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A116415
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a(n)=5a(n-1)-3a(n-2).
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row sums of A116414.
Binomial transform of the sequence A006190 - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Nov 23 2007
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REFERENCES
| Sergio Falcon and Angel Plaza: "On k-Fibonacci sequences and polynomials and their derivatives". doi:10.1016/j.chaos.2007.03.2007
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FORMULA
| 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 (sfalcon(AT)dma.ulpgc.es), 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. [From Milan R. Janjic (agnus(AT)blic.net), 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 Adamson, Feb 18 2011
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MATHEMATICA
| Join[{a=1, b=5}, Table[c=5*b-3*a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 5, 3) for n in xrange(1, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A001906, A007070, A084326.
Sequence in context: A053154 A141222 A127360 * A026861 A026888 A083586
Adjacent sequences: A116412 A116413 A116414 * A116416 A116417 A116418
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 13 2006
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