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 A309040 #20 Oct 31 2024 21:55:41
%S A309040 2,6,5,4,19,3,71,14,53,11,989,13,3691,41,145,194,51409,51,191861,181,
%T A309040 2017,571,2672279,193,524899,2131,140453,2521,138907099,241,518408351,
%U A309040 37634,391249,29681,5352481,2701,26947261171,110771,5449393,37441,375326930089
%N A309040 a(n) = MPR2(n, 4), where MPR2(n, x) is the (monic) minimal polynomial of 2*cos(2*Pi/n) as defined in A232624.
%F A309040 By the comment in A232624, we have: A001353(n) = Product_{k|2n, k>=3} MPR2(k, 4) = Product_{k|2n, k>=3} a(k).
%F A309040 a(n) = Product_{0<=m<=n/2, gcd(m, n)=1} (4 - 2*cos(2Pi*m/n)).
%F A309040 If 4 divides n, then a(n) = Product_{k|(n/2)} A001353((n/2)/k)^mu(k) = A306825(n/2), where mu = A008683. For odd n > 1, a(n)*a(2n) = Product_{k|n} A001353(n/k)^mu(k) = A306825(n). [Corrected by _Jianing Song_, Oct 31 2024]
%F A309040 Let b(n) = MPR2(n, -4)*(-1)^A023022(n) for n > 2, then a(n) = b(2n) for odd n, a(n) = b(n/2) for n congruent to 4 modulo 2, a(n) = b(n) for n divisible by 4.
%e A309040 MPR2(15, x) = x^4 - x^3 - 4x^2 + 4x + 1, so a(15) = MPR2(15, 4) = 145.
%t A309040 a[n_] := (p = MinimalPolynomial[2*Cos[2*(Pi/n)], 4]; p); Table[a[n], {n, 1, 40}]
%Y A309040 Cf. A001353, A232624, A306825.
%Y A309040 Cf. also A023022, A008683.
%K A309040 nonn
%O A309040 1,1
%A A309040 _Jianing Song_, Jul 08 2019