A100048 A Chebyshev transform of the Pell numbers.

0, 1, 2, 2, 4, 9, 16, 29, 56, 106, 198, 373, 704, 1325, 2494, 4698, 8848, 16661, 31376, 59089, 111276, 209554, 394634, 743177, 1399552, 2635641, 4963450, 9347186, 17602652, 33149377, 62427024, 117562789, 221394656, 416931194, 785166286

Offset: 0

Author

Paul Barry, Oct 31 2004

Keywords

• easy
• nonn

Comments

A Chebyshev transform of the Pell numbers A000129: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. Case P1 = 2, P2 = -1, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. - Peter Bala, Mar 24 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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 (2,-1,2,-1).

Crossrefs

Cf. A099443, A011655, A100047.

Programs

Mathematica

LinearRecurrence[{2,-1,2,-1},{0,1,2,2},40] (* Harvey P. Dale, Jun 07 2015 *)

PARI

my(x='x+O('x^50)); Vec(x*(1-x^2)/(1-2*x+x^2-2*x^3+x^4)) \\ G. C. Greubel, Aug 08 2017

Formula

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

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

a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A000129(n-2*k)/(n-k).

From Peter Bala, Mar 24 2014: (Start)

a(n) = |u(n)|^2, where {u(n)} is the Lucas-type sequence defined by the recurrence u(0) = 0, u(1) = 1 and u(n) = P*u(n-1) - u(n-2) for n >= 2, where P = 1/2*(sqrt(7) + i).

a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(2))/2 and beta = (1 - sqrt(2))/2 and T(n,x) denotes the Chebyshev polynomial of the first kind.

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

The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/2*(sqrt(7) + i)*x + x^2) and x/(1 - 1/2*(sqrt(7) - i)*x + x^2). (End)