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

1,1

Links

Jianing Song, Table of n, a(n) for n = 1..3498

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.

Prod_{d|n, d>=3} a(d) is the lower-left entry of [4, 1; -1, 0]^(n/2) for even n, and Prod_{d|n, d>=2} a(d) is the sum of the first row of [4, 1; -1, 0]^((n-1)/2) for odd n. - Jianing Song, Dec 13 2025

Example

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

Mathematica

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

Prog

(PARI) Prod(n) = my(M = [4, 1; -1, 0]); if(n%2==0, (M^(n/2))[2, 1], (M^((n-1)/2) * [1; 1])[1, 1])
a(n) = if(n==1, return(2)); if(n==2, return(6)); my(P = 1); fordiv(n, d, P *= Prod(n/d)^moebius(d)); P \\ Jianing Song, Dec 13 2025

Crossrefs

Cf. A001353, A232624, A306825.

Cf. also A023022, A008683.

Sequence in context: A388152 A171897 A105029 * A316134 A273621 A190124

Adjacent sequences: A309037 A309038 A309039 * A309041 A309042 A309043

Keyword

nonn

Author

Jianing Song, Jul 08 2019

Status

approved