A116415 a(n) = 5*a(n-1) - 3*a(n-2).
1, 5, 22, 95, 409, 1760, 7573, 32585, 140206, 603275, 2595757, 11168960, 48057529, 206780765, 889731238, 3828313895, 16472375761, 70876937120, 304967558317, 1312206980225, 5646132226174, 24294040190195, 104531804272453
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Sergio Falcón, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
- Sergio Falcón and Ángel Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, Volume 39, Issue 3, 15 February 2009, Pages 1005-1019.
- Index entries for linear recurrences with constant coefficients, signature (5,-3).
Programs
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Magma
I:=[1,5]; [n le 2 select I[n] else 5*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 16 2015
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Mathematica
Join[{a=1,b=5},Table[c=5*b-3*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *) LinearRecurrence[{5,-3},{1,5},40] (* Harvey P. Dale, Jun 19 2012 *) -
PARI
Vec(1/(1-5*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015
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Sage
[lucas_number1(n,5,3) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009
Formula
G.f.: 1/(1 - 5*x + 3*x^2).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n-j,k)*C(k+j,j)*3^j.
a(n) = (1/sqrt(13))*(((5+sqrt(13))/2)^n - ((5-sqrt(13))/2)^n). - Sergio Falcon, Nov 23 2007
If p[i] = (3^i-1)/2, and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, May 08 2010
a(n) = 4*a(n-1) + a(n-2) + a(n-3) + ... + a(0) + 1. These expansions with the partial sums on one side can be generated en masse by taking the g.f. of the partial sum and its partial fraction, 1/(1-x)/(1 - 5*x + 3*x^2) = -1/(1-x)+(2-3*x)/(1 - 5*x + 3*x^2) and reading this as a(0) + a(1) + ... + a(n) = -1 + 2*a(n)- 3*a(n-1). - Gary W. Adamson, Feb 18 2011
Comments