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A100048
A Chebyshev transform of the Pell numbers.
6
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
0,3
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
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
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)
MATHEMATICA
LinearRecurrence[{2, -1, 2, -1}, {0, 1, 2, 2}, 40] (* Harvey P. Dale, Jun 07 2015 *)
PROG
(PARI) x='x+O('x^50); Vec(x(1-x^2)/(1-2x+x^2-2x^3+x^4)) \\ G. C. Greubel, Aug 08 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 31 2004
STATUS
approved