A241606 A linear divisibility sequence of the fourth order related to A003779.

1, 11, 95, 781, 6336, 51205, 413351, 3335651, 26915305, 217172736, 1752296281, 14138673395, 114079985111, 920471087701, 7426955448000, 59925473898301, 483517428660911, 3901330906652795, 31478457514091281, 253988526230055936

Offset: 1

Author

Peter Bala, Apr 26 2014

Keywords

• nonn
• easy

Comments

A003779, which counts spanning trees in the graph P_5 x P_n, is a linear divisibility sequence of order 16. It factors into two fourth-order linear divisibility sequences; this sequence is one of the factors, the other is A143699.

The present sequence is the case P1 = 11, P2 = 23, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy.

Links

• 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 (11,-25,11,-1).

Crossrefs

Cf. A003779, A100047, A143699.

Programs

Mathematica

a[n_] := ChebyshevU[n-1, 1/4*(7-Sqrt[5])]*ChebyshevU[n-1, 1/4*(7+Sqrt[5])]; Table[a[n]//Round, {n, 1, 20}] (* Jean-François Alcover, Apr 28 2014, after Peter Bala *)

Formula

O.g.f. x*(1 - x^2)/(1 - 11*x + 25*x^2 - 11*x^3 + x^4).

a(n) = A003779(n)/A143699(n).

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(11 + sqrt(29)), beta = 1/4*(11 - sqrt(29)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.

a(n)= U(n-1,1/4*(7 - sqrt(5)))*U(n-1,1/4*(7 + sqrt(5))), n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = the bottom left entry of the 2X2 matrix T(n,M), where M is the 2 X 2 matrix [0, -23/4; 1, 11/2].

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

a(n) = 11*a(n-1) - 25*a(n-2) + 11*a(n-3) - a(n-4). - Vaclav Kotesovec, Apr 28 2014