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A116415
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a(n) = 5a(n-1) - 3a(n-2).
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10
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1, 5, 22, 95, 409, 1760, 7573, 32585, 140206, 603275, 2595757, 11168960, 48057529, 206780765, 889731238, 3828313895, 16472375761, 70876937120, 304967558317, 1312206980225, 5646132226174, 24294040190195, 104531804272453
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OFFSET
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0,2
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COMMENTS
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Row sums of A116414.
Binomial transform of the sequence A006190. - Sergio Falcon, Nov 23 2007
a(n+1) equals the number of words of length n over {0,1,2,3,4} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015
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LINKS
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Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Sergio Falcon and Angel 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).
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FORMULA
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G.f.: 1/(1 - 5x + 3x^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 - 5x + 3x^2) = -1/(1-x)+(2-3x)/(1 - 5x + 3x^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
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MATHEMATICA
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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 *)
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PROG
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(Sage) [lucas_number1(n, 5, 3) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009
(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
(PARI) Vec(1/(1-5*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015
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CROSSREFS
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Cf. A001906, A007070, A084326.
Sequence in context: A053154 A141222 A127360 * A026861 A026888 A266430
Adjacent sequences: A116412 A116413 A116414 * A116416 A116417 A116418
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Feb 13 2006
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STATUS
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approved
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