A161498 Expansion of x*(1-x)*(1+x)/(1-13*x+36*x^2-13*x^3+x^4).

1, 13, 132, 1261, 11809, 109824, 1018849, 9443629, 87504516, 810723277, 7510988353, 69584925696, 644660351425, 5972359368781, 55329992188548, 512595960817837, 4748863783286881, 43995092132369664, 407585519020921249

Offset

1,2

Comments

Proposed by R. Guy in the seqfan list, Mar 29 2009.

The sequence is the case P1 = 13, P2 = 34, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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 (13,-36,13,-1)

Formula

a(n) = A139400(n) / ( A001906(n)*A001353(n)*A004254(n) ).

a(n) = 13*a(n-1)-36*a(n-2)+13*a(n-3)-a(n-4).

a(n) = A187732(n)-A187732(n-2). - R. J. Mathar, Mar 18 2011

From Peter Bala, Apr 03 2014: (Start)

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = 1/4*(13 + sqrt(33)), beta = 1/4*(13 - sqrt(33)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n) = U(n-1,1/2*(4 + sqrt(3) ))*U(n-1,1/2*(4 - sqrt(3))) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -17/2; 1, 13/2].

See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

Mathematica

CoefficientList[Series[(1 - x)*(1 + x)/(1 - 13*x + 36*x^2 - 13*x^3 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 19 2012 *)

Prog

(Magma) I:=[1, 13, 132, 1261]; [n le 4 select I[n] else 13*Self(n-1)-36*Self(n-2)+13*Self(n-3)-Self(n-4): n in [1..20]]; // Vincenzo Librandi, Dec 19 2012

Crossrefs

Cf. A006238, A100047.

Sequence in context: A048554 A293580 A295349 * A037715 A037617 A295603

Adjacent sequences: A161495 A161496 A161497 * A161499 A161500 A161501

Keyword

nonn,easy

Author

R. J. Mathar, Jun 11 2009

Status

approved