A152098 Quartic product sequence: m = 4; p = 4^3; a(n) = Product_{k=1..(n-1)/2} ( 1 + m*cos(k*Pi/n)^2 + p*cos(k*Pi/n)^4 ).

1, 1, 1, 6, 19, 61, 240, 769, 2869, 9774, 34831, 121969, 428640, 1509301, 5297641, 18644406, 65502139, 230343541, 809678160, 2846468089, 10006911469, 35178340254, 123671565271, 434760784009, 1528407648960, 5373087522061

Offset

0,4

Comments

Ratio at n=30 is 3.515489857847214.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,9,4,-16).

Formula

a(n) = Product_{k=1..(n-1)/2} ( 1 + m*cos(k*Pi/n)^2 + p*cos(k*Pi/n)^4 ), with m=4 and p=64.

G.f.: (1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4). - Colin Barker, Oct 23 2013

Maple

seq(coeff(series((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 26 2019

Mathematica

m=4; l=4^3; Table[Product[1 +m*Cos[k*Pi/n]^2 +l*Cos[k*Pi/n]^4, {k, (n -1)/2}], {n, 0, 30}]//Round
LinearRecurrence[{1, 9, 4, -16}, {1, 1, 1, 6, 19}, 30] (* G. C. Greubel, Sep 26 2019 *)

Prog

(PARI) a(n) = round(prod(k=1, (n-1)/2, 1+4*cos(k*Pi/n)^2+4^3*cos(k*Pi/n)^4)) \\ Colin Barker, Oct 23 2013
(PARI) my(x='x+O('x^30)); Vec((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3 +16*x^4)) \\ G. C. Greubel, Sep 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)* (1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4) )); // G. C. Greubel, Sep 26 2019
(Sage)
def A152098_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x)*(1+x-8*x^2-16*x^3)/(1-x-9*x^2-4*x^3+16*x^4)).list()
A152098_list(30) # G. C. Greubel, Sep 26 2019
(GAP) a:=[1, 1, 6, 19];; for n in [5..30] do a[n]:=a[n-1]+9*a[n-2]+4*a[n-3] -16*a[n-4]; od; Concatenation([1], a); # G. C. Greubel, Sep 26 2019

Crossrefs

Sequence in context: A125069 A080926 A184189 * A041673 A137195 A055916

Adjacent sequences: A152095 A152096 A152097 * A152099 A152100 A152101

Keyword

nonn,easy

Author

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

Status

approved