*y^15+x^8*y^16-11*x^10*y^13+17*x^9*y^14-2*x^8*y^ 15+2*x^10*y^12-13*x^9*y^13-7*x^8*y^14+8*x^9*y^12+27*x^8*y^13+x^7*y^14-4*x^9*y^ 11-34*x^8*y^12-3*x^7*y^13+x^9*y^10+19*x^8*y^11+4*x^7*y^12-4*x^8*y^10-2*x^7*y^11 -3*x^6*y^12-3*x^7*y^10+11*x^6*y^11+5*x^7*y^9-17*x^6*y^10-2*x^7*y^8+13*x^6*y^9-6 *x^5*y^10-4*x^6*y^8+18*x^5*y^9-16*x^5*y^8+2*x^5*y^7-2*x^4*y^8+2*x^5*y^6+4*x^4*y ^7-3*x^4*y^6+x^3*y^6+x^4*y^4-3*x^3*y^5+3*x^3*y^4-x^3*y^3+5*x^2*y^4-6*x^2*y^3+x^ 2*y^2+2*x*y^2-2*x*y+1)/(x^9*y^16-4*x^9*y^15+5*x^9*y^14-x^8*y^14-5*x^9*y^12+4*x^ 8*y^13+4*x^9*y^11-6*x^8*y^12-x^9*y^10+4*x^8*y^11-x^7*y^12-x^8*y^10+2*x^7*y^11+x ^7*y^10-4*x^7*y^9-x^6*y^10+2*x^7*y^8+x^6*y^8+2*x^5*y^9-2*x^5*y^7+x^5*y^6-x^4*y^ 7+5*x^4*y^6-3*x^4*y^5-x^4*y^4-3*x^3*y^5+x^3*y^3+x^2*y^4-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\ bors of a random word of length n tends to n times, 8 6 5 4 3 2 2 (a - 4 a - 2 a + 12 a + 3 a - 12 a + 3) ----------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.523127339 BTW the ratio for words with, 500, letters is, 1.521956075 ------------------------------------------------ "Theorem Number 47" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 12 7 11 6 12 7 10 ) | ) C(m, n) x y | = - (x y - 4 x y + 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 11 7 9 6 10 7 8 6 9 5 10 6 8 - 7 x y - 4 x y + 11 x y + x y - 11 x y + 7 x y + 7 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 7 - 20 x y - 2 x y + 21 x y - 10 x y + 4 x y + 2 x y - 10 x y 4 6 4 5 3 6 3 5 3 4 3 3 2 4 + 10 x y - 4 x y - 3 x y + 3 x y + x y - x y - 5 x y 2 3 2 / 7 13 7 12 7 11 + 4 x y - 2 x y + 2 x y - 1) / (2 x y - 5 x y + 4 x y / 7 10 6 11 7 9 6 10 7 8 6 9 6 8 - 2 x y + 2 x y + 2 x y - x y - x y - 3 x y + x y 5 9 6 7 5 8 5 7 5 6 4 7 4 5 3 5 - 2 x y + x y + 5 x y - 2 x y - x y - 3 x y + 3 x y + x y 3 4 3 3 2 4 2 3 2 - 3 x y + x y - x y + 2 x y - x y - 2 x y + 1) and in Maple notation -(x^7*y^12-4*x^7*y^11+2*x^6*y^12+6*x^7*y^10-7*x^6*y^11-4*x^7*y^9+11*x^6*y^10+x^ 7*y^8-11*x^6*y^9+7*x^5*y^10+7*x^6*y^8-20*x^5*y^9-2*x^6*y^7+21*x^5*y^8-10*x^5*y^ 7+4*x^4*y^8+2*x^5*y^6-10*x^4*y^7+10*x^4*y^6-4*x^4*y^5-3*x^3*y^6+3*x^3*y^5+x^3*y ^4-x^3*y^3-5*x^2*y^4+4*x^2*y^3-2*x*y^2+2*x*y-1)/(2*x^7*y^13-5*x^7*y^12+4*x^7*y^ 11-2*x^7*y^10+2*x^6*y^11+2*x^7*y^9-x^6*y^10-x^7*y^8-3*x^6*y^9+x^6*y^8-2*x^5*y^9 +x^6*y^7+5*x^5*y^8-2*x^5*y^7-x^5*y^6-3*x^4*y^7+3*x^4*y^5+x^3*y^5-3*x^3*y^4+x^3* y^3-x^2*y^4+2*x^2*y^3-x*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, 5 4 3 2 2 (a + a + a - 6 a + 3) --------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.516641784 BTW the ratio for words with, 500, letters is, 1.515435701 ------------------------------------------------ "Theorem Number 48" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 12 7 11 6 12 7 10 ) | ) C(m, n) x y | = - (x y - 4 x y + 2 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 6 11 7 9 6 10 7 8 6 9 5 10 6 8 - 7 x y - 4 x y + 11 x y + x y - 11 x y + 7 x y + 7 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 7 - 20 x y - 2 x y + 21 x y - 10 x y + 4 x y + 2 x y - 10 x y 4 6 4 5 3 6 3 5 3 4 3 3 2 4 + 10 x y - 4 x y - 3 x y + 3 x y + x y - x y - 5 x y 2 3 2 / 7 13 7 12 7 11 + 4 x y - 2 x y + 2 x y - 1) / (2 x y - 5 x y + 4 x y / 7 10 6 11 7 9 6 10 7 8 6 9 6 8 - 2 x y + 2 x y + 2 x y - x y - x y - 3 x y + x y 5 9 6 7 5 8 5 7 5 6 4 7 4 5 3 5 - 2 x y + x y + 5 x y - 2 x y - x y - 3 x y + 3 x y + x y 3 4 3 3 2 4 2 3 2 - 3 x y + x y - x y + 2 x y - x y - 2 x y + 1) and in Maple notation -(x^7*y^12-4*x^7*y^11+2*x^6*y^12+6*x^7*y^10-7*x^6*y^11-4*x^7*y^9+11*x^6*y^10+x^ 7*y^8-11*x^6*y^9+7*x^5*y^10+7*x^6*y^8-20*x^5*y^9-2*x^6*y^7+21*x^5*y^8-10*x^5*y^ 7+4*x^4*y^8+2*x^5*y^6-10*x^4*y^7+10*x^4*y^6-4*x^4*y^5-3*x^3*y^6+3*x^3*y^5+x^3*y ^4-x^3*y^3-5*x^2*y^4+4*x^2*y^3-2*x*y^2+2*x*y-1)/(2*x^7*y^13-5*x^7*y^12+4*x^7*y^ 11-2*x^7*y^10+2*x^6*y^11+2*x^7*y^9-x^6*y^10-x^7*y^8-3*x^6*y^9+x^6*y^8-2*x^5*y^9 +x^6*y^7+5*x^5*y^8-2*x^5*y^7-x^5*y^6-3*x^4*y^7+3*x^4*y^5+x^3*y^5-3*x^3*y^4+x^3* y^3-x^2*y^4+2*x^2*y^3-x*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, 5 4 3 2 2 (a + a + a - 6 a + 3) --------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.516641784 BTW the ratio for words with, 500, letters is, 1.515435701 ------------------------------------------------ "Theorem Number 49" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 6 9 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 10 6 8 5 9 5 8 5 7 4 8 5 6 4 6 + 5 x y + x y - 15 x y + 16 x y - 7 x y - x y + x y + x y 3 5 3 4 2 4 2 3 2 2 2 / 6 11 - x y + x y + 4 x y - 2 x y - x y + x y - x y + 1) / (x y / 6 10 6 9 6 8 5 9 5 8 5 7 5 6 4 7 + x y - 5 x y + 3 x y - x y - x y + 3 x y - x y - x y 4 6 3 5 3 4 2 4 2 3 2 2 2 + x y + 3 x y - 2 x y - x y - x y + x y + 2 x y + x y - 1) and in Maple notation -(2*x^6*y^12-7*x^6*y^11+9*x^6*y^10-5*x^6*y^9+5*x^5*y^10+x^6*y^8-15*x^5*y^9+16*x ^5*y^8-7*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^6-x^3*y^5+x^3*y^4+4*x^2*y^4-2*x^2*y^3-x^ 2*y^2+x*y^2-x*y+1)/(x^6*y^11+x^6*y^10-5*x^6*y^9+3*x^6*y^8-x^5*y^9-x^5*y^8+3*x^5 *y^7-x^5*y^6-x^4*y^7+x^4*y^6+3*x^3*y^5-2*x^3*y^4-x^2*y^4-x^2*y^3+x^2*y^2+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, 5 4 3 2 2 (a - a - a + 6 a - 3) - --------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.508501487 BTW the ratio for words with, 500, letters is, 1.507392527 ------------------------------------------------ "Theorem Number 50" 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, 3, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 6 12 6 11 6 10 6 9 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y - 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 10 6 8 5 9 5 8 5 7 4 8 5 6 4 6 + 5 x y + x y - 15 x y + 16 x y - 7 x y - x y + x y + x y 3 5 3 4 2 4 2 3 2 2 2 / 6 11 - x y + x y + 4 x y - 2 x y - x y + x y - x y + 1) / (x y / 6 10 6 9 6 8 5 9 5 8 5 7 5 6 4 7 + x y - 5 x y + 3 x y - x y - x y + 3 x y - x y - x y 4 6 3 5 3 4 2 4 2 3 2 2 2 + x y + 3 x y - 2 x y - x y - x y + x y + 2 x y + x y - 1) and in Maple notation -(2*x^6*y^12-7*x^6*y^11+9*x^6*y^10-5*x^6*y^9+5*x^5*y^10+x^6*y^8-15*x^5*y^9+16*x ^5*y^8-7*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^6-x^3*y^5+x^3*y^4+4*x^2*y^4-2*x^2*y^3-x^ 2*y^2+x*y^2-x*y+1)/(x^6*y^11+x^6*y^10-5*x^6*y^9+3*x^6*y^8-x^5*y^9-x^5*y^8+3*x^5 *y^7-x^5*y^6-x^4*y^7+x^4*y^6+3*x^3*y^5-2*x^3*y^4-x^2*y^4-x^2*y^3+x^2*y^2+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, 5 4 3 2 2 (a - a - a + 6 a - 3) - --------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.508501487 BTW the ratio for words with, 500, letters is, 1.507392527 ------------------------------------------------ "Theorem Number 51" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 13 24 13 23 13 22 ) | ) C(m, n) x y | = (x y - 8 x y + 28 x y / | / | ----- | ----- | m = 0 \ n = 0 / 13 21 12 22 13 20 12 21 13 19 - 56 x y + 2 x y + 70 x y - 14 x y - 56 x y 12 20 13 18 12 19 11 20 13 17 12 18 + 42 x y + 28 x y - 70 x y - x y - 8 x y + 70 x y 11 19 13 16 12 17 11 18 12 16 11 17 + 6 x y + x y - 42 x y - 15 x y + 14 x y + 20 x y 10 18 12 15 11 16 10 17 11 15 10 16 + x y - 2 x y - 15 x y - 6 x y + 6 x y + 15 x y 9 17 11 14 10 15 9 16 10 14 9 15 - 2 x y - x y - 20 x y + 14 x y + 15 x y - 38 x y 10 13 9 14 8 15 10 12 9 13 8 14 - 6 x y + 53 x y - 2 x y + x y - 42 x y + 8 x y 9 12 8 13 7 14 9 11 8 12 7 13 + 20 x y - 16 x y + x y - 6 x y + 22 x y - 4 x y 9 10 8 11 7 12 8 10 7 11 6 12 8 9 + x y - 20 x y + 7 x y + 10 x y - 4 x y - x y - 2 x y 7 10 6 11 7 9 6 10 7 8 6 9 - 5 x y + 6 x y + 8 x y - 19 x y - 3 x y + 30 x y 5 10 6 8 5 9 6 7 5 8 6 6 5 7 - 6 x y - 23 x y + 22 x y + 8 x y - 30 x y - x y + 20 x y 4 8 5 6 4 7 5 5 4 6 4 5 3 6 - 4 x y - 8 x y + 10 x y + 2 x y - 11 x y + 6 x y + 2 x y 4 4 3 5 3 4 2 4 2 3 2 2 2 - x y - 4 x y + 2 x y + 5 x y - 6 x y + x y + 2 x y - 2 x y / 9 16 9 15 9 14 9 13 8 14 9 12 + 1) / (x y - 4 x y + 7 x y - 8 x y + 2 x y + 7 x y / 8 13 9 11 8 12 9 10 8 11 7 12 8 10 - 8 x y - 4 x y + 14 x y + x y - 14 x y + x y + 8 x y 7 11 8 9 7 10 7 9 6 10 6 8 5 9 - 4 x y - 2 x y + 5 x y - 2 x y + x y - 2 x y - 2 x y 6 7 5 8 6 6 5 7 5 6 4 7 5 5 + 2 x y - 2 x y - x y + 6 x y - 5 x y + 2 x y + 2 x y 4 6 4 5 3 6 4 4 3 5 3 4 2 4 2 2 - 7 x y + 6 x y + x y - x y + 2 x y - x y - x y + x y 2 - x y - 2 x y + 1) and in Maple notation (x^13*y^24-8*x^13*y^23+28*x^13*y^22-56*x^13*y^21+2*x^12*y^22+70*x^13*y^20-14*x^ 12*y^21-56*x^13*y^19+42*x^12*y^20+28*x^13*y^18-70*x^12*y^19-x^11*y^20-8*x^13*y^ 17+70*x^12*y^18+6*x^11*y^19+x^13*y^16-42*x^12*y^17-15*x^11*y^18+14*x^12*y^16+20 *x^11*y^17+x^10*y^18-2*x^12*y^15-15*x^11*y^16-6*x^10*y^17+6*x^11*y^15+15*x^10*y ^16-2*x^9*y^17-x^11*y^14-20*x^10*y^15+14*x^9*y^16+15*x^10*y^14-38*x^9*y^15-6*x^ 10*y^13+53*x^9*y^14-2*x^8*y^15+x^10*y^12-42*x^9*y^13+8*x^8*y^14+20*x^9*y^12-16* x^8*y^13+x^7*y^14-6*x^9*y^11+22*x^8*y^12-4*x^7*y^13+x^9*y^10-20*x^8*y^11+7*x^7* y^12+10*x^8*y^10-4*x^7*y^11-x^6*y^12-2*x^8*y^9-5*x^7*y^10+6*x^6*y^11+8*x^7*y^9-\ 19*x^6*y^10-3*x^7*y^8+30*x^6*y^9-6*x^5*y^10-23*x^6*y^8+22*x^5*y^9+8*x^6*y^7-30* x^5*y^8-x^6*y^6+20*x^5*y^7-4*x^4*y^8-8*x^5*y^6+10*x^4*y^7+2*x^5*y^5-11*x^4*y^6+ 6*x^4*y^5+2*x^3*y^6-x^4*y^4-4*x^3*y^5+2*x^3*y^4+5*x^2*y^4-6*x^2*y^3+x^2*y^2+2*x *y^2-2*x*y+1)/(x^9*y^16-4*x^9*y^15+7*x^9*y^14-8*x^9*y^13+2*x^8*y^14+7*x^9*y^12-\ 8*x^8*y^13-4*x^9*y^11+14*x^8*y^12+x^9*y^10-14*x^8*y^11+x^7*y^12+8*x^8*y^10-4*x^ 7*y^11-2*x^8*y^9+5*x^7*y^10-2*x^7*y^9+x^6*y^10-2*x^6*y^8-2*x^5*y^9+2*x^6*y^7-2* x^5*y^8-x^6*y^6+6*x^5*y^7-5*x^5*y^6+2*x^4*y^7+2*x^5*y^5-7*x^4*y^6+6*x^4*y^5+x^3 *y^6-x^4*y^4+2*x^3*y^5-x^3*y^4-x^2*y^4+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\ bors of a random word of length n tends to n times, 8 6 5 4 3 2 2 (a - 4 a - 2 a + 13 a + 2 a - 12 a + 3) ----------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.496554483 BTW the ratio for words with, 500, letters is, 1.495593660 ------------------------------------------------ "Theorem Number 52" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 14 25 14 24 14 23 ) | ) C(m, n) x y | = (x y - 8 x y + 28 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 22 14 21 13 22 14 20 13 21 12 22 - 56 x y + 70 x y - x y - 56 x y + 7 x y + x y 14 19 13 20 12 21 14 18 13 19 + 28 x y - 21 x y - 7 x y - 8 x y + 35 x y 12 20 14 17 13 18 12 19 13 17 12 18 + 21 x y + x y - 35 x y - 35 x y + 21 x y + 35 x y 13 16 12 17 13 15 12 16 11 17 10 18 - 7 x y - 21 x y + x y + 7 x y + x y + x y 12 15 11 16 10 17 11 15 10 16 11 14 - x y - 5 x y - 7 x y + 10 x y + 20 x y - 10 x y 10 15 9 16 11 13 10 14 9 15 8 16 - 31 x y + x y + 5 x y + 29 x y - 7 x y - x y 11 12 10 13 9 14 8 15 10 12 9 13 - x y - 17 x y + 20 x y + 5 x y + 6 x y - 30 x y 8 14 10 11 9 12 8 13 7 14 9 11 - 11 x y - x y + 25 x y + 13 x y + x y - 11 x y 8 12 7 13 9 10 8 11 7 12 8 10 - 9 x y - x y + 2 x y + 5 x y - 7 x y - 3 x y 7 11 6 12 8 9 7 10 6 11 7 9 6 10 7 8 + 16 x y - x y + x y - 11 x y + x y + x y + x y + x y 6 9 5 10 6 8 5 9 6 7 5 8 6 6 - 2 x y + 3 x y + 3 x y - 9 x y - 3 x y + 8 x y + x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 4 - x y - 2 x y - x y + 3 x y - 2 x y + x y - x y + x y 2 4 2 3 2 2 2 / 11 19 11 18 + 3 x y - 2 x y - x y + x y - x y + 1) / (x y - 5 x y / 11 17 11 16 10 17 11 15 10 16 11 14 + 10 x y - 10 x y - 2 x y + 5 x y + 8 x y - x y 10 15 9 16 10 14 9 15 10 13 9 14 - 12 x y + x y + 8 x y - 2 x y - 2 x y - x y 9 13 8 14 9 12 8 13 9 11 8 12 9 10 + 4 x y - 2 x y - x y + 4 x y - 2 x y + x y + x y 8 11 7 12 8 10 7 11 8 9 7 10 6 11 - 7 x y + x y + 5 x y - 3 x y - x y + 2 x y + x y 7 9 7 8 6 8 5 9 5 8 6 6 5 7 4 8 + x y - x y - 2 x y - 2 x y - x y + x y + 2 x y - x y 4 7 4 6 4 5 3 6 3 5 3 4 2 3 2 2 + 3 x y - x y - x y + 2 x y - 2 x y + 2 x y + x y - x y 2 - 2 x y - x y + 1) and in Maple notation (x^14*y^25-8*x^14*y^24+28*x^14*y^23-56*x^14*y^22+70*x^14*y^21-x^13*y^22-56*x^14 *y^20+7*x^13*y^21+x^12*y^22+28*x^14*y^19-21*x^13*y^20-7*x^12*y^21-8*x^14*y^18+ 35*x^13*y^19+21*x^12*y^20+x^14*y^17-35*x^13*y^18-35*x^12*y^19+21*x^13*y^17+35*x ^12*y^18-7*x^13*y^16-21*x^12*y^17+x^13*y^15+7*x^12*y^16+x^11*y^17+x^10*y^18-x^ 12*y^15-5*x^11*y^16-7*x^10*y^17+10*x^11*y^15+20*x^10*y^16-10*x^11*y^14-31*x^10* y^15+x^9*y^16+5*x^11*y^13+29*x^10*y^14-7*x^9*y^15-x^8*y^16-x^11*y^12-17*x^10*y^ 13+20*x^9*y^14+5*x^8*y^15+6*x^10*y^12-30*x^9*y^13-11*x^8*y^14-x^10*y^11+25*x^9* y^12+13*x^8*y^13+x^7*y^14-11*x^9*y^11-9*x^8*y^12-x^7*y^13+2*x^9*y^10+5*x^8*y^11 -7*x^7*y^12-3*x^8*y^10+16*x^7*y^11-x^6*y^12+x^8*y^9-11*x^7*y^10+x^6*y^11+x^7*y^ 9+x^6*y^10+x^7*y^8-2*x^6*y^9+3*x^5*y^10+3*x^6*y^8-9*x^5*y^9-3*x^6*y^7+8*x^5*y^8 +x^6*y^6-x^5*y^7-2*x^4*y^8-x^5*y^6+3*x^4*y^7-2*x^4*y^6+x^4*y^5-x^3*y^6+x^3*y^4+ 3*x^2*y^4-2*x^2*y^3-x^2*y^2+x*y^2-x*y+1)/(x^11*y^19-5*x^11*y^18+10*x^11*y^17-10 *x^11*y^16-2*x^10*y^17+5*x^11*y^15+8*x^10*y^16-x^11*y^14-12*x^10*y^15+x^9*y^16+ 8*x^10*y^14-2*x^9*y^15-2*x^10*y^13-x^9*y^14+4*x^9*y^13-2*x^8*y^14-x^9*y^12+4*x^ 8*y^13-2*x^9*y^11+x^8*y^12+x^9*y^10-7*x^8*y^11+x^7*y^12+5*x^8*y^10-3*x^7*y^11-x ^8*y^9+2*x^7*y^10+x^6*y^11+x^7*y^9-x^7*y^8-2*x^6*y^8-2*x^5*y^9-x^5*y^8+x^6*y^6+ 2*x^5*y^7-x^4*y^8+3*x^4*y^7-x^4*y^6-x^4*y^5+2*x^3*y^6-2*x^3*y^5+2*x^3*y^4+x^2*y ^3-x^2*y^2-2*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, 8 7 6 5 4 3 2 2 (a + a - 4 a - 3 a + 13 a + 2 a - 12 a + 3) ---------------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.491777274 BTW the ratio for words with, 500, letters is, 1.490875895 ------------------------------------------------ "Theorem Number 53" 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, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 14 25 14 24 14 23 ) | ) C(m, n) x y | = (x y - 8 x y + 28 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 22 14 21 13 22 14 20 13 21 12 22 - 56 x y + 70 x y - x y - 56 x y + 7 x y + x y 14 19 13 20 12 21 14 18 13 19 + 28 x y - 21 x y - 7 x y - 8 x y + 35 x y 12 20 14 17 13 18 12 19 13 17 12 18 + 21 x y + x y - 35 x y - 35 x y + 21 x y + 35 x y 13 16 12 17 13 15 12 16 11 17 10 18 - 7 x y - 21 x y + x y + 7 x y + x y + x y 12 15 11 16 10 17 11 15 10 16 11 14 - x y - 5 x y - 7 x y + 10 x y + 20 x y - 10 x y 10 15 9 16 11 13 10 14 9 15 8 16 - 31 x y + x y + 5 x y + 29 x y - 7 x y - x y 11 12 10 13 9 14 8 15 10 12 9 13 - x y - 17 x y + 20 x y + 5 x y + 6 x y - 30 x y 8 14 10 11 9 12 8 13 7 14 9 11 - 11 x y - x y + 25 x y + 13 x y + x y - 11 x y 8 12 7 13 9 10 8 11 7 12 8 10 - 9 x y - x y + 2 x y + 5 x y - 7 x y - 3 x y 7 11 6 12 8 9 7 10 6 11 7 9 6 10 7 8 + 16 x y - x y + x y - 11 x y + x y + x y + x y + x y 6 9 5 10 6 8 5 9 6 7 5 8 6 6 - 2 x y + 3 x y + 3 x y - 9 x y - 3 x y + 8 x y + x y 5 7 4 8 5 6 4 7 4 6 4 5 3 6 3 4 - x y - 2 x y - x y + 3 x y - 2 x y + x y - x y + x y 2 4 2 3 2 2 2 / 11 19 11 18 + 3 x y - 2 x y - x y + x y - x y + 1) / (x y - 5 x y / 11 17 11 16 10 17 11 15 10 16 11 14 + 10 x y - 10 x y - 2 x y + 5 x y + 8 x y - x y 10 15 9 16 10 14 9 15 10 13 9 14 - 12 x y + x y + 8 x y - 2 x y - 2 x y - x y 9 13 8 14 9 12 8 13 9 11 8 12 9 10 + 4 x y - 2 x y - x y + 4 x y - 2 x y + x y + x y 8 11 7 12 8 10 7 11 8 9 7 10 6 11 - 7 x y + x y + 5 x y - 3 x y - x y + 2 x y + x y 7 9 7 8 6 8 5 9 5 8 6 6 5 7 4 8 + x y - x y - 2 x y - 2 x y - x y + x y + 2 x y - x y 4 7 4 6 4 5 3 6 3 5 3 4 2 3 2 2 + 3 x y - x y - x y + 2 x y - 2 x y + 2 x y + x y - x y 2 - 2 x y - x y + 1) and in Maple notation (x^14*y^25-8*x^14*y^24+28*x^14*y^23-56*x^14*y^22+70*x^14*y^21-x^13*y^22-56*x^14 *y^20+7*x^13*y^21+x^12*y^22+28*x^14*y^19-21*x^13*y^20-7*x^12*y^21-8*x^14*y^18+ 35*x^13*y^19+21*x^12*y^20+x^14*y^17-35*x^13*y^18-35*x^12*y^19+21*x^13*y^17+35*x ^12*y^18-7*x^13*y^16-21*x^12*y^17+x^13*y^15+7*x^12*y^16+x^11*y^17+x^10*y^18-x^ 12*y^15-5*x^11*y^16-7*x^10*y^17+10*x^11*y^15+20*x^10*y^16-10*x^11*y^14-31*x^10* y^15+x^9*y^16+5*x^11*y^13+29*x^10*y^14-7*x^9*y^15-x^8*y^16-x^11*y^12-17*x^10*y^ 13+20*x^9*y^14+5*x^8*y^15+6*x^10*y^12-30*x^9*y^13-11*x^8*y^14-x^10*y^11+25*x^9* y^12+13*x^8*y^13+x^7*y^14-11*x^9*y^11-9*x^8*y^12-x^7*y^13+2*x^9*y^10+5*x^8*y^11 -7*x^7*y^12-3*x^8*y^10+16*x^7*y^11-x^6*y^12+x^8*y^9-11*x^7*y^10+x^6*y^11+x^7*y^ 9+x^6*y^10+x^7*y^8-2*x^6*y^9+3*x^5*y^10+3*x^6*y^8-9*x^5*y^9-3*x^6*y^7+8*x^5*y^8 +x^6*y^6-x^5*y^7-2*x^4*y^8-x^5*y^6+3*x^4*y^7-2*x^4*y^6+x^4*y^5-x^3*y^6+x^3*y^4+ 3*x^2*y^4-2*x^2*y^3-x^2*y^2+x*y^2-x*y+1)/(x^11*y^19-5*x^11*y^18+10*x^11*y^17-10 *x^11*y^16-2*x^10*y^17+5*x^11*y^15+8*x^10*y^16-x^11*y^14-12*x^10*y^15+x^9*y^16+ 8*x^10*y^14-2*x^9*y^15-2*x^10*y^13-x^9*y^14+4*x^9*y^13-2*x^8*y^14-x^9*y^12+4*x^ 8*y^13-2*x^9*y^11+x^8*y^12+x^9*y^10-7*x^8*y^11+x^7*y^12+5*x^8*y^10-3*x^7*y^11-x ^8*y^9+2*x^7*y^10+x^6*y^11+x^7*y^9-x^7*y^8-2*x^6*y^8-2*x^5*y^9-x^5*y^8+x^6*y^6+ 2*x^5*y^7-x^4*y^8+3*x^4*y^7-x^4*y^6-x^4*y^5+2*x^3*y^6-2*x^3*y^5+2*x^3*y^4+x^2*y ^3-x^2*y^2-2*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, 8 7 6 5 4 3 2 2 (a + a - 4 a - 3 a + 13 a + 2 a - 12 a + 3) ---------------------------------------------------- 4 2 5 a - 6 a + 3 5 3 where a is the root of the polynomial, x - 2 x + 3 x - 1, and in decimals this is, 1.491777274 BTW the ratio for words with, 500, letters is, 1.490875895 ------------------------------------------------ "Theorem Number 54" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 22 12 21 12 20 ) | ) C(m, n) x y | = (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 19 11 20 11 19 12 17 11 18 12 16 - 14 x y + x y - 6 x y + 14 x y + 14 x y - 14 x y 11 17 10 18 12 15 10 17 12 14 11 15 - 14 x y - x y + 6 x y + 7 x y - x y + 14 x y 10 16 11 14 10 15 9 16 11 13 10 14 - 21 x y - 14 x y + 35 x y - 3 x y + 6 x y - 35 x y 9 15 8 16 11 12 10 13 9 14 8 15 + 16 x y - x y - x y + 21 x y - 35 x y + 5 x y 10 12 9 13 8 14 10 11 9 12 8 13 - 7 x y + 40 x y - 14 x y + x y - 25 x y + 24 x y 7 14 9 11 8 12 7 13 9 10 8 11 - 2 x y + 8 x y - 22 x y + 10 x y - x y + 7 x y 7 12 8 10 7 11 6 12 8 9 7 10 - 23 x y + 4 x y + 26 x y - 2 x y - 4 x y - 8 x y 6 11 8 8 7 9 6 10 7 8 6 9 5 10 + 12 x y + x y - 10 x y - 24 x y + 9 x y + 15 x y - x y 7 7 6 8 5 9 6 7 5 8 6 6 4 8 - 2 x y + 7 x y + 5 x y - 11 x y - 7 x y + 3 x y + 4 x y 5 6 4 7 5 5 4 6 4 5 3 6 4 4 + 6 x y - 6 x y - 3 x y + 2 x y - 2 x y + 5 x y + 2 x y 3 5 3 4 3 3 2 4 2 3 2 2 2 - 9 x y + 5 x y - x y + 6 x y - 6 x y + x y + 2 x y - 2 x y / 11 18 11 17 11 16 10 17 10 16 + 1) / (x y - 4 x y + 5 x y - x y + 4 x y / 11 14 10 15 11 13 9 15 11 12 10 13 - 5 x y - 5 x y + 4 x y - x y - x y + 5 x y 9 14 10 12 9 13 8 14 10 11 9 12 8 13 + 2 x y - 4 x y + 2 x y + x y + x y - 8 x y - 3 x y 9 11 8 12 9 10 8 11 8 10 7 11 8 9 + 7 x y - x y - 2 x y + 10 x y - 9 x y + 2 x y + x y 7 10 6 11 8 8 7 9 6 10 7 8 6 9 - 9 x y + x y + x y + 8 x y - 2 x y + x y + 4 x y 7 7 6 8 5 9 6 7 6 6 5 7 4 8 5 6 - 2 x y - 3 x y + x y - 3 x y + 3 x y + 2 x y - x y + x y 4 7 5 5 4 6 4 5 3 6 4 4 3 5 3 3 + x y - 3 x y - 3 x y + x y - x y + 2 x y + x y - x y 2 2 2 + x y - x y - 2 x y + 1) and in Maple notation (x^12*y^22-6*x^12*y^21+14*x^12*y^20-14*x^12*y^19+x^11*y^20-6*x^11*y^19+14*x^12* y^17+14*x^11*y^18-14*x^12*y^16-14*x^11*y^17-x^10*y^18+6*x^12*y^15+7*x^10*y^17-x ^12*y^14+14*x^11*y^15-21*x^10*y^16-14*x^11*y^14+35*x^10*y^15-3*x^9*y^16+6*x^11* y^13-35*x^10*y^14+16*x^9*y^15-x^8*y^16-x^11*y^12+21*x^10*y^13-35*x^9*y^14+5*x^8 *y^15-7*x^10*y^12+40*x^9*y^13-14*x^8*y^14+x^10*y^11-25*x^9*y^12+24*x^8*y^13-2*x ^7*y^14+8*x^9*y^11-22*x^8*y^12+10*x^7*y^13-x^9*y^10+7*x^8*y^11-23*x^7*y^12+4*x^ 8*y^10+26*x^7*y^11-2*x^6*y^12-4*x^8*y^9-8*x^7*y^10+12*x^6*y^11+x^8*y^8-10*x^7*y ^9-24*x^6*y^10+9*x^7*y^8+15*x^6*y^9-x^5*y^10-2*x^7*y^7+7*x^6*y^8+5*x^5*y^9-11*x ^6*y^7-7*x^5*y^8+3*x^6*y^6+4*x^4*y^8+6*x^5*y^6-6*x^4*y^7-3*x^5*y^5+2*x^4*y^6-2* x^4*y^5+5*x^3*y^6+2*x^4*y^4-9*x^3*y^5+5*x^3*y^4-x^3*y^3+6*x^2*y^4-6*x^2*y^3+x^2 *y^2+2*x*y^2-2*x*y+1)/(x^11*y^18-4*x^11*y^17+5*x^11*y^16-x^10*y^17+4*x^10*y^16-\ 5*x^11*y^14-5*x^10*y^15+4*x^11*y^13-x^9*y^15-x^11*y^12+5*x^10*y^13+2*x^9*y^14-4 *x^10*y^12+2*x^9*y^13+x^8*y^14+x^10*y^11-8*x^9*y^12-3*x^8*y^13+7*x^9*y^11-x^8*y ^12-2*x^9*y^10+10*x^8*y^11-9*x^8*y^10+2*x^7*y^11+x^8*y^9-9*x^7*y^10+x^6*y^11+x^ 8*y^8+8*x^7*y^9-2*x^6*y^10+x^7*y^8+4*x^6*y^9-2*x^7*y^7-3*x^6*y^8+x^5*y^9-3*x^6* y^7+3*x^6*y^6+2*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^7-3*x^5*y^5-3*x^4*y^6+x^4*y^5-x^3 *y^6+2*x^4*y^4+x^3*y^5-x^3*y^3+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\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (a + a - 2 a + 3 a - 8 a + 2 a - 6 a + 3) - -------------------------------------------------- 6 4 3 2 5 a + 2 a + 2 a - 6 a + 2 a - 3 5 3 2 where a is the root of the polynomial, x - x + x - 3 x + 1, and in decimals this is, 1.490979609 BTW the ratio for words with, 500, letters is, 1.490107855 ------------------------------------------------ "Theorem Number 55" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 11 20 12 18 11 19 12 17 - 14 x y + x y - 6 x y + 14 x y + 14 x y - 14 x y 11 18 10 19 12 16 10 18 12 15 11 16 - 14 x y + x y + 6 x y - 7 x y - x y + 14 x y 10 17 11 15 10 16 11 14 10 15 9 16 + 21 x y - 14 x y - 35 x y + 6 x y + 35 x y - 3 x y 11 13 10 14 9 15 8 16 10 13 9 14 - x y - 21 x y + 16 x y - x y + 7 x y - 35 x y 8 15 10 12 9 13 8 14 9 12 8 13 + 5 x y - x y + 40 x y - 17 x y - 25 x y + 38 x y 7 14 9 11 8 12 7 13 9 10 8 11 - 2 x y + 8 x y - 49 x y + 8 x y - x y + 35 x y 7 12 8 10 7 11 8 9 7 10 6 11 - 19 x y - 13 x y + 32 x y + 2 x y - 32 x y + 2 x y 7 9 6 10 7 8 6 9 5 10 6 8 + 16 x y - 11 x y - 3 x y + 19 x y - 2 x y - 13 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 6 + 8 x y + 3 x y - 13 x y + 10 x y + 2 x y - 3 x y - 6 x y 4 5 3 6 3 5 3 4 2 4 2 3 2 2 + 4 x y + 4 x y - 6 x y + 2 x y + 6 x y - 6 x y + x y 2 / 11 19 11 18 11 17 10 18 + 2 x y - 2 x y + 1) / (x y - 4 x y + 5 x y - x y / 10 17 11 15 10 16 11 14 9 16 11 13 + 4 x y - 5 x y - 5 x y + 4 x y - x y - x y 10 14 9 15 10 13 9 14 10 12 9 13 + 5 x y + 3 x y - 4 x y - 3 x y + x y + 2 x y 8 14 9 12 8 13 9 11 8 12 9 10 8 11 - x y - 3 x y + 2 x y + 3 x y - 2 x y - x y + 4 x y 7 12 8 10 7 11 8 9 7 10 6 11 7 9 - x y - 5 x y + x y + 2 x y - 4 x y + 2 x y + 7 x y 6 10 7 8 6 9 6 8 5 9 6 7 5 8 5 7 - 2 x y - 3 x y + x y - 4 x y + x y + 3 x y + 2 x y + x y 5 6 4 7 4 6 4 5 3 4 2 2 2 - 3 x y - 2 x y - 2 x y + 4 x y - x y + x y - x y - 2 x y + 1 ) and in Maple notation (x^12*y^23-6*x^12*y^22+14*x^12*y^21-14*x^12*y^20+x^11*y^21-6*x^11*y^20+14*x^12* y^18+14*x^11*y^19-14*x^12*y^17-14*x^11*y^18+x^10*y^19+6*x^12*y^16-7*x^10*y^18-x ^12*y^15+14*x^11*y^16+21*x^10*y^17-14*x^11*y^15-35*x^10*y^16+6*x^11*y^14+35*x^ 10*y^15-3*x^9*y^16-x^11*y^13-21*x^10*y^14+16*x^9*y^15-x^8*y^16+7*x^10*y^13-35*x ^9*y^14+5*x^8*y^15-x^10*y^12+40*x^9*y^13-17*x^8*y^14-25*x^9*y^12+38*x^8*y^13-2* x^7*y^14+8*x^9*y^11-49*x^8*y^12+8*x^7*y^13-x^9*y^10+35*x^8*y^11-19*x^7*y^12-13* x^8*y^10+32*x^7*y^11+2*x^8*y^9-32*x^7*y^10+2*x^6*y^11+16*x^7*y^9-11*x^6*y^10-3* x^7*y^8+19*x^6*y^9-2*x^5*y^10-13*x^6*y^8+8*x^5*y^9+3*x^6*y^7-13*x^5*y^8+10*x^5* y^7+2*x^4*y^8-3*x^5*y^6-6*x^4*y^6+4*x^4*y^5+4*x^3*y^6-6*x^3*y^5+2*x^3*y^4+6*x^2 *y^4-6*x^2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(x^11*y^19-4*x^11*y^18+5*x^11*y^17-x^10 *y^18+4*x^10*y^17-5*x^11*y^15-5*x^10*y^16+4*x^11*y^14-x^9*y^16-x^11*y^13+5*x^10 *y^14+3*x^9*y^15-4*x^10*y^13-3*x^9*y^14+x^10*y^12+2*x^9*y^13-x^8*y^14-3*x^9*y^ 12+2*x^8*y^13+3*x^9*y^11-2*x^8*y^12-x^9*y^10+4*x^8*y^11-x^7*y^12-5*x^8*y^10+x^7 *y^11+2*x^8*y^9-4*x^7*y^10+2*x^6*y^11+7*x^7*y^9-2*x^6*y^10-3*x^7*y^8+x^6*y^9-4* x^6*y^8+x^5*y^9+3*x^6*y^7+2*x^5*y^8+x^5*y^7-3*x^5*y^6-2*x^4*y^7-2*x^4*y^6+4*x^4 *y^5-x^3*y^4+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\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (a + a - 2 a + 3 a - 8 a + 2 a - 6 a + 3) - -------------------------------------------------- 6 4 3 2 5 a + 2 a + 2 a - 6 a + 2 a - 3 5 3 2 where a is the root of the polynomial, x - x + x - 3 x + 1, and in decimals this is, 1.490979609 BTW the ratio for words with, 500, letters is, 1.490107855 ------------------------------------------------ "Theorem Number 56" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 23 12 22 12 21 ) | ) C(m, n) x y | = (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 20 11 21 11 20 12 18 11 19 12 17 - 14 x y + x y - 6 x y + 14 x y + 14 x y - 14 x y 11 18 10 19 12 16 10 18 12 15 11 16 - 14 x y + x y + 6 x y - 7 x y - x y + 14 x y 10 17 11 15 10 16 11 14 10 15 9 16 + 21 x y - 14 x y - 35 x y + 6 x y + 35 x y - 3 x y 11 13 10 14 9 15 8 16 10 13 9 14 - x y - 21 x y + 16 x y - x y + 7 x y - 35 x y 8 15 10 12 9 13 8 14 9 12 8 13 + 5 x y - x y + 40 x y - 17 x y - 25 x y + 38 x y 7 14 9 11 8 12 7 13 9 10 8 11 - 2 x y + 8 x y - 49 x y + 8 x y - x y + 35 x y 7 12 8 10 7 11 8 9 7 10 6 11 - 19 x y - 13 x y + 32 x y + 2 x y - 32 x y + 2 x y 7 9 6 10 7 8 6 9 5 10 6 8 + 16 x y - 11 x y - 3 x y + 19 x y - 2 x y - 13 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 6 + 8 x y + 3 x y - 13 x y + 10 x y + 2 x y - 3 x y - 6 x y 4 5 3 6 3 5 3 4 2 4 2 3 2 2 + 4 x y + 4 x y - 6 x y + 2 x y + 6 x y - 6 x y + x y 2 / 11 19 11 18 11 17 10 18 + 2 x y - 2 x y + 1) / (x y - 4 x y + 5 x y - x y / 10 17 11 15 10 16 11 14 9 16 11 13 + 4 x y - 5 x y - 5 x y + 4 x y - x y - x y 10 14 9 15 10 13 9 14 10 12 9 13 + 5 x y + 3 x y - 4 x y - 3 x y + x y + 2 x y 8 14 9 12 8 13 9 11 8 12 9 10 8 11 - x y - 3 x y + 2 x y + 3 x y - 2 x y - x y + 4 x y 7 12 8 10 7 11 8 9 7 10 6 11 7 9 - x y - 5 x y + x y + 2 x y - 4 x y + 2 x y + 7 x y 6 10 7 8 6 9 6 8 5 9 6 7 5 8 5 7 - 2 x y - 3 x y + x y - 4 x y + x y + 3 x y + 2 x y + x y 5 6 4 7 4 6 4 5 3 4 2 2 2 - 3 x y - 2 x y - 2 x y + 4 x y - x y + x y - x y - 2 x y + 1 ) and in Maple notation (x^12*y^23-6*x^12*y^22+14*x^12*y^21-14*x^12*y^20+x^11*y^21-6*x^11*y^20+14*x^12* y^18+14*x^11*y^19-14*x^12*y^17-14*x^11*y^18+x^10*y^19+6*x^12*y^16-7*x^10*y^18-x ^12*y^15+14*x^11*y^16+21*x^10*y^17-14*x^11*y^15-35*x^10*y^16+6*x^11*y^14+35*x^ 10*y^15-3*x^9*y^16-x^11*y^13-21*x^10*y^14+16*x^9*y^15-x^8*y^16+7*x^10*y^13-35*x ^9*y^14+5*x^8*y^15-x^10*y^12+40*x^9*y^13-17*x^8*y^14-25*x^9*y^12+38*x^8*y^13-2* x^7*y^14+8*x^9*y^11-49*x^8*y^12+8*x^7*y^13-x^9*y^10+35*x^8*y^11-19*x^7*y^12-13* x^8*y^10+32*x^7*y^11+2*x^8*y^9-32*x^7*y^10+2*x^6*y^11+16*x^7*y^9-11*x^6*y^10-3* x^7*y^8+19*x^6*y^9-2*x^5*y^10-13*x^6*y^8+8*x^5*y^9+3*x^6*y^7-13*x^5*y^8+10*x^5* y^7+2*x^4*y^8-3*x^5*y^6-6*x^4*y^6+4*x^4*y^5+4*x^3*y^6-6*x^3*y^5+2*x^3*y^4+6*x^2 *y^4-6*x^2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(x^11*y^19-4*x^11*y^18+5*x^11*y^17-x^10 *y^18+4*x^10*y^17-5*x^11*y^15-5*x^10*y^16+4*x^11*y^14-x^9*y^16-x^11*y^13+5*x^10 *y^14+3*x^9*y^15-4*x^10*y^13-3*x^9*y^14+x^10*y^12+2*x^9*y^13-x^8*y^14-3*x^9*y^ 12+2*x^8*y^13+3*x^9*y^11-2*x^8*y^12-x^9*y^10+4*x^8*y^11-x^7*y^12-5*x^8*y^10+x^7 *y^11+2*x^8*y^9-4*x^7*y^10+2*x^6*y^11+7*x^7*y^9-2*x^6*y^10-3*x^7*y^8+x^6*y^9-4* x^6*y^8+x^5*y^9+3*x^6*y^7+2*x^5*y^8+x^5*y^7-3*x^5*y^6-2*x^4*y^7-2*x^4*y^6+4*x^4 *y^5-x^3*y^4+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\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (a + a - 2 a + 3 a - 8 a + 2 a - 6 a + 3) - -------------------------------------------------- 6 4 3 2 5 a + 2 a + 2 a - 6 a + 2 a - 3 5 3 2 where a is the root of the polynomial, x - x + x - 3 x + 1, and in decimals this is, 1.490979609 BTW the ratio for words with, 500, letters is, 1.490107855 ------------------------------------------------ "Theorem Number 57" 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, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 12 22 12 21 12 20 ) | ) C(m, n) x y | = (x y - 6 x y + 14 x y / | / | ----- | ----- | m = 0 \ n = 0 / 12 19 11 20 11 19 12 17 11 18 12 16 - 14 x y + x y - 6 x y + 14 x y + 14 x y - 14 x y 11 17 10 18 12 15 10 17 12 14 11 15 - 14 x y - x y + 6 x y + 7 x y - x y + 14 x y 10 16 11 14 10 15 9 16 11 13 10 14 - 21 x y - 14 x y + 35 x y - 3 x y + 6 x y - 35 x y 9 15 8 16 11 12 10 13 9 14 8 15 + 16 x y - x y - x y + 21 x y - 35 x y + 5 x y 10 12 9 13 8 14 10 11 9 12 8 13 - 7 x y + 40 x y - 14 x y + x y - 25 x y + 24 x y 7 14 9 11 8 12 7 13 9 10 8 11 - 2 x y + 8 x y - 22 x y + 10 x y - x y + 7 x y 7 12 8 10 7 11 6 12 8 9 7 10 - 23 x y + 4 x y + 26 x y - 2 x y - 4 x y - 8 x y 6 11 8 8 7 9 6 10 7 8 6 9 5 10 + 12 x y + x y - 10 x y - 24 x y + 9 x y + 15 x y - x y 7 7 6 8 5 9 6 7 5 8 6 6 4 8 - 2 x y + 7 x y + 5 x y - 11 x y - 7 x y + 3 x y + 4 x y 5 6 4 7 5 5 4 6 4 5 3 6 4 4 + 6 x y - 6 x y - 3 x y + 2 x y - 2 x y + 5 x y + 2 x y 3 5 3 4 3 3 2 4 2 3 2 2 2 - 9 x y + 5 x y - x y + 6 x y - 6 x y + x y + 2 x y - 2 x y / 11 18 11 17 11 16 10 17 10 16 + 1) / (x y - 4 x y + 5 x y - x y + 4 x y / 11 14 10 15 11 13 9 15 11 12 10 13 - 5 x y - 5 x y + 4 x y - x y - x y + 5 x y 9 14 10 12 9 13 8 14 10 11 9 12 8 13 + 2 x y - 4 x y + 2 x y + x y + x y - 8 x y - 3 x y 9 11 8 12 9 10 8 11 8 10 7 11 8 9 + 7 x y - x y - 2 x y + 10 x y - 9 x y + 2 x y + x y 7 10 6 11 8 8 7 9 6 10 7 8 6 9 - 9 x y + x y + x y + 8 x y - 2 x y + x y + 4 x y 7 7 6 8 5 9 6 7 6 6 5 7 4 8 5 6 - 2 x y - 3 x y + x y - 3 x y + 3 x y + 2 x y - x y + x y 4 7 5 5 4 6 4 5 3 6 4 4 3 5 3 3 + x y - 3 x y - 3 x y + x y - x y + 2 x y + x y - x y 2 2 2 + x y - x y - 2 x y + 1) and in Maple notation (x^12*y^22-6*x^12*y^21+14*x^12*y^20-14*x^12*y^19+x^11*y^20-6*x^11*y^19+14*x^12* y^17+14*x^11*y^18-14*x^12*y^16-14*x^11*y^17-x^10*y^18+6*x^12*y^15+7*x^10*y^17-x ^12*y^14+14*x^11*y^15-21*x^10*y^16-14*x^11*y^14+35*x^10*y^15-3*x^9*y^16+6*x^11* y^13-35*x^10*y^14+16*x^9*y^15-x^8*y^16-x^11*y^12+21*x^10*y^13-35*x^9*y^14+5*x^8 *y^15-7*x^10*y^12+40*x^9*y^13-14*x^8*y^14+x^10*y^11-25*x^9*y^12+24*x^8*y^13-2*x ^7*y^14+8*x^9*y^11-22*x^8*y^12+10*x^7*y^13-x^9*y^10+7*x^8*y^11-23*x^7*y^12+4*x^ 8*y^10+26*x^7*y^11-2*x^6*y^12-4*x^8*y^9-8*x^7*y^10+12*x^6*y^11+x^8*y^8-10*x^7*y ^9-24*x^6*y^10+9*x^7*y^8+15*x^6*y^9-x^5*y^10-2*x^7*y^7+7*x^6*y^8+5*x^5*y^9-11*x ^6*y^7-7*x^5*y^8+3*x^6*y^6+4*x^4*y^8+6*x^5*y^6-6*x^4*y^7-3*x^5*y^5+2*x^4*y^6-2* x^4*y^5+5*x^3*y^6+2*x^4*y^4-9*x^3*y^5+5*x^3*y^4-x^3*y^3+6*x^2*y^4-6*x^2*y^3+x^2 *y^2+2*x*y^2-2*x*y+1)/(x^11*y^18-4*x^11*y^17+5*x^11*y^16-x^10*y^17+4*x^10*y^16-\ 5*x^11*y^14-5*x^10*y^15+4*x^11*y^13-x^9*y^15-x^11*y^12+5*x^10*y^13+2*x^9*y^14-4 *x^10*y^12+2*x^9*y^13+x^8*y^14+x^10*y^11-8*x^9*y^12-3*x^8*y^13+7*x^9*y^11-x^8*y ^12-2*x^9*y^10+10*x^8*y^11-9*x^8*y^10+2*x^7*y^11+x^8*y^9-9*x^7*y^10+x^6*y^11+x^ 8*y^8+8*x^7*y^9-2*x^6*y^10+x^7*y^8+4*x^6*y^9-2*x^7*y^7-3*x^6*y^8+x^5*y^9-3*x^6* y^7+3*x^6*y^6+2*x^5*y^7-x^4*y^8+x^5*y^6+x^4*y^7-3*x^5*y^5-3*x^4*y^6+x^4*y^5-x^3 *y^6+2*x^4*y^4+x^3*y^5-x^3*y^3+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\ bors of a random word of length n tends to n times, 8 7 6 5 4 3 2 2 (a + a - 2 a + 3 a - 8 a + 2 a - 6 a + 3) - -------------------------------------------------- 6 4 3 2 5 a + 2 a + 2 a - 6 a + 2 a - 3 5 3 2 where a is the root of the polynomial, x - x + x - 3 x + 1, and in decimals this is, 1.490979609 BTW the ratio for words with, 500, letters is, 1.490107855 ------------------------------------------------ "Theorem Number 58" 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, 2, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 6 11 7 9 6 10 6 9 5 10 6 8 - 5 x y + 4 x y + x y - 13 x y + 16 x y - 2 x y - 9 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 7 + 5 x y + 2 x y - 6 x y + 5 x y - 4 x y - 2 x y + 5 x y 4 6 3 5 3 4 2 4 2 3 2 / 8 14 - x y + x y - x y + 4 x y - 3 x y + x y - x y + 1) / (x y / 8 13 8 12 8 11 7 11 7 10 6 11 7 9 - 3 x y + 3 x y - x y + x y - 2 x y - x y + x y 6 10 6 9 6 8 6 7 5 8 5 7 5 6 4 7 + x y - x y + 3 x y - 2 x y - x y - x y + 2 x y + 2 x y 4 5 3 5 3 4 2 4 2 - 2 x y - 2 x y + 3 x y - x y + 2 x y + x y - 1) and in Maple notation -(2*x^7*y^13-7*x^7*y^12+9*x^7*y^11-5*x^7*y^10+4*x^6*y^11+x^7*y^9-13*x^6*y^10+16 *x^6*y^9-2*x^5*y^10-9*x^6*y^8+5*x^5*y^9+2*x^6*y^7-6*x^5*y^8+5*x^5*y^7-4*x^4*y^8 -2*x^5*y^6+5*x^4*y^7-x^4*y^6+x^3*y^5-x^3*y^4+4*x^2*y^4-3*x^2*y^3+x*y^2-x*y+1)/( x^8*y^14-3*x^8*y^13+3*x^8*y^12-x^8*y^11+x^7*y^11-2*x^7*y^10-x^6*y^11+x^7*y^9+x^ 6*y^10-x^6*y^9+3*x^6*y^8-2*x^6*y^7-x^5*y^8-x^5*y^7+2*x^5*y^6+2*x^4*y^7-2*x^4*y^ 5-2*x^3*y^5+3*x^3*y^4-x^2*y^4+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 2 (a - 6 a + 3) ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.481357750 BTW the ratio for words with, 500, letters is, 1.480448037 ------------------------------------------------ "Theorem Number 59" 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, 2, 1]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 ) | ) C(m, n) x y | = - (2 x y - 7 x y + 9 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 6 11 7 9 6 10 6 9 5 10 6 8 - 5 x y + 4 x y + x y - 13 x y + 16 x y - 2 x y - 9 x y 5 9 6 7 5 8 5 7 4 8 5 6 4 7 + 5 x y + 2 x y - 6 x y + 5 x y - 4 x y - 2 x y + 5 x y 4 6 3 5 3 4 2 4 2 3 2 / 8 14 - x y + x y - x y + 4 x y - 3 x y + x y - x y + 1) / (x y / 8 13 8 12 8 11 7 11 7 10 6 11 7 9 - 3 x y + 3 x y - x y + x y - 2 x y - x y + x y 6 10 6 9 6 8 6 7 5 8 5 7 5 6 4 7 + x y - x y + 3 x y - 2 x y - x y - x y + 2 x y + 2 x y 4 5 3 5 3 4 2 4 2 - 2 x y - 2 x y + 3 x y - x y + 2 x y + x y - 1) and in Maple notation -(2*x^7*y^13-7*x^7*y^12+9*x^7*y^11-5*x^7*y^10+4*x^6*y^11+x^7*y^9-13*x^6*y^10+16 *x^6*y^9-2*x^5*y^10-9*x^6*y^8+5*x^5*y^9+2*x^6*y^7-6*x^5*y^8+5*x^5*y^7-4*x^4*y^8 -2*x^5*y^6+5*x^4*y^7-x^4*y^6+x^3*y^5-x^3*y^4+4*x^2*y^4-3*x^2*y^3+x*y^2-x*y+1)/( x^8*y^14-3*x^8*y^13+3*x^8*y^12-x^8*y^11+x^7*y^11-2*x^7*y^10-x^6*y^11+x^7*y^9+x^ 6*y^10-x^6*y^9+3*x^6*y^8-2*x^6*y^7-x^5*y^8-x^5*y^7+2*x^5*y^6+2*x^4*y^7-2*x^4*y^ 5-2*x^3*y^5+3*x^3*y^4-x^2*y^4+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 2 (a - 6 a + 3) ------------------------- 2 2 (3 a - 2 a + 3) (a + 1) 3 2 where a is the root of the polynomial, x - x + 3 x - 1, and in decimals this is, 1.481357750 BTW the ratio for words with, 500, letters is, 1.480448037 ------------------------------------------------ "Theorem Number 60" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 6 12 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 6 11 7 9 6 10 7 8 6 9 5 10 - 10 x y + 9 x y + 5 x y - 16 x y - x y + 14 x y - 3 x y 6 8 5 9 6 7 5 8 4 8 5 6 4 7 4 6 - 6 x y + 8 x y + x y - 6 x y + 2 x y + x y - 3 x y + x y 4 5 3 6 4 4 3 5 3 4 3 3 2 4 2 3 - x y + 3 x y + x y - 7 x y + 5 x y - x y + 5 x y - 6 x y 2 2 2 / 5 8 5 7 5 6 4 7 + x y + 2 x y - 2 x y + 1) / (2 x y - 5 x y + 4 x y - 2 x y / 5 5 4 6 4 5 3 5 3 3 2 4 2 2 2 - x y + 6 x y - 4 x y - 3 x y + x y + x y - x y + x y + 2 x y - 1) and in Maple notation -(x^7*y^13-5*x^7*y^12+10*x^7*y^11-2*x^6*y^12-10*x^7*y^10+9*x^6*y^11+5*x^7*y^9-\ 16*x^6*y^10-x^7*y^8+14*x^6*y^9-3*x^5*y^10-6*x^6*y^8+8*x^5*y^9+x^6*y^7-6*x^5*y^8 +2*x^4*y^8+x^5*y^6-3*x^4*y^7+x^4*y^6-x^4*y^5+3*x^3*y^6+x^4*y^4-7*x^3*y^5+5*x^3* y^4-x^3*y^3+5*x^2*y^4-6*x^2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(2*x^5*y^8-5*x^5*y^7+4 *x^5*y^6-2*x^4*y^7-x^5*y^5+6*x^4*y^6-4*x^4*y^5-3*x^3*y^5+x^3*y^3+x^2*y^4-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\ bors of a random word of length n tends to n times, 4 3 2 2 (4 a + 3 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.467307928 BTW the ratio for words with, 500, letters is, 1.466755616 ------------------------------------------------ "Theorem Number 61" 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, 2, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 7 13 7 12 7 11 6 12 ) | ) C(m, n) x y | = - (x y - 5 x y + 10 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 10 6 11 7 9 6 10 7 8 6 9 5 10 - 10 x y + 9 x y + 5 x y - 16 x y - x y + 14 x y - 3 x y 6 8 5 9 6 7 5 8 4 8 5 6 4 7 4 6 - 6 x y + 8 x y + x y - 6 x y + 2 x y + x y - 3 x y + x y 4 5 3 6 4 4 3 5 3 4 3 3 2 4 2 3 - x y + 3 x y + x y - 7 x y + 5 x y - x y + 5 x y - 6 x y 2 2 2 / 5 8 5 7 5 6 4 7 + x y + 2 x y - 2 x y + 1) / (2 x y - 5 x y + 4 x y - 2 x y / 5 5 4 6 4 5 3 5 3 3 2 4 2 2 2 - x y + 6 x y - 4 x y - 3 x y + x y + x y - x y + x y + 2 x y - 1) and in Maple notation -(x^7*y^13-5*x^7*y^12+10*x^7*y^11-2*x^6*y^12-10*x^7*y^10+9*x^6*y^11+5*x^7*y^9-\ 16*x^6*y^10-x^7*y^8+14*x^6*y^9-3*x^5*y^10-6*x^6*y^8+8*x^5*y^9+x^6*y^7-6*x^5*y^8 +2*x^4*y^8+x^5*y^6-3*x^4*y^7+x^4*y^6-x^4*y^5+3*x^3*y^6+x^4*y^4-7*x^3*y^5+5*x^3* y^4-x^3*y^3+5*x^2*y^4-6*x^2*y^3+x^2*y^2+2*x*y^2-2*x*y+1)/(2*x^5*y^8-5*x^5*y^7+4 *x^5*y^6-2*x^4*y^7-x^5*y^5+6*x^4*y^6-4*x^4*y^5-3*x^3*y^5+x^3*y^3+x^2*y^4-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\ bors of a random word of length n tends to n times, 4 3 2 2 (4 a + 3 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.467307928 BTW the ratio for words with, 500, letters is, 1.466755616 ------------------------------------------------ "Theorem Number 62" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 ) | ) C(m, n) x y | = (4 x y - 12 x y + 12 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 6 4 5 2 4 2 3 2 - 2 x y + 2 x y + 2 x y - 2 x y - 4 x y + 4 x y - x y + x y / - 1) / ( / 4 7 4 6 3 5 3 4 2 4 2 3 2 2 x y - 2 x y - 4 x y + 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation (4*x^5*y^10-12*x^5*y^9+12*x^5*y^8-4*x^5*y^7-2*x^4*y^8+2*x^4*y^7+2*x^4*y^6-2*x^4 *y^5-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1)/(2*x^4*y^7-2*x^4*y^6-4*x^3*y^5+2*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, 4 3 2 2 (4 a + 2 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.422649729 BTW the ratio for words with, 500, letters is, 1.422519224 ------------------------------------------------ "Theorem Number 63" 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, 3, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 ) | ) C(m, n) x y | = (4 x y - 12 x y + 12 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 6 4 5 2 4 2 3 2 - 2 x y + 2 x y + 2 x y - 2 x y - 4 x y + 4 x y - x y + x y / - 1) / ( / 4 7 4 6 3 5 3 4 2 4 2 3 2 2 x y - 2 x y - 4 x y + 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation (4*x^5*y^10-12*x^5*y^9+12*x^5*y^8-4*x^5*y^7-2*x^4*y^8+2*x^4*y^7+2*x^4*y^6-2*x^4 *y^5-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1)/(2*x^4*y^7-2*x^4*y^6-4*x^3*y^5+2*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, 4 3 2 2 (4 a + 2 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.422649729 BTW the ratio for words with, 500, letters is, 1.422519224 ------------------------------------------------ "Theorem Number 64" 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, 3, 2]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 ) | ) C(m, n) x y | = (4 x y - 12 x y + 12 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 6 4 5 2 4 2 3 2 - 2 x y + 2 x y + 2 x y - 2 x y - 4 x y + 4 x y - x y + x y / - 1) / ( / 4 7 4 6 3 5 3 4 2 4 2 3 2 2 x y - 2 x y - 4 x y + 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation (4*x^5*y^10-12*x^5*y^9+12*x^5*y^8-4*x^5*y^7-2*x^4*y^8+2*x^4*y^7+2*x^4*y^6-2*x^4 *y^5-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1)/(2*x^4*y^7-2*x^4*y^6-4*x^3*y^5+2*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, 4 3 2 2 (4 a + 2 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.422649729 BTW the ratio for words with, 500, letters is, 1.422519224 ------------------------------------------------ "Theorem Number 65" 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, 1, 3]}, and h\ aving m neighbors (i.e. Hamming distance 1) that also obey this property\ , then infinity /infinity \ ----- | ----- | \ | \ n m| 5 10 5 9 5 8 5 7 ) | ) C(m, n) x y | = (4 x y - 12 x y + 12 x y - 4 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 8 4 7 4 6 4 5 2 4 2 3 2 - 2 x y + 2 x y + 2 x y - 2 x y - 4 x y + 4 x y - x y + x y / - 1) / ( / 4 7 4 6 3 5 3 4 2 4 2 3 2 2 x y - 2 x y - 4 x y + 2 x y - x y + x y + 2 x y + x y - 1) and in Maple notation (4*x^5*y^10-12*x^5*y^9+12*x^5*y^8-4*x^5*y^7-2*x^4*y^8+2*x^4*y^7+2*x^4*y^6-2*x^4 *y^5-4*x^2*y^4+4*x^2*y^3-x*y^2+x*y-1)/(2*x^4*y^7-2*x^4*y^6-4*x^3*y^5+2*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, 4 3 2 2 (4 a + 2 a - 12 a + 3) - --------------------------- 2 3 (2 a - 1) 3 where a is the root of the polynomial, 2 x - 3 x + 1, and in decimals this is, 1.422649729 BTW the ratio for words with, 500, letters is, 1.422519224 ---------------------------------- This ends this paper that took, 977.273, seconds to produce ------------------------------