A152096 Quartic product sequence: a(n) = Product_{k=1..(n-1)/2} (1 + m*cos(k*Pi/n)^2 + q*cos(k*Pi/n)^4), with m=12 and q = 3*4^3.

1, 1, 1, 16, 55, 355, 1888, 9829, 57145, 294064, 1683055, 8893147, 49635520, 267601933, 1472118817, 8012384080, 43823300455, 239288418067, 1306681029664, 7139564615413, 38980858167625, 212971742938096, 1162967620577311

Offset: 0

Author

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

Keywords

• nonn

Comments

Limiting ratio at n=30 equals 5.461866286689612.

Exact value of this limit is (1 + sqrt(205) + sqrt(2*(7+sqrt(205))))/4 = 5.46185461429652018724... - Vaclav Kotesovec, Nov 30 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1 + x*(1-12*x^2)/(1-x-27*x^2-12*x^3+144*x^4) )); // G. C. Greubel, May 15 2019

Mathematica

With[{m = 3*4, q = 3*4^3}, Table[Round[Product[1 + m*Cos[k*Pi/n]^2 + q*Cos[k*Pi/n]^4, {k, 1, (n-1)/2}]], {n, 0, 30}]] (* modified by G. C. Greubel, May 15 2019 *)
CoefficientList[Series[1+x*(1-12*x^2)/(1-x-27*x^2-12*x^3+144*x^4), {x, 0, 22}], x] (* Vaclav Kotesovec, Nov 30 2012 *)

PARI

my(x='x+O('x^30)); Vec(1 + x*(1-12*x^2)/(1-x-27*x^2-12*x^3 +144*x^4)) \\ G. C. Greubel, May 15 2019

Sage

(1 + x*(1-12*x^2)/(1-x-27*x^2-12*x^3+144*x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 15 2019

Formula

a(n) = Product_{k=1..(n-1)/2} (1 + m*cos(k*Pi/n)^2 + q*cos(k*Pi/n)^4), with m=3*4 and q = 3*4^3.

G.f.: 1 + x*(1-12*x^2)/(1-x-27*x^2-12*x^3+144*x^4). - Vaclav Kotesovec, Nov 30 2012