OFFSET
1,1
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, a(n)*a(2n) = Product_{k|n} A001353(n/k)^mu(k) = A306825(n).
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Jul 08 2019
STATUS
approved