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.

2, 6, 5, 4, 19, 3, 71, 14, 53, 11, 989, 13, 3691, 41, 145, 194, 51409, 51, 191861, 181, 2017, 571, 2672279, 193, 524899, 2131, 140453, 2521, 138907099, 241, 518408351, 37634, 391249, 29681, 5352481, 2701, 26947261171, 110771, 5449393, 37441, 375326930089

Offset: 1

Author

Jianing Song, Jul 08 2019

Keywords

• nonn

Examples

			MPR2(15, x) = x^4 - x^3 - 4x^2 + 4x + 1, so a(15) = MPR2(15, 4) = 145.
		

Crossrefs

Cf. A001353, A232624, A306825.

Cf. also A023022, A008683.

Programs

Mathematica

a[n_] := (p = MinimalPolynomial[2*Cos[2*(Pi/n)], 4]; p); Table[a[n], {n, 1, 40}]

Formula

By the comment in A232624, we have: A001353(n) = Product_{k|2n, k>=3} MPR2(k, 4) = Product_{k|2n, k>=3} a(k).

a(n) = Product_{0<=m<=n/2, gcd(m, n)=1} (4 - 2*cos(2Pi*m/n)).

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]

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.