A100048 - OEIS

%I #23 Aug 08 2017 02:56:55

%S 0,1,2,2,4,9,16,29,56,106,198,373,704,1325,2494,4698,8848,16661,31376,

%T 59089,111276,209554,394634,743177,1399552,2635641,4963450,9347186,

%U 17602652,33149377,62427024,117562789,221394656,416931194,785166286

%N A Chebyshev transform of the Pell numbers.

%C 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)).

%C 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

%H G. C. Greubel, <a href="/A100048/b100048.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="https://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-1).

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

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

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

%F From _Peter Bala_, Mar 24 2014: (Start)

%F 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).

%F 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.

%F 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].

%F 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)

%t LinearRecurrence[{2,-1,2,-1},{0,1,2,2},40] (* _Harvey P. Dale_, Jun 07 2015 *)

%o (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

%Y Cf. A099443, A011655, A100047.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 31 2004