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

Author

R. J. Mathar, Jun 11 2009

Keywords

• nonn
• easy

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)

Crossrefs

Cf. A006238, A100047.

Programs

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

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 *)

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)