A152118 a(n) = Product_{k=1..floor((n-1)/2)} (4 + 4*cos(k*Pi/n)^2).

1, 1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149, 53789260175, 259717522849

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Nov 24 2008

Keywords

• nonn
• easy

Comments

Product_{k=1..floor((n-1)/2)} (m + 4*cos(k*Pi/n)^2) for m=1,2,3,4 respectively give A000045, A002530, A136211 and this sequence.

Apparently the same as A041011 after the initial term. - R. J. Mathar, Nov 27 2008

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A000045, A002530, A136211.

Cf. A041011 (essentially the same).

Programs

Maple

with(combinat); a := n -> `if`(n = 0, 1, fibonacci(n, 2)/2^((n + 1) mod 2)); seq(a(n), n = 0 .. 31); # Miles Wilson, Aug 04 2024

Mathematica

a = Table[Product[4 + 4*Cos[k*Pi/n]^2, {k, 1, (n - 1)/2}], {n, 0, 30}]; FullSimplify[ExpandAll[a]] Round[%]
Join[{1}, LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 6}, 20]] (* G. C. Greubel, Mar 28 2016 *)

PARI

a(n) = round(prod(k=1, (n-1)/2, 4 + 4*cos(k*Pi/n)^2)) \\ Colin Barker, Oct 23 2013

PARI

Vec((x^4-x^3-5*x^2+x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, Mar 28 2016

Formula

From Colin Barker, Oct 23 2013: (Start)

a(n) = 6*a(n-2)-a(n-4) for n>4.

G.f.: (x^4-x^3-5*x^2+x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). (End)

a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n)) / (8*sqrt(2)) for n>0. - Colin Barker, Mar 28 2016

E.g.f.: (1/(2*sqrt(2)))*(2*sqrt(2) + (2*cosh(x) + sinh(x))*sinh(sqrt(2)*x)). - G. C. Greubel, Mar 28 2016