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 A181871 #27 Jul 22 2025 08:42:48
%S A181871 0,2,0,2,-3,0,4,-2,2,5,0,-20,0,16,-3,0,4,-7,0,56,0,-112,0,64,-2,0,4,
%T A181871 -3,0,36,0,-96,0,64,5,0,-20,0,16,-11,0,220,0,-1232,0,2816,0,-2816,0,
%U A181871 1024,-1,2,13,0,-364,0,2912,0,-9984,0,16640,0,-13312,0,4096,-7,0,56,0,-112,0,64,1,0,-32,0,224,0,-448,0,256,2,0,-16,0,16,17,0,-816,0,11424,0,-71808,0,239360,0,-452608,0,487424,0,-278528,0,65536,-3,0,36,0,-96,0,64
%N A181871 Coefficient array for integer polynomial version of minimal polynomials of sin(2*Pi/n). Rising powers of x.
%C A181871 The sequence of row lengths of this array is A093819(n)+1: [2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, ...].
%C A181871 pi(n,x) := Sum_{m=0..d(n)} a(n,m)*x^m, n >= 1, is related to the (monic) minimal polynomial of sin(2*Pi/n), called Pi(n,x), by pi(n,x) = (2^d(n))*Pi(n,x), with the degree sequence d(n)=A093819(n), and Pi(n,x) is given in A181872/A181873.
%C A181871 Pi(n,x)=Psi(c(n),x) with the minimal polynomials Psi(n,x) of cos(2*Pi/n), and c(n):=A178182(n).
%C A181871 The minimal polynomials of sin(2*Pi/n) are, e.g., treated in the Lehmer and Niven references. (Note the mistake in the Lehmer references explained in the W. Lang link.) The fundamental polynomials Psi(n,x) are also studied in the Watkins-Zeitlin reference, where a recurrence is given.
%C A181871 See A231188 for the (monic and integer) minimal polynomials of 2*sin(2*Pi/n). = _Wolfdieter Lang_, Nov 30 2013
%D A181871 I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons..
%H A181871 Wolfdieter Lang, <a href="/A181872/a181872.pdf">A181872/A181873. Minimal polynomials for sin(2Pi/n), and pi(n,x) polynomials..</a>
%H A181871 D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A Note on Trigonometric Algebraic Numbers</a>, Am. Math. Monthly 40 (3) (1933) 165-6.
%H A181871 W. Watkins and J. Zeitlin, <a href="http://www.jstor.org/stable/2324301">The Minimal Polynomial of cos(2Pi/n)</a>, Am. Math. Monthly 100,5 (1993) 471-4.
%F A181871 a(n,m) = [x^m]pi(n,x), n >= 1, m=0..A093819(n), and pi(n,x) defined above in the comments.
%e A181871 [0, 2], [0, 2], [-3, 0, 4], [-2, 2], [5, 0, -20, 0, 16], [-3, 0, 4], [-7, 0, 56, 0, -112, 0, 64], [-2, 0, 4], [-3, 0, 36, 0, -96, 0, 64], [5, 0, -20, 0, 16], ...
%e A181871 pi(2,x) = (2^1)*Pi(2,x) = 2*Psi(c(2),x) = 2*Psi(4,x) = 2*x.
%t A181871 ro[n_] := (cc = CoefficientList[ p = MinimalPolynomial[ Sin[2*(Pi/n)], x], x]; 2^Exponent[p, x]*(cc/Last[cc])); Flatten[ Table[ ro[n], {n, 1, 18}]] (* _Jean-François Alcover_, Sep 28 2011 *)
%Y A181871 Cf. A181877 (cos(2*Pi/n) case), A231188 (2*sin(2*Pi/n) case).
%K A181871 sign,easy,tabf
%O A181871 1,2
%A A181871 _Wolfdieter Lang_, Jan 14 2011