This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255237 #18 Apr 12 2018 14:27:38 %S A255237 1,2,1,0,1,-1,1,-1,0,1,0,-1,1,-2,0,1,0,0,-1,1,0,0,-1,0,1,-1,-1,0,1,2, %T A255237 0,-2,0,1,0,0,0,0,-1,1,-1,0,-1,0,1,0,0,0,0,0,-1,1,-2,0,2,0,-2,0,1,-1, %U A255237 -2,-1,1,1 %N A255237 Array of conversion coefficients for the minimal polynomials C of 2 cos(Pi/n) in terms of Chebyshev's S-polynomials. %C A255237 The row length sequence is 1, 1 + A055034(n), n >= 1. %C A255237 For the minimal polynomial C(n, x) of the algebraic number rho(n) := 2*cos(Pi/n) (the length ratio of the smallest diagonal and the side of a regular n-gon) see the coefficient array in A187360. The coefficient triangle of Chebyshev's S-polynomials is given in A049310. %C A255237 The conversion is C(n, x) = sum(T(n, m)*S(m, x), m=0..delta(n)), for n >= 0 with C(0, x) := 1 (undefined product), delta(0) = 0 and delta(n) = A055034(n), n >= 1. %C A255237 Originally Ahmet Zahid KÜÇÜK observed the structure for prime n. The precise formula for odd primes prime(n) = A000040(n), n >= 2, is C(prime(n), x) = S((prime(n)-1)/2, x) - S((prime(n)-3)/2, x). %C A255237 This is equivalent to C(prime(n),x) = (-1)^((p(n)-1)/2)*S(prime(n)-1,I*sqrt(x-2)), with I^2 = -1. %C A255237 Proof: The known identity S(n, x) - S(n-1, x) = (-1)^n*S(2*n, I*sqrt(x-2)) (from bisection). The degrees of the monic polynomials of both sides match, as do the known zeros. %C A255237 The row sums give 1, 3, 1, 0, 0, 0, -1, 0, 0, -1, 1, 0, -1, 0, -1, -2, 0, 0, -1, 0 ... %C A255237 The alternating row sums give 1, 1, -1, -2, 0, 2, -1, -2, 0, -1, 1, -2, -1, 2, -1, 0, 0, 2, -1, -2, ... %C A255237 For the reverse problem, the factorization of S polynomials into C polynomials see a Apr 12 2018 comment in A049310. - _Wolfdieter Lang_, Apr 12 2018 %F A255237 The conversion is C(n, x) = sum(T(n, m)*S(m, x), m = 0..delta(n)), that is %F A255237 T(n, m) = [S(m, x)] C(n, x), n >= 0, m = 0, ..., delta(n), with C(0, x) := 1, delta(0) = 0 and delta(n) = A055034(n), n >= 1. For the C and S polynomials see A187360 and A049310, respectively. %F A255237 For n >= 2: T(prime(n), (prime(n) -1)/2) = +1, T(prime(n), (prime(n) -3)/2) = -1 and T(prime(n), m) = 0 otherwise. %e A255237 The array T(n, m) begins: %e A255237 n\m 0 1 2 3 4 5 6 ... %e A255237 0: 1 %e A255237 1: 2 1 %e A255237 2: 0 1 %e A255237 3: -1 1 %e A255237 4: -1 0 1 %e A255237 5: 0 -1 1 %e A255237 6: -2 0 1 %e A255237 7: 0 0 -1 1 %e A255237 8: 0 0 -1 0 1 %e A255237 9: -1 -1 0 1 %e A255237 10: 2 0 -2 0 1 %e A255237 11: 0 0 0 0 -1 1 %e A255237 12: -1 0 -1 0 1 %e A255237 13: 0 0 0 0 0 -1 1 %e A255237 14: -2 0 2 0 -2 0 1 %e A255237 15: -1 -2 -1 1 1 %e A255237 ... %e A255237 n=0: C(0, x) = 1 = 1*S(0, x), %e A255237 n=1: C(1, x) = 2 + x = 2*S(0, x) + 1*S(1, x), %e A255237 n=2: C(2, x) = x = 0*S(0, x) + 1*S(1, x), %e A255237 n=3: C(3, x) = -1 + x = -1*S(0, x) + 1*S(1, x), %e A255237 n=4: C(4, x) = -2 + x^2 = -1*S(0, x) + 0 + 1*S(2, x) = -1 + (-1 + x^2), ... %Y A255237 Cf. A187360, A055034, A049310. %K A255237 sign,easy,tabf %O A255237 0,2 %A A255237 Ahmet Zahid KÜÇÜK and _Wolfdieter Lang_, Mar 11 2015