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 A230079 #9 Nov 04 2013 05:30:33
%S A230079 1,-1,1,2,1,-1,-3,0,1,3,3,-4,-1,1,0,4,0,-1,-3,6,4,-5,-1,1,4,4,-1,-1,
%T A230079 -4,10,10,-15,-6,7,1,-1,5,10,-20,-15,21,7,-8,-1,1,0,12,0,-19,0,8,0,-1,
%U A230079 -8,-8,6,6,-1,-1,6,15,-35,-35,56,28,-36,-9,10,1,-1
%N A230079 Table a(n,m) of coefficients of inverses of rho(A230078(n)), n>=2, with rho(k):= 2*cos(Pi/k), in the power basis of Q(rho(A230078(n))).
%C A230079 The length of row n is delta(A230078(n)), n>=2, with delta(k) = A055034(k).
%C A230079 The power base of the algebraic number field Q(rho(k)), with rho(k):= 2*cos(Pi/k), k >= 2, is <1, rho(k), rho(k)^2, ..., rho(k)^(delta(k)-1)>. Q(rho(k))-integers have integer coefficients in this basis. A230078(n), n >= 2, gives precisely the numbers k for which the inverse 1/rho(k) is a Q(rho(k))-integer. The present table a(n,m) lists these integer coefficients for 1/rho(A230078(n)), n >= 2, m = 0, 1, ..., delta(A230078(n))-1. delta(k) is the degree of the minimal polynomial C(k, x) of rho(k) (see A187360).
%C A230079 In general, 1/rho(k) = -(sum(c(k, m+1)*rho(k)^m, m=0..delta(k)-1))/c(k, 0), k >= 2, with the coefficients c(k ,m) of the minimal polynomial C(k, x) given in A187360(k, m). c(k ,0) = C(k, x=0) is +1 or -1 if and only if k is from {A230078(n), n>=2}, leading to a Q(rho(k))-integer.
%F A230079 a(n,m) = -c(b(n), m+1)/c(b(n), 0), with b(n) := A230078(n), for n>=2 and m= 0, 1, ... , delta(b(n)) -1, with delta(k) = A055034(k), and c(k, m) = A187360(k, m) (see a comment above on the minimal C polynomials).
%e A230079 The table a(n,m) begins, with b(n):=A230078(n):
%e A230079 n, b(n)\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A230079 2, 3: 1
%e A230079 3, 5: -1 1
%e A230079 4, 7: 2 1 -1
%e A230079 5, 9: -3 0 1
%e A230079 6, 11: 3 3 -4 -1 1
%e A230079 7, 12: 0 4 0 -1
%e A230079 8, 13: -3 6 4 -5 -1 1
%e A230079 9, 15: 4 4 -1 -1
%e A230079 10, 17: -4 10 10 -15 -6 7 1 -1
%e A230079 11, 19: 5 10 -20 -15 21 7 -8 -1 1
%e A230079 12, 20: 0 12 0 -19 0 8 0 -1
%e A230079 13, 21: -8 -8 6 6 -1 -1
%e A230079 14, 23: 6 15 -35 -35 56 28 -36 -9 10 1 -1
%e A230079 15, 24: 0 16 0 -20 0 8 0 -1
%e A230079 ...
%e A230079 n=2: C(3, x) = x - 1, delta(3) =1, 1/rho(3) = 1, a rational integer.
%e A230079 n=3: C(5, x) =x^2 - x -1, delta(5) = 2, a(3,0) = - c(5, 1)/c(5, 0) = -(-1)/(-1) = -1, a(3,1) = - c(5, 2)/c(5, 0) = -1/(-1) = +1.
%e A230079 n =3: rho(5) = tau := (1 + sqrt(5))/2 (golden section); 1/rho(5) = -1*1 + 1*rho(5).
%e A230079 n= 4: rho(7) = 2*cos(Pi/7), (approximately 1.801937736); 1/rho(7) = 2*1 + 1*rho(7) - 1*rho(7)^2, (approximately 0.5549581320).
%e A230079 n=10: rho(17) = 2*cos(Pi/17), (approximately 1.965946199); 1/rho(17) = -4*1 + 10*rho(17) + 10*rho(17)^2 - 15*rho(17)^3 - 6*rho(17)^4 + 7*rho(17)^5 + 1*rho(17)^6 -1*rho(17)^7, (approximately 0.5086609190).
%Y A230079 Cf. A055034, A187360, A230078.
%K A230079 sign,tabf,easy
%O A230079 2,4
%A A230079 _Wolfdieter Lang_, Nov 02 2013