Counting Words in the Alphabet, {1, 2}, That Avoid A Certain set of , 1, consecutive subwords of length, 3, For all the possibilities, According to their Number of Good Neighbors By Shalosh B. Ekhad ------------------------------------------------ "Theorem Number 1" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| ) | ) C(m, n) x y | = / | / | ----- | ----- | m = 0 \ n = 0 / 5 5 5 4 5 3 3 3 3 2 2 2 2 x y - 2 x y + x y + x y - x y - 2 x y + x y - x y - 1 - ------------------------------------------------------------------ 6 5 6 4 6 3 4 3 4 2 3 3 2 x y - 2 x y + x y + x y - x y - x y - x y - x y + 1 and in Maple notation -(x^5*y^5-2*x^5*y^4+x^5*y^3+x^3*y^3-x^3*y^2-2*x^2*y^2+x^2*y-x*y-1)/(x^6*y^5-2*x ^6*y^4+x^6*y^3+x^4*y^3-x^4*y^2-x^3*y^3-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 (2 a + 1) ----------------------------- 2 2 (3 a + 2 a + 1) (a + a + 1) 3 2 where a is the root of the polynomial, x + x + x - 1, and in decimals this is, 0.7631601556 BTW the ratio for words with, 500, letters is, 0.7623086554 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 5 5 4 5 3 4 4 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 3 4 2 4 3 2 3 2 2 2 - 4 x y + 3 x y - x y - x y + x y - 2 x y + x y - x y + x - 1) / 6 5 6 4 6 3 5 3 5 2 4 3 4 3 2 / (x y - 2 x y + x y + x y - x y - x y + x y - x y + x y / - x y - x + 1) and in Maple notation -(x^5*y^5-2*x^5*y^4+x^5*y^3+2*x^4*y^4-4*x^4*y^3+3*x^4*y^2-x^4*y-x^3*y^2+x^3*y-2 *x^2*y^2+x^2*y-x*y+x-1)/(x^6*y^5-2*x^6*y^4+x^6*y^3+x^5*y^3-x^5*y^2-x^4*y^3+x^4* y-x^3*y+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 2 (a - 1) - ------------------------- 2 2 (3 a - 2 a + 2) (a + 1) 3 2 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.5557511628 BTW the ratio for words with, 500, letters is, 0.5561976731 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 5 5 4 5 3 5 2 4 3 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 2 4 2 2 2 / + 2 x y - x y - 2 x y + 2 x y - x y + x - 1) / ( / 4 2 4 3 2 x y - x y - x y + x y + x - 1) and in Maple notation (x^5*y^5-3*x^5*y^4+3*x^5*y^3-x^5*y^2-x^4*y^3+2*x^4*y^2-x^4*y-2*x^2*y^2+2*x^2*y- x*y+x-1)/(x^4*y^2-x^4*y-x^3*y^2+x*y+x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5533963303 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2]}, and having m nei\ ghbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 5 5 5 4 5 3 5 2 4 3 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y - x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 4 2 4 2 2 2 / + 2 x y - x y - 2 x y + 2 x y - x y + x - 1) / ( / 3 2 3 2 (x y - x y - x y - x y + 1) (x - 1)) and in Maple notation (x^5*y^5-3*x^5*y^4+3*x^5*y^3-x^5*y^2-x^4*y^3+2*x^4*y^2-x^4*y-2*x^2*y^2+2*x^2*y- x*y+x-1)/(x^3*y^2-x^3*y-x^2*y-x*y+1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.5533963303 [This estimate was obtained by looking at words of length, 500, ] ---------------------------------- This ends this paper that took, 0.637, seconds to produce ------------------------------ Counting Words in the Alphabet, {1, 2}, That Avoid A Certain set of , 1, consecutive subwords of length, 4, For all the possibilities, According to their Number of Good Neighbors By Shalosh B. Ekhad ------------------------------------------------ "Theorem Number 1" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 9 6 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 7 7 6 7 5 5 5 5 4 5 3 4 4 4 3 - x y + 2 x y - x y - 2 x y + 3 x y - x y - 2 x y + 2 x y 3 3 3 2 2 2 / 10 9 10 8 10 7 + 2 x y - x y + x y + x y + 1) / (x y - 3 x y + 3 x y / 10 6 8 7 8 6 8 5 6 6 6 5 5 4 5 3 - x y - x y + 2 x y - x y - x y + x y - 2 x y + 2 x y 4 4 3 2 2 2 + x y + x y + x y + x y - 1) and in Maple notation -(x^9*y^9-3*x^9*y^8+3*x^9*y^7-x^9*y^6-x^7*y^7+2*x^7*y^6-x^7*y^5-2*x^5*y^5+3*x^5 *y^4-x^5*y^3-2*x^4*y^4+2*x^4*y^3+2*x^3*y^3-x^3*y^2+x^2*y^2+x*y+1)/(x^10*y^9-3*x ^10*y^8+3*x^10*y^7-x^10*y^6-x^8*y^7+2*x^8*y^6-x^8*y^5-x^6*y^6+x^6*y^5-2*x^5*y^4 +2*x^5*y^3+x^4*y^4+x^3*y^2+x^2*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 2 2 (3 a + 2 a + 1) ------------------------------------------- 6 5 4 3 2 4 a + 7 a + 9 a + 10 a + 6 a + 3 a + 1 4 3 2 where a is the root of the polynomial, x + x + x + x - 1, and in decimals this is, 0.8673142246 BTW the ratio for words with, 500, letters is, 0.8661704526 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 8 8 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 6 8 7 9 5 8 6 7 7 8 5 7 6 - 4 x y - 4 x y + x y + 6 x y + 2 x y - 4 x y - 7 x y 8 4 7 5 6 6 7 4 6 5 7 3 6 4 5 5 + x y + 9 x y + x y - 5 x y - 4 x y + x y + 5 x y - x y 6 3 5 4 5 3 4 4 5 2 4 3 4 2 3 3 - 2 x y + x y + x y - x y - x y + 2 x y - x y - 3 x y 3 2 3 2 2 2 / 7 5 7 4 + 4 x y - x y - 2 x y + 2 x y - x y + x - 1) / (x y - 2 x y / 6 5 7 3 6 4 6 3 5 3 5 2 4 3 4 2 - x y + x y + 2 x y - x y + x y - x y - 2 x y + x y 3 3 3 + x y - x y + x y + x - 1) and in Maple notation (x^9*y^9-4*x^9*y^8+6*x^9*y^7+x^8*y^8-4*x^9*y^6-4*x^8*y^7+x^9*y^5+6*x^8*y^6+2*x^ 7*y^7-4*x^8*y^5-7*x^7*y^6+x^8*y^4+9*x^7*y^5+x^6*y^6-5*x^7*y^4-4*x^6*y^5+x^7*y^3 +5*x^6*y^4-x^5*y^5-2*x^6*y^3+x^5*y^4+x^5*y^3-x^4*y^4-x^5*y^2+2*x^4*y^3-x^4*y^2-\ 3*x^3*y^3+4*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^7*y^5-2*x^7*y^4-x^6*y^5 +x^7*y^3+2*x^6*y^4-x^6*y^3+x^5*y^3-x^5*y^2-2*x^4*y^3+x^4*y^2+x^3*y^3-x^3*y+x*y+ x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.7629624354 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 8 8 ) | ) C(m, n) x y | = (x y - 4 x y + 6 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 6 8 7 9 5 8 6 7 7 8 5 7 6 - 4 x y - 4 x y + x y + 6 x y + 2 x y - 4 x y - 7 x y 8 4 7 5 6 6 7 4 6 5 7 3 6 4 5 5 + x y + 9 x y + x y - 5 x y - 4 x y + x y + 5 x y - x y 6 3 5 4 5 3 4 4 5 2 4 3 4 2 3 3 - 2 x y + x y + x y - x y - x y + 2 x y - x y - 3 x y 3 2 3 2 2 2 / + 4 x y - x y - 2 x y + 2 x y - x y + x - 1) / ((x - 1) / 6 5 6 4 6 3 4 3 4 2 3 3 2 (x y - 2 x y + x y + x y - x y - x y - x y - x y + 1)) and in Maple notation (x^9*y^9-4*x^9*y^8+6*x^9*y^7+x^8*y^8-4*x^9*y^6-4*x^8*y^7+x^9*y^5+6*x^8*y^6+2*x^ 7*y^7-4*x^8*y^5-7*x^7*y^6+x^8*y^4+9*x^7*y^5+x^6*y^6-5*x^7*y^4-4*x^6*y^5+x^7*y^3 +5*x^6*y^4-x^5*y^5-2*x^6*y^3+x^5*y^4+x^5*y^3-x^4*y^4-x^5*y^2+2*x^4*y^3-x^4*y^2-\ 3*x^3*y^3+4*x^3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x-1)/(x^6*y^5-2*x^6*y^4+x ^6*y^3+x^4*y^3-x^4*y^2-x^3*y^3-x^2*y-x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.7629624354 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 6 6 6 5 6 4 5 5 ) | ) C(m, n) x y | = - (x y - 2 x y + x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 5 4 5 3 4 4 4 3 3 3 3 2 2 2 + 4 x y - 2 x y - 2 x y + 2 x y + 2 x y - 2 x y + x y + 1) / 6 6 6 4 5 5 5 3 4 4 4 3 3 2 2 2 / (x y - x y - 2 x y + 2 x y + x y - 2 x y + 2 x y - x y / + 2 x y - 1) and in Maple notation -(x^6*y^6-2*x^6*y^5+x^6*y^4-2*x^5*y^5+4*x^5*y^4-2*x^5*y^3-2*x^4*y^4+2*x^4*y^3+2 *x^3*y^3-2*x^3*y^2+x^2*y^2+1)/(x^6*y^6-x^6*y^4-2*x^5*y^5+2*x^5*y^3+x^4*y^4-2*x^ 4*y^3+2*x^3*y^2-x^2*y^2+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 5 4 3 2 a - 2 a + a - 4 a + 2 a + 1 - -------------------------------- 3 2 2 (2 a - 3 a + a - 1) (a + 1) 4 3 2 where a is the root of the polynomial, x - 2 x + x - 2 x + 1, and in decimals this is, 0.7547496112 BTW the ratio for words with, 500, letters is, 0.7546115187 ------------------------------------------------ "Theorem Number 5" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 7 7 6 7 5 6 6 ) | ) C(m, n) x y | = - (x y - 3 x y + 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 4 6 5 7 3 6 4 5 5 6 3 5 4 5 3 - 3 x y - 2 x y + x y + 2 x y - x y - x y + x y + x y 4 4 5 2 4 3 4 2 3 3 3 2 3 2 2 - 2 x y - x y + 3 x y - x y - 4 x y + 4 x y - x y - 2 x y 2 / 8 6 8 5 8 4 7 4 6 5 + 2 x y - x y + x - 1) / (x y - 2 x y + x y + x y - x y / 7 3 6 4 5 3 4 4 5 2 4 3 4 2 3 - x y + x y - x y - x y + x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^7*y^7-3*x^7*y^6+4*x^7*y^5+x^6*y^6-3*x^7*y^4-2*x^6*y^5+x^7*y^3+2*x^6*y^4-x^5 *y^5-x^6*y^3+x^5*y^4+x^5*y^3-2*x^4*y^4-x^5*y^2+3*x^4*y^3-x^4*y^2-4*x^3*y^3+4*x^ 3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4-x^6*y ^5-x^7*y^3+x^6*y^4-x^5*y^3-x^4*y^4+x^5*y^2+x^4*y^3-x^4*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 5 3 2 (a - 2 a + 1) -------------------------- 3 2 3 (4 a - 3 a + 2) (a + 1) 4 3 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.7280670770 BTW the ratio for words with, 500, letters is, 0.7281000451 ------------------------------------------------ "Theorem Number 6" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 7 7 7 6 7 5 6 6 ) | ) C(m, n) x y | = - (x y - 3 x y + 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 4 6 5 7 3 6 4 5 5 6 3 5 4 5 3 - 3 x y - 2 x y + x y + 2 x y - x y - x y + x y + x y 4 4 5 2 4 3 4 2 3 3 3 2 3 2 2 - 2 x y - x y + 3 x y - x y - 4 x y + 4 x y - x y - 2 x y 2 / 8 6 8 5 8 4 7 4 6 5 + 2 x y - x y + x - 1) / (x y - 2 x y + x y + x y - x y / 7 3 6 4 5 3 4 4 5 2 4 3 4 2 3 - x y + x y - x y - x y + x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^7*y^7-3*x^7*y^6+4*x^7*y^5+x^6*y^6-3*x^7*y^4-2*x^6*y^5+x^7*y^3+2*x^6*y^4-x^5 *y^5-x^6*y^3+x^5*y^4+x^5*y^3-2*x^4*y^4-x^5*y^2+3*x^4*y^3-x^4*y^2-4*x^3*y^3+4*x^ 3*y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^8*y^6-2*x^8*y^5+x^8*y^4+x^7*y^4-x^6*y ^5-x^7*y^3+x^6*y^4-x^5*y^3-x^4*y^4+x^5*y^2+x^4*y^3-x^4*y^2+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 5 3 2 (a - 2 a + 1) -------------------------- 3 2 3 (4 a - 3 a + 2) (a + 1) 4 3 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.7280670770 BTW the ratio for words with, 500, letters is, 0.7281000451 ------------------------------------------------ "Theorem Number 7" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 8 8 8 7 8 6 7 7 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 5 7 6 8 4 7 5 6 6 7 4 6 5 6 4 - 4 x y + 6 x y + x y - 6 x y + x y + 2 x y - 2 x y + x y 4 4 4 3 4 2 3 3 3 2 / 5 4 - 2 x y + 4 x y - 2 x y + 2 x y - 2 x y + 1) / (2 x y / 5 3 4 4 4 3 4 2 3 3 3 2 - 2 x y + x y - 4 x y + 2 x y - 2 x y + 2 x y + 2 x y - 1) and in Maple notation -(x^8*y^8-4*x^8*y^7+6*x^8*y^6-2*x^7*y^7-4*x^8*y^5+6*x^7*y^6+x^8*y^4-6*x^7*y^5+x ^6*y^6+2*x^7*y^4-2*x^6*y^5+x^6*y^4-2*x^4*y^4+4*x^4*y^3-2*x^4*y^2+2*x^3*y^3-2*x^ 3*y^2+1)/(2*x^5*y^4-2*x^5*y^3+x^4*y^4-4*x^4*y^3+2*x^4*y^2-2*x^3*y^3+2*x^3*y^2+2 *x*y-1) As the length of the word goes to infinity, the average number of good neigh\ 6 4 3 a + a - 4 a + 1 bors of a random word of length n tends to n times, - ------------------ 3 2 a - 1 4 where a is the root of the polynomial, x - 2 x + 1, and in decimals this is, 0.6931508323 BTW the ratio for words with, 500, letters is, 0.6939901218 ------------------------------------------------ "Theorem Number 8" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 1]}, and having m \ neighbors (i.e. Hamming distance 1) that also obey this property, then infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 8 8 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 6 8 7 8 6 7 7 8 5 7 6 8 4 7 5 - x y - 4 x y + 6 x y + x y - 4 x y - 4 x y + x y + 6 x y 6 6 7 4 6 5 7 3 6 4 5 5 5 4 - 2 x y - 4 x y + 3 x y + x y - x y - 2 x y + 4 x y 5 3 4 4 4 3 4 2 3 3 3 / 10 9 - 2 x y - 2 x y + 4 x y - 2 x y + 2 x y - x y + 1) / (x y / 10 8 10 7 9 8 10 6 9 7 9 6 9 5 7 6 - 3 x y + 3 x y - x y - x y + 3 x y - 3 x y + x y - x y 7 4 6 5 7 3 6 4 4 4 3 3 3 + 2 x y + x y - x y - x y + x y - 2 x y + x y + 2 x y - 1) and in Maple notation -(x^9*y^9-3*x^9*y^8+3*x^9*y^7+x^8*y^8-x^9*y^6-4*x^8*y^7+6*x^8*y^6+x^7*y^7-4*x^8 *y^5-4*x^7*y^6+x^8*y^4+6*x^7*y^5-2*x^6*y^6-4*x^7*y^4+3*x^6*y^5+x^7*y^3-x^6*y^4-\ 2*x^5*y^5+4*x^5*y^4-2*x^5*y^3-2*x^4*y^4+4*x^4*y^3-2*x^4*y^2+2*x^3*y^3-x^3*y+1)/ (x^10*y^9-3*x^10*y^8+3*x^10*y^7-x^9*y^8-x^10*y^6+3*x^9*y^7-3*x^9*y^6+x^9*y^5-x^ 7*y^6+2*x^7*y^4+x^6*y^5-x^7*y^3-x^6*y^4+x^4*y^4-2*x^3*y^3+x^3*y+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 3 2 (2 a - 1) - -------------------------- 3 2 3 (4 a - 3 a + 2) (a + 1) 4 3 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.6844701446 BTW the ratio for words with, 500, letters is, 0.6850191346 ---------------------------------- This ends this paper that took, 7.138, seconds to produce ------------------------------ Counting Words in the Alphabet, {1, 2}, That Avoid A Certain set of , 1, consecutive subwords of length, 5, For all the possibilities, According to their Number of Good Neighbors By Shalosh B. Ekhad ------------------------------------------------ "Theorem Number 1" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 14 14 14 13 14 12 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 11 14 10 11 11 11 10 11 9 11 8 9 9 - 4 x y + x y + x y - 3 x y + 3 x y - x y - 3 x y 9 8 9 7 8 8 9 6 8 7 8 6 6 6 + 7 x y - 5 x y - 2 x y + x y + 4 x y - 2 x y - 2 x y 6 5 6 4 5 5 5 4 4 4 4 3 3 3 + 3 x y - x y - 2 x y + 2 x y + 3 x y - 2 x y + 2 x y 3 2 2 2 / 15 14 15 13 15 12 - x y + x y + x y + 1) / (x y - 4 x y + 6 x y / 15 11 15 10 12 11 12 10 12 9 12 8 10 10 - 4 x y + x y + x y - 3 x y + 3 x y - x y - x y 10 9 10 8 10 7 9 8 9 7 9 6 7 7 7 6 + x y + x y - x y - 2 x y + 4 x y - 2 x y - x y + x y 6 5 6 4 5 5 5 4 4 4 3 2 2 2 - 2 x y + 2 x y + 2 x y - x y + x y + x y + x y + x y - 1) and in Maple notation -(x^14*y^14-4*x^14*y^13+6*x^14*y^12-4*x^14*y^11+x^14*y^10+x^11*y^11-3*x^11*y^10 +3*x^11*y^9-x^11*y^8-3*x^9*y^9+7*x^9*y^8-5*x^9*y^7-2*x^8*y^8+x^9*y^6+4*x^8*y^7-\ 2*x^8*y^6-2*x^6*y^6+3*x^6*y^5-x^6*y^4-2*x^5*y^5+2*x^5*y^4+3*x^4*y^4-2*x^4*y^3+2 *x^3*y^3-x^3*y^2+x^2*y^2+x*y+1)/(x^15*y^14-4*x^15*y^13+6*x^15*y^12-4*x^15*y^11+ x^15*y^10+x^12*y^11-3*x^12*y^10+3*x^12*y^9-x^12*y^8-x^10*y^10+x^10*y^9+x^10*y^8 -x^10*y^7-2*x^9*y^8+4*x^9*y^7-2*x^9*y^6-x^7*y^7+x^7*y^6-2*x^6*y^5+2*x^6*y^4+2*x ^5*y^5-x^5*y^4+x^4*y^4+x^3*y^2+x^2*y^2+x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 3 2 2 (4 a + 3 a + 2 a + 1) ----------------------------------------------------- 4 3 2 4 3 2 (5 a + 4 a + 3 a + 2 a + 1) (a + a + a + a + 1) 5 4 3 2 where a is the root of the polynomial, x + x + x + x + x - 1, and in decimals this is, 0.9241477666 BTW the ratio for words with, 500, letters is, 0.9227677954 ------------------------------------------------ "Theorem Number 2" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 8 9 7 8 8 9 6 8 7 8 6 7 7 - 6 x y + 6 x y - x y - 2 x y + 2 x y - x y + 2 x y 7 6 7 5 6 6 6 5 6 4 5 5 5 4 - 4 x y + 2 x y + 3 x y - 4 x y + x y + 2 x y - 4 x y 5 3 4 4 4 3 2 2 / 11 10 11 9 + 2 x y - 3 x y + 2 x y - x y - 1) / (x y - 2 x y / 10 10 11 8 10 9 10 8 9 9 10 7 9 8 + x y + x y - 4 x y + 5 x y - 2 x y - 2 x y + 6 x y 9 7 8 8 9 6 8 7 8 6 7 7 7 6 - 6 x y + 3 x y + 2 x y - 6 x y + 3 x y - 2 x y + 6 x y 7 5 6 6 6 5 6 4 5 5 5 4 5 3 - 4 x y + x y - 4 x y + 3 x y - 2 x y + 3 x y - 2 x y 4 4 4 3 3 3 2 2 + 3 x y - 2 x y - 2 x y + x y - 2 x y + 1) and in Maple notation -(x^10*y^10-2*x^10*y^9+x^10*y^8+2*x^9*y^9-6*x^9*y^8+6*x^9*y^7-x^8*y^8-2*x^9*y^6 +2*x^8*y^7-x^8*y^6+2*x^7*y^7-4*x^7*y^6+2*x^7*y^5+3*x^6*y^6-4*x^6*y^5+x^6*y^4+2* x^5*y^5-4*x^5*y^4+2*x^5*y^3-3*x^4*y^4+2*x^4*y^3-x^2*y^2-1)/(x^11*y^10-2*x^11*y^ 9+x^10*y^10+x^11*y^8-4*x^10*y^9+5*x^10*y^8-2*x^9*y^9-2*x^10*y^7+6*x^9*y^8-6*x^9 *y^7+3*x^8*y^8+2*x^9*y^6-6*x^8*y^7+3*x^8*y^6-2*x^7*y^7+6*x^7*y^6-4*x^7*y^5+x^6* y^6-4*x^6*y^5+3*x^6*y^4-2*x^5*y^5+3*x^5*y^4-2*x^5*y^3+3*x^4*y^4-2*x^4*y^3-2*x^3 *y^3+x^2*y^2-2*x*y+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 4 2 2 (a + 2 a - 2 a - 1) - -------------------------------------------- 4 3 2 4 2 (5 a - 4 a + 6 a - 2 a + 2) (a + a + 1) 5 4 3 2 where a is the root of the polynomial, x - x + 2 x - x + 2 x - 1, and in decimals this is, 0.8706468852 BTW the ratio for words with, 500, letters is, 0.8698157702 ------------------------------------------------ "Theorem Number 3" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 1, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 14 14 14 13 14 12 13 13 ) | ) C(m, n) x y | = (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 11 13 12 14 10 13 11 14 9 13 10 - 10 x y - 5 x y + 5 x y + 10 x y - x y - 10 x y 12 11 13 9 12 10 13 8 12 9 11 10 12 8 - x y + 5 x y + 4 x y - x y - 6 x y - x y + 4 x y 11 9 10 10 12 7 11 8 10 9 11 7 10 8 + 4 x y - x y - x y - 6 x y + 3 x y + 4 x y - 3 x y 9 9 11 6 10 7 9 8 9 7 8 8 9 6 - 4 x y - x y + x y + 13 x y - 15 x y - 3 x y + 7 x y 8 7 9 5 8 6 7 7 8 5 7 6 7 5 + 10 x y - x y - 11 x y - x y + 4 x y + 5 x y - 7 x y 6 6 7 4 6 5 6 4 6 3 4 4 4 3 - x y + 3 x y + 4 x y - 5 x y + 2 x y + 3 x y - 4 x y 4 2 3 3 3 2 2 2 2 / 11 9 + x y + 2 x y - 2 x y + x y - x y + x y - x + 1) / (x y / 11 8 10 9 11 7 10 8 11 6 10 7 10 6 - 3 x y - x y + 3 x y + 3 x y - x y - 3 x y + x y 9 7 9 6 8 7 9 5 8 6 8 5 7 6 7 5 - x y + 2 x y + x y - x y - 2 x y + x y - x y + x y 6 6 6 5 6 4 6 3 5 4 5 3 4 4 4 2 + x y - x y - 2 x y + 2 x y + 3 x y - 2 x y - x y + x y 2 2 2 - x y + x y - x y - x + 1) and in Maple notation (x^14*y^14-5*x^14*y^13+10*x^14*y^12+x^13*y^13-10*x^14*y^11-5*x^13*y^12+5*x^14*y ^10+10*x^13*y^11-x^14*y^9-10*x^13*y^10-x^12*y^11+5*x^13*y^9+4*x^12*y^10-x^13*y^ 8-6*x^12*y^9-x^11*y^10+4*x^12*y^8+4*x^11*y^9-x^10*y^10-x^12*y^7-6*x^11*y^8+3*x^ 10*y^9+4*x^11*y^7-3*x^10*y^8-4*x^9*y^9-x^11*y^6+x^10*y^7+13*x^9*y^8-15*x^9*y^7-\ 3*x^8*y^8+7*x^9*y^6+10*x^8*y^7-x^9*y^5-11*x^8*y^6-x^7*y^7+4*x^8*y^5+5*x^7*y^6-7 *x^7*y^5-x^6*y^6+3*x^7*y^4+4*x^6*y^5-5*x^6*y^4+2*x^6*y^3+3*x^4*y^4-4*x^4*y^3+x^ 4*y^2+2*x^3*y^3-2*x^3*y^2+x^2*y^2-x^2*y+x*y-x+1)/(x^11*y^9-3*x^11*y^8-x^10*y^9+ 3*x^11*y^7+3*x^10*y^8-x^11*y^6-3*x^10*y^7+x^10*y^6-x^9*y^7+2*x^9*y^6+x^8*y^7-x^ 9*y^5-2*x^8*y^6+x^8*y^5-x^7*y^6+x^7*y^5+x^6*y^6-x^6*y^5-2*x^6*y^4+2*x^6*y^3+3*x ^5*y^4-2*x^5*y^3-x^4*y^4+x^4*y^2-x^2*y^2+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.8665649408 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 4" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 2, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 14 14 14 13 14 12 13 13 ) | ) C(m, n) x y | = (x y - 5 x y + 10 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 14 11 13 12 14 10 13 11 14 9 13 10 - 10 x y - 5 x y + 5 x y + 10 x y - x y - 10 x y 12 11 13 9 12 10 13 8 12 9 11 10 12 8 - x y + 5 x y + 4 x y - x y - 6 x y - x y + 4 x y 11 9 10 10 12 7 11 8 10 9 11 7 10 8 + 4 x y - x y - x y - 6 x y + 3 x y + 4 x y - 3 x y 9 9 11 6 10 7 9 8 9 7 8 8 9 6 - 4 x y - x y + x y + 13 x y - 15 x y - 3 x y + 7 x y 8 7 9 5 8 6 7 7 8 5 7 6 7 5 + 10 x y - x y - 11 x y - x y + 4 x y + 5 x y - 7 x y 6 6 7 4 6 5 6 4 6 3 4 4 4 3 - x y + 3 x y + 4 x y - 5 x y + 2 x y + 3 x y - 4 x y 4 2 3 3 3 2 2 2 2 / 10 9 + x y + 2 x y - 2 x y + x y - x y + x y - x + 1) / ((x y / 10 8 10 7 10 6 8 7 8 6 8 5 6 6 6 5 - 3 x y + 3 x y - x y - x y + 2 x y - x y - x y + x y 5 4 5 3 4 4 3 2 2 2 - 2 x y + 2 x y + x y + x y + x y + x y - 1) (x - 1)) and in Maple notation (x^14*y^14-5*x^14*y^13+10*x^14*y^12+x^13*y^13-10*x^14*y^11-5*x^13*y^12+5*x^14*y ^10+10*x^13*y^11-x^14*y^9-10*x^13*y^10-x^12*y^11+5*x^13*y^9+4*x^12*y^10-x^13*y^ 8-6*x^12*y^9-x^11*y^10+4*x^12*y^8+4*x^11*y^9-x^10*y^10-x^12*y^7-6*x^11*y^8+3*x^ 10*y^9+4*x^11*y^7-3*x^10*y^8-4*x^9*y^9-x^11*y^6+x^10*y^7+13*x^9*y^8-15*x^9*y^7-\ 3*x^8*y^8+7*x^9*y^6+10*x^8*y^7-x^9*y^5-11*x^8*y^6-x^7*y^7+4*x^8*y^5+5*x^7*y^6-7 *x^7*y^5-x^6*y^6+3*x^7*y^4+4*x^6*y^5-5*x^6*y^4+2*x^6*y^3+3*x^4*y^4-4*x^4*y^3+x^ 4*y^2+2*x^3*y^3-2*x^3*y^2+x^2*y^2-x^2*y+x*y-x+1)/(x^10*y^9-3*x^10*y^8+3*x^10*y^ 7-x^10*y^6-x^8*y^7+2*x^8*y^6-x^8*y^5-x^6*y^6+x^6*y^5-2*x^5*y^4+2*x^5*y^3+x^4*y^ 4+x^3*y^2+x^2*y^2+x*y-1)/(x-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, approximaley to, 0.8665649408 [This estimate was obtained by looking at words of length, 500, ] ------------------------------------------------ "Theorem Number 5" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = - (x y - 2 x y + x y + 5 x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 8 9 7 8 8 9 6 8 7 8 6 7 7 - 11 x y + 7 x y + 8 x y - x y - 17 x y + 11 x y + 9 x y 8 5 7 6 7 5 6 6 7 4 6 5 6 4 - 2 x y - 18 x y + 12 x y + 3 x y - 3 x y - 8 x y + 8 x y 5 5 6 3 5 4 5 3 4 4 5 2 4 3 4 2 - x y - 3 x y - x y + 3 x y - 4 x y - x y + 2 x y + x y 3 3 3 2 2 2 2 / 11 10 11 9 - 2 x y + x y - x y + x y - x y + x - 1) / (x y - 2 x y / 10 10 11 8 10 9 10 8 9 9 10 7 9 8 + x y + x y + x y - 5 x y + 2 x y + 3 x y - x y 9 7 8 8 9 6 8 7 8 6 8 5 7 6 - 3 x y + 2 x y + 2 x y - 3 x y + 3 x y - 2 x y - 2 x y 7 5 6 6 7 4 6 5 6 4 5 5 6 3 5 3 + 4 x y - x y - 2 x y - x y + x y - x y + x y - x y 5 2 4 2 3 2 2 2 2 + x y - x y + x y - x y + x y - x y - x + 1) and in Maple notation -(x^10*y^10-2*x^10*y^9+x^10*y^8+5*x^9*y^9-11*x^9*y^8+7*x^9*y^7+8*x^8*y^8-x^9*y^ 6-17*x^8*y^7+11*x^8*y^6+9*x^7*y^7-2*x^8*y^5-18*x^7*y^6+12*x^7*y^5+3*x^6*y^6-3*x ^7*y^4-8*x^6*y^5+8*x^6*y^4-x^5*y^5-3*x^6*y^3-x^5*y^4+3*x^5*y^3-4*x^4*y^4-x^5*y^ 2+2*x^4*y^3+x^4*y^2-2*x^3*y^3+x^3*y^2-x^2*y^2+x^2*y-x*y+x-1)/(x^11*y^10-2*x^11* y^9+x^10*y^10+x^11*y^8+x^10*y^9-5*x^10*y^8+2*x^9*y^9+3*x^10*y^7-x^9*y^8-3*x^9*y ^7+2*x^8*y^8+2*x^9*y^6-3*x^8*y^7+3*x^8*y^6-2*x^8*y^5-2*x^7*y^6+4*x^7*y^5-x^6*y^ 6-2*x^7*y^4-x^6*y^5+x^6*y^4-x^5*y^5+x^6*y^3-x^5*y^3+x^5*y^2-x^4*y^2+x^3*y^2-x^2 *y^2+x^2*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 6 4 3 2 (a - 3 a + 2 a + 1) -------------------------------------- 4 3 2 4 3 (5 a + 4 a - 3 a + 2) (a + a + 1) 5 4 3 where a is the root of the polynomial, x + x - x + 2 x - 1, and in decimals this is, 0.8518579974 BTW the ratio for words with, 500, letters is, 0.8511967760 ------------------------------------------------ "Theorem Number 6" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 8 8 ) | ) C(m, n) x y | = - (x y - 3 x y + 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 6 8 7 9 5 8 6 7 7 8 5 7 6 - 3 x y - 3 x y + x y + 5 x y - 2 x y - 4 x y + 4 x y 8 4 7 5 6 6 6 5 6 4 5 5 6 3 + x y - 2 x y - 4 x y + 7 x y - 2 x y - 4 x y - x y 5 4 5 3 4 4 4 3 4 2 3 3 3 2 + 7 x y - 3 x y - 6 x y + 7 x y - 2 x y - 3 x y + 4 x y 3 2 2 2 / 10 8 10 7 9 8 - x y - 2 x y + 2 x y - x y + x - 1) / (x y - 2 x y - x y / 10 6 9 7 9 6 8 6 7 7 8 5 7 6 8 4 + x y + 3 x y - 2 x y - 2 x y - x y + 3 x y + x y - x y 7 5 7 4 6 4 5 5 6 3 5 4 5 3 4 3 + x y - x y - x y - 2 x y + x y + 2 x y - x y + x y 3 3 3 - x y + x y - x y - x + 1) and in Maple notation -(x^9*y^9-3*x^9*y^8+4*x^9*y^7+x^8*y^8-3*x^9*y^6-3*x^8*y^7+x^9*y^5+5*x^8*y^6-2*x ^7*y^7-4*x^8*y^5+4*x^7*y^6+x^8*y^4-2*x^7*y^5-4*x^6*y^6+7*x^6*y^5-2*x^6*y^4-4*x^ 5*y^5-x^6*y^3+7*x^5*y^4-3*x^5*y^3-6*x^4*y^4+7*x^4*y^3-2*x^4*y^2-3*x^3*y^3+4*x^3 *y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^10*y^8-2*x^10*y^7-x^9*y^8+x^10*y^6+3*x ^9*y^7-2*x^9*y^6-2*x^8*y^6-x^7*y^7+3*x^8*y^5+x^7*y^6-x^8*y^4+x^7*y^5-x^7*y^4-x^ 6*y^4-2*x^5*y^5+x^6*y^3+2*x^5*y^4-x^5*y^3+x^4*y^3-x^3*y^3+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 6 4 2 (a + a - 3 a + 1) -------------------------- 4 3 4 (5 a - 4 a + 2) (a + 1) 5 4 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.8422657512 BTW the ratio for words with, 500, letters is, 0.8417645687 ------------------------------------------------ "Theorem Number 7" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 9 9 9 8 9 7 8 8 ) | ) C(m, n) x y | = - (x y - 3 x y + 4 x y + x y / | / | ----- | ----- | m = 0 \ n = 0 / 9 6 8 7 9 5 8 6 7 7 8 5 7 6 - 3 x y - 3 x y + x y + 5 x y - 2 x y - 4 x y + 4 x y 8 4 7 5 6 6 6 5 6 4 5 5 6 3 + x y - 2 x y - 4 x y + 7 x y - 2 x y - 4 x y - x y 5 4 5 3 4 4 4 3 4 2 3 3 3 2 + 7 x y - 3 x y - 6 x y + 7 x y - 2 x y - 3 x y + 4 x y 3 2 2 2 / 10 8 10 7 9 8 - x y - 2 x y + 2 x y - x y + x - 1) / (x y - 2 x y - x y / 10 6 9 7 9 6 8 6 7 7 8 5 7 6 8 4 + x y + 3 x y - 2 x y - 2 x y - x y + 3 x y + x y - x y 7 5 7 4 6 4 5 5 6 3 5 4 5 3 4 3 + x y - x y - x y - 2 x y + x y + 2 x y - x y + x y 3 3 3 - x y + x y - x y - x + 1) and in Maple notation -(x^9*y^9-3*x^9*y^8+4*x^9*y^7+x^8*y^8-3*x^9*y^6-3*x^8*y^7+x^9*y^5+5*x^8*y^6-2*x ^7*y^7-4*x^8*y^5+4*x^7*y^6+x^8*y^4-2*x^7*y^5-4*x^6*y^6+7*x^6*y^5-2*x^6*y^4-4*x^ 5*y^5-x^6*y^3+7*x^5*y^4-3*x^5*y^3-6*x^4*y^4+7*x^4*y^3-2*x^4*y^2-3*x^3*y^3+4*x^3 *y^2-x^3*y-2*x^2*y^2+2*x^2*y-x*y+x-1)/(x^10*y^8-2*x^10*y^7-x^9*y^8+x^10*y^6+3*x ^9*y^7-2*x^9*y^6-2*x^8*y^6-x^7*y^7+3*x^8*y^5+x^7*y^6-x^8*y^4+x^7*y^5-x^7*y^4-x^ 6*y^4-2*x^5*y^5+x^6*y^3+2*x^5*y^4-x^5*y^3+x^4*y^3-x^3*y^3+x^3*y-x*y-x+1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 7 6 4 2 (a + a - 3 a + 1) -------------------------- 4 3 4 (5 a - 4 a + 2) (a + 1) 5 4 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.8422657512 BTW the ratio for words with, 500, letters is, 0.8417645687 ------------------------------------------------ "Theorem Number 8" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 1, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 8 8 8 7 8 6 7 7 ) | ) C(m, n) x y | = - (x y - 2 x y + x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 6 7 5 6 6 6 5 5 5 5 4 5 3 + 4 x y - 2 x y - 2 x y + 2 x y - 2 x y + 4 x y - 2 x y 4 4 4 3 3 3 3 2 / 8 8 8 6 7 6 + 2 x y - 2 x y + 2 x y - x y + 1) / (x y - x y - 4 x y / 7 5 6 6 6 5 6 4 5 4 5 3 4 4 + 4 x y - 2 x y + 6 x y - 4 x y - 3 x y + 2 x y + 2 x y 3 3 3 2 - 2 x y + x y + 2 x y - 1) and in Maple notation -(x^8*y^8-2*x^8*y^7+x^8*y^6-2*x^7*y^7+4*x^7*y^6-2*x^7*y^5-2*x^6*y^6+2*x^6*y^5-2 *x^5*y^5+4*x^5*y^4-2*x^5*y^3+2*x^4*y^4-2*x^4*y^3+2*x^3*y^3-x^3*y^2+1)/(x^8*y^8- x^8*y^6-4*x^7*y^6+4*x^7*y^5-2*x^6*y^6+6*x^6*y^5-4*x^6*y^4-3*x^5*y^4+2*x^5*y^3+2 *x^4*y^4-2*x^3*y^3+x^3*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 4 3 2 (a - 2 a + a - 5 a + 2 a + 1) - ------------------------------------ 4 3 2 3 (5 a - 8 a + 3 a - 2) (a + 1) 5 4 3 where a is the root of the polynomial, x - 2 x + x - 2 x + 1, and in decimals this is, 0.8354766504 BTW the ratio for words with, 500, letters is, 0.8350914388 ------------------------------------------------ "Theorem Number 9" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 1, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 8 8 8 7 8 6 7 7 ) | ) C(m, n) x y | = - (x y - 2 x y + x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 7 6 7 5 6 6 6 5 5 5 5 4 5 3 + 4 x y - 2 x y - 2 x y + 2 x y - 2 x y + 4 x y - 2 x y 4 4 4 3 3 3 3 2 / 8 8 8 6 7 6 + 2 x y - 2 x y + 2 x y - x y + 1) / (x y - x y - 4 x y / 7 5 6 6 6 5 6 4 5 4 5 3 4 4 + 4 x y - 2 x y + 6 x y - 4 x y - 3 x y + 2 x y + 2 x y 3 3 3 2 - 2 x y + x y + 2 x y - 1) and in Maple notation -(x^8*y^8-2*x^8*y^7+x^8*y^6-2*x^7*y^7+4*x^7*y^6-2*x^7*y^5-2*x^6*y^6+2*x^6*y^5-2 *x^5*y^5+4*x^5*y^4-2*x^5*y^3+2*x^4*y^4-2*x^4*y^3+2*x^3*y^3-x^3*y^2+1)/(x^8*y^8- x^8*y^6-4*x^7*y^6+4*x^7*y^5-2*x^6*y^6+6*x^6*y^5-4*x^6*y^4-3*x^5*y^4+2*x^5*y^3+2 *x^4*y^4-2*x^3*y^3+x^3*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 4 3 2 (a - 2 a + a - 5 a + 2 a + 1) - ------------------------------------ 4 3 2 3 (5 a - 8 a + 3 a - 2) (a + 1) 5 4 3 where a is the root of the polynomial, x - 2 x + x - 2 x + 1, and in decimals this is, 0.8354766504 BTW the ratio for words with, 500, letters is, 0.8350914388 ------------------------------------------------ "Theorem Number 10" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 1, 2, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 10 6 9 7 8 8 9 6 8 7 - 4 x y + 6 x y + x y - 6 x y + x y + 2 x y - 2 x y 8 6 5 5 5 4 5 3 4 4 4 3 / 8 7 + x y - 2 x y + 4 x y - 2 x y + 2 x y - 2 x y + 1) / (x y / 8 6 7 7 8 5 7 6 7 5 6 5 6 4 5 4 - 2 x y + x y + x y - 2 x y + x y + 2 x y - 2 x y - 3 x y 5 3 4 4 4 3 + 2 x y - 2 x y + 2 x y + 2 x y - 1) and in Maple notation -(x^10*y^10-4*x^10*y^9+6*x^10*y^8-2*x^9*y^9-4*x^10*y^7+6*x^9*y^8+x^10*y^6-6*x^9 *y^7+x^8*y^8+2*x^9*y^6-2*x^8*y^7+x^8*y^6-2*x^5*y^5+4*x^5*y^4-2*x^5*y^3+2*x^4*y^ 4-2*x^4*y^3+1)/(x^8*y^7-2*x^8*y^6+x^7*y^7+x^8*y^5-2*x^7*y^6+x^7*y^5+2*x^6*y^5-2 *x^6*y^4-3*x^5*y^4+2*x^5*y^3-2*x^4*y^4+2*x^4*y^3+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 5 4 2 (a + a - 5 a + 1) - ---------------------- 4 5 a - 2 5 where a is the root of the polynomial, x - 2 x + 1, and in decimals this is, 0.8311554128 BTW the ratio for words with, 500, letters is, 0.8309998372 ------------------------------------------------ "Theorem Number 11" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = - (x y - 4 x y + 6 x y - 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 10 6 9 7 8 8 9 6 8 7 - 4 x y + 6 x y + x y - 6 x y + x y + 2 x y - 2 x y 8 6 5 5 5 4 5 3 4 4 4 3 / 8 7 + x y - 2 x y + 4 x y - 2 x y + 2 x y - 2 x y + 1) / (x y / 8 6 7 7 8 5 7 6 7 5 6 5 6 4 5 4 - 2 x y + x y + x y - 2 x y + x y + 2 x y - 2 x y - 3 x y 5 3 4 4 4 3 + 2 x y - 2 x y + 2 x y + 2 x y - 1) and in Maple notation -(x^10*y^10-4*x^10*y^9+6*x^10*y^8-2*x^9*y^9-4*x^10*y^7+6*x^9*y^8+x^10*y^6-6*x^9 *y^7+x^8*y^8+2*x^9*y^6-2*x^8*y^7+x^8*y^6-2*x^5*y^5+4*x^5*y^4-2*x^5*y^3+2*x^4*y^ 4-2*x^4*y^3+1)/(x^8*y^7-2*x^8*y^6+x^7*y^7+x^8*y^5-2*x^7*y^6+x^7*y^5+2*x^6*y^5-2 *x^6*y^4-3*x^5*y^4+2*x^5*y^3-2*x^4*y^4+2*x^4*y^3+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 5 4 2 (a + a - 5 a + 1) - ---------------------- 4 5 a - 2 5 where a is the root of the polynomial, x - 2 x + 1, and in decimals this is, 0.8311554128 BTW the ratio for words with, 500, letters is, 0.8309998372 ------------------------------------------------ "Theorem Number 12" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 2, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 8 8 8 7 8 6 7 7 ) | ) C(m, n) x y | = - (3 x y - 8 x y + 8 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 5 7 6 7 5 6 6 7 4 6 5 6 4 5 5 - 3 x y + x y + x y - x y - x y + 2 x y - x y + 2 x y 5 4 5 3 4 4 4 3 4 2 3 3 3 2 - 4 x y + 2 x y - 3 x y + x y + x y - 2 x y + 2 x y - 1) / 9 9 9 8 9 7 8 8 9 6 8 7 8 6 / (x y - 3 x y + 4 x y - x y - 2 x y + 2 x y - 2 x y / 7 7 8 5 7 6 7 5 6 6 7 4 6 5 6 4 - x y + x y + 3 x y - 3 x y + 3 x y + x y - 6 x y + 3 x y 5 5 5 4 4 4 4 3 4 2 3 3 3 2 - 2 x y + x y - x y + 3 x y - x y + 2 x y - 2 x y - 2 x y + 1) and in Maple notation -(3*x^8*y^8-8*x^8*y^7+8*x^8*y^6-x^7*y^7-3*x^8*y^5+x^7*y^6+x^7*y^5-x^6*y^6-x^7*y ^4+2*x^6*y^5-x^6*y^4+2*x^5*y^5-4*x^5*y^4+2*x^5*y^3-3*x^4*y^4+x^4*y^3+x^4*y^2-2* x^3*y^3+2*x^3*y^2-1)/(x^9*y^9-3*x^9*y^8+4*x^9*y^7-x^8*y^8-2*x^9*y^6+2*x^8*y^7-2 *x^8*y^6-x^7*y^7+x^8*y^5+3*x^7*y^6-3*x^7*y^5+3*x^6*y^6+x^7*y^4-6*x^6*y^5+3*x^6* y^4-2*x^5*y^5+x^5*y^4-x^4*y^4+3*x^4*y^3-x^4*y^2+2*x^3*y^3-2*x^3*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, 7 4 2 (a - 3 a + 1) -------------------------- 4 3 4 (5 a - 4 a + 2) (a + 1) 5 4 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.8224486940 BTW the ratio for words with, 500, letters is, 0.8222294163 ------------------------------------------------ "Theorem Number 13" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 1, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 8 8 8 7 8 6 7 7 ) | ) C(m, n) x y | = - (3 x y - 8 x y + 8 x y - x y / | / | ----- | ----- | m = 0 \ n = 0 / 8 5 7 6 7 5 6 6 7 4 6 5 6 4 5 5 - 3 x y + x y + x y - x y - x y + 2 x y - x y + 2 x y 5 4 5 3 4 4 4 3 4 2 3 3 3 2 - 4 x y + 2 x y - 3 x y + x y + x y - 2 x y + 2 x y - 1) / 9 9 9 8 9 7 8 8 9 6 8 7 8 6 / (x y - 3 x y + 4 x y - x y - 2 x y + 2 x y - 2 x y / 7 7 8 5 7 6 7 5 6 6 7 4 6 5 6 4 - x y + x y + 3 x y - 3 x y + 3 x y + x y - 6 x y + 3 x y 5 5 5 4 4 4 4 3 4 2 3 3 3 2 - 2 x y + x y - x y + 3 x y - x y + 2 x y - 2 x y - 2 x y + 1) and in Maple notation -(3*x^8*y^8-8*x^8*y^7+8*x^8*y^6-x^7*y^7-3*x^8*y^5+x^7*y^6+x^7*y^5-x^6*y^6-x^7*y ^4+2*x^6*y^5-x^6*y^4+2*x^5*y^5-4*x^5*y^4+2*x^5*y^3-3*x^4*y^4+x^4*y^3+x^4*y^2-2* x^3*y^3+2*x^3*y^2-1)/(x^9*y^9-3*x^9*y^8+4*x^9*y^7-x^8*y^8-2*x^9*y^6+2*x^8*y^7-2 *x^8*y^6-x^7*y^7+x^8*y^5+3*x^7*y^6-3*x^7*y^5+3*x^6*y^6+x^7*y^4-6*x^6*y^5+3*x^6* y^4-2*x^5*y^5+x^5*y^4-x^4*y^4+3*x^4*y^3-x^4*y^2+2*x^3*y^3-2*x^3*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, 7 4 2 (a - 3 a + 1) -------------------------- 4 3 4 (5 a - 4 a + 2) (a + 1) 5 4 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.8224486940 BTW the ratio for words with, 500, letters is, 0.8222294163 ------------------------------------------------ "Theorem Number 14" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 1, 2, 1, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 9 7 8 8 9 6 8 7 8 6 7 7 - x y - 6 x y + 6 x y + x y - 2 x y - 4 x y + 5 x y - x y 8 5 7 6 7 5 6 6 7 4 6 5 6 4 4 4 - 2 x y + x y + x y - x y - x y + 2 x y - x y - 2 x y 4 3 3 3 3 2 2 2 2 / 10 10 10 9 + 2 x y - 2 x y + 2 x y - x y + x y - 1) / (x y - 3 x y / 10 8 10 7 8 8 8 7 8 6 7 7 7 6 7 5 + 3 x y - x y - x y + 2 x y - x y + x y - 2 x y + 2 x y 6 6 7 4 6 5 6 4 5 5 5 4 4 4 4 3 - x y - x y + 2 x y - x y - 2 x y + x y + 2 x y - 2 x y 2 2 2 - x y + x y + 2 x y - 1) and in Maple notation (x^10*y^10-3*x^10*y^9+3*x^10*y^8+2*x^9*y^9-x^10*y^7-6*x^9*y^8+6*x^9*y^7+x^8*y^8 -2*x^9*y^6-4*x^8*y^7+5*x^8*y^6-x^7*y^7-2*x^8*y^5+x^7*y^6+x^7*y^5-x^6*y^6-x^7*y^ 4+2*x^6*y^5-x^6*y^4-2*x^4*y^4+2*x^4*y^3-2*x^3*y^3+2*x^3*y^2-x^2*y^2+x^2*y-1)/(x ^10*y^10-3*x^10*y^9+3*x^10*y^8-x^10*y^7-x^8*y^8+2*x^8*y^7-x^8*y^6+x^7*y^7-2*x^7 *y^6+2*x^7*y^5-x^6*y^6-x^7*y^4+2*x^6*y^5-x^6*y^4-2*x^5*y^5+x^5*y^4+2*x^4*y^4-2* x^4*y^3-x^2*y^2+x^2*y+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 8 7 6 4 2 (a + a + a - 5 a + 1) - --------------------------- 4 5 a - 2 5 where a is the root of the polynomial, x - 2 x + 1, and in decimals this is, 0.8214235170 BTW the ratio for words with, 500, letters is, 0.8213048474 ------------------------------------------------ "Theorem Number 15" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 1, 2, 2]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 10 10 10 9 10 8 9 9 ) | ) C(m, n) x y | = (x y - 3 x y + 3 x y + 2 x y / | / | ----- | ----- | m = 0 \ n = 0 / 10 7 9 8 9 7 8 8 9 6 8 7 8 6 7 7 - x y - 6 x y + 6 x y + x y - 2 x y - 4 x y + 5 x y - x y 8 5 7 6 7 5 6 6 7 4 6 5 6 4 4 4 - 2 x y + x y + x y - x y - x y + 2 x y - x y - 2 x y 4 3 3 3 3 2 2 2 2 / 10 10 10 9 + 2 x y - 2 x y + 2 x y - x y + x y - 1) / (x y - 3 x y / 10 8 10 7 8 8 8 7 8 6 7 7 7 6 7 5 + 3 x y - x y - x y + 2 x y - x y + x y - 2 x y + 2 x y 6 6 7 4 6 5 6 4 5 5 5 4 4 4 4 3 - x y - x y + 2 x y - x y - 2 x y + x y + 2 x y - 2 x y 2 2 2 - x y + x y + 2 x y - 1) and in Maple notation (x^10*y^10-3*x^10*y^9+3*x^10*y^8+2*x^9*y^9-x^10*y^7-6*x^9*y^8+6*x^9*y^7+x^8*y^8 -2*x^9*y^6-4*x^8*y^7+5*x^8*y^6-x^7*y^7-2*x^8*y^5+x^7*y^6+x^7*y^5-x^6*y^6-x^7*y^ 4+2*x^6*y^5-x^6*y^4-2*x^4*y^4+2*x^4*y^3-2*x^3*y^3+2*x^3*y^2-x^2*y^2+x^2*y-1)/(x ^10*y^10-3*x^10*y^9+3*x^10*y^8-x^10*y^7-x^8*y^8+2*x^8*y^7-x^8*y^6+x^7*y^7-2*x^7 *y^6+2*x^7*y^5-x^6*y^6-x^7*y^4+2*x^6*y^5-x^6*y^4-2*x^5*y^5+x^5*y^4+2*x^4*y^4-2* x^4*y^3-x^2*y^2+x^2*y+2*x*y-1) As the length of the word goes to infinity, the average number of good neigh\ bors of a random word of length n tends to n times, 8 7 6 4 2 (a + a + a - 5 a + 1) - --------------------------- 4 5 a - 2 5 where a is the root of the polynomial, x - 2 x + 1, and in decimals this is, 0.8214235170 BTW the ratio for words with, 500, letters is, 0.8213048474 ------------------------------------------------ "Theorem Number 16" Let C(m,n) be the number of words of length n in the alphabet, {1, 2}, avoiding consecutive substrings in the set, {[1, 2, 2, 2, 1]}, and having\ m neighbors (i.e. Hamming distance 1) that also obey this property, the\ n infinity /infinity \ ----- | ----- | \ | \ n m| 12 12 12 11 12 10 ) | ) C(m, n) x y | = - (x y - 3 x y + 3 x y / | / | ----- | ----- | m = 0 \ n = 0 / 11 11 12 9 11 10 11 9 10 10 11 8 10 9 + x y - x y - 4 x y + 6 x y + x y - 4 x y - 4 x y 11 7 10 8 9 9 10 7 9 8 10 6 9 7 + x y + 6 x y + x y - 4 x y - 4 x y + x y + 6 x y 8 8 9 6 8 7 9 5 8 6 7 7 7 6 - 2 x y - 4 x y + 3 x y + x y - x y - 2 x y + 4 x y 7 5 6 6 6 5 6 4 5 5 5 4 5 3 - 2 x y - 2 x y + 4 x y - 2 x y - 2 x y + 4 x y - 2 x y 4 4 4 3 4 2 / 13 12 13 11 13 10 + 3 x y - x y - x y + 1) / (x y - 3 x y + 3 x y / 12 11 13 9 12 10 12 9 12 8 9 8 9 6 - x y - x y + 3 x y - 3 x y + x y - x y + 2 x y 8 7 9 5 8 6 5 5 5 4 4 4 4 3 4 2 + x y - x y - x y + 2 x y - x y - 3 x y + x y + x y + 2 x y - 1) and in Maple notation -(x^12*y^12-3*x^12*y^11+3*x^12*y^10+x^11*y^11-x^12*y^9-4*x^11*y^10+6*x^11*y^9+x ^10*y^10-4*x^11*y^8-4*x^10*y^9+x^11*y^7+6*x^10*y^8+x^9*y^9-4*x^10*y^7-4*x^9*y^8 +x^10*y^6+6*x^9*y^7-2*x^8*y^8-4*x^9*y^6+3*x^8*y^7+x^9*y^5-x^8*y^6-2*x^7*y^7+4*x ^7*y^6-2*x^7*y^5-2*x^6*y^6+4*x^6*y^5-2*x^6*y^4-2*x^5*y^5+4*x^5*y^4-2*x^5*y^3+3* x^4*y^4-x^4*y^3-x^4*y^2+1)/(x^13*y^12-3*x^13*y^11+3*x^13*y^10-x^12*y^11-x^13*y^ 9+3*x^12*y^10-3*x^12*y^9+x^12*y^8-x^9*y^8+2*x^9*y^6+x^8*y^7-x^9*y^5-x^8*y^6+2*x ^5*y^5-x^5*y^4-3*x^4*y^4+x^4*y^3+x^4*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 2 (3 a - 1) - -------------------------- 4 3 4 (5 a - 4 a + 2) (a + 1) 5 4 where a is the root of the polynomial, x - x + 2 x - 1, and in decimals this is, 0.8121977054 BTW the ratio for words with, 500, letters is, 0.8121447128 ---------------------------------- This ends this paper that took, 162.506, seconds to produce this took, 162.507, seconds.