A152094 Quartic product sequence: a(n) = Product_{k=1..floor((n-1)/2)} (1 + m*cos(k*Pi/n)^2 + q*cos(k*Pi/n)^4 ), with m = 2*4, q=2*4^3.

1, 1, 1, 11, 37, 179, 869, 3683, 18389, 80179, 385029, 1739651, 8134709, 37397203, 173097317, 799986979, 3694294933, 17085418099, 78904394437, 364797113027, 1685324681973, 7789441113619, 35993781049381, 166339303316579

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Limiting ratio at n=30: 4.621205928975311.

Exact value of this limit is (1 + sqrt(137) + sqrt(2*(5 + sqrt(137))))/4 = 4.621196599954103646539033792... - 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*(1-8*x^2)/(1-x-18*x^2-8*x^3+64*x^4) )); // G. C. Greubel, May 08 2019

Mathematica

Table[Product[1 +8*Cos[k*Pi/n]^2 +128*Cos[k*Pi/n]^4, {k,1,(n-1)/2}], {n, 0, 30}]//Round (* modified by G. C. Greubel, May 08 2019 *)
CoefficientList[Series[1+x*(1-8*x^2)/(1-x-18*x^2-8*x^3+64*x^4), {x, 0, 23}], x] (* Vaclav Kotesovec, Nov 30 2012 *)

PARI

my(x='x+O('x^30)); Vec(1 + x*(1-8*x^2)/(1-x-18*x^2-8*x^3+64*x^4)) \\ G. C. Greubel, May 08 2019

Sage

(1 + x*(1-8*x^2)/(1-x-18*x^2-8*x^3+64*x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 08 2019

Formula

G.f.: 1 + x*(1-8*x^2)/(1-x-18*x^2-8*x^3+64*x^4). - Vaclav Kotesovec, Nov 30 2012