A152090 a(n) = 2^n*Product_{k=1..floor((n-1)/2)} (1 + 2*cos(k*Pi/n)^2 + 4*cos(k*Pi/n)^4).

1, 1, 1, 3, 7, 16, 39, 91, 217, 513, 1216, 2881, 6825, 16171, 38311, 90768, 215047, 509491, 1207089, 2859841, 6775552, 16052673, 38032081, 90105811, 213479175, 505776016, 1198287271, 2838988683, 6726147337, 15935624641, 37754768064

Offset

0,4

Comments

Limiting ratio after n=30 terms is 2.369205407038926.

With a(0)=0, this is a divisibility sequence with g.f. x(1-x^2)/(1 - x - 3x^2 - x^3 + x^4). The limiting ratio is the largest zero of 1 - x - 3x^2 - x^3 + x^4. - T. D. Noe, Dec 22 2008

The sequence is the case P1 = 1, P2 = -5, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

Links

Table of n, a(n) for n=0..30.

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

Index entries for linear recurrences with constant coefficients, signature (1,3,1,-1).

Formula

From Colin Barker, Jan 05 2014: (Start)

a(n) = a(n-1) +3*a(n-2) +a(n-3) -a(n-4) for n>4.

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

From Peter Bala, Mar 25 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(1 + sqrt(21)), beta = 1/4*(1 - sqrt(21)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = U(n-1,1/4*(1 + sqrt(-3)))*U(n-1,1/4*(1 - sqrt(-3))) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/4; 1, 1/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

Mathematica

bb = Table[FullSimplify[ExpandAll[Product[1 + 4*Cos[k*Pi/n]^2 + 16*Cos[k*Pi/n]^4, {k, 1, (n - 1)/2}]]], {n, 0, 30}]
LinearRecurrence[{1, 3, 1, -1}, {1, 1, 1, 3, 7}, 50] (* G. C. Greubel, Aug 08 2017 *)

Prog

(PARI) Vec((x^4-2*x^3-3*x^2+1)/(x^4-x^3-3*x^2-x+1) + O(x^100)) \\ Colin Barker, Jan 05 2014

Crossrefs

Cf. A100047.

Sequence in context: A196154 A227235 A304937 * A190528 A203611 A176604

Adjacent sequences: A152087 A152088 A152089 * A152091 A152092 A152093

Keyword

nonn,easy

Author

Roger L. Bagula and Gary W. Adamson, Nov 23 2008

Status

approved