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A006645
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Self-convolution of Pell numbers (A000129).
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7
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0, 0, 1, 4, 14, 44, 131, 376, 1052, 2888, 7813, 20892, 55338, 145428, 379655, 985520, 2545720, 6547792, 16777993, 42847988, 109099078, 277040572, 701794187, 1773851304, 4474555476, 11266301976, 28318897549, 71070913036, 178106093666
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OFFSET
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0,4
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REFERENCES
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R. P. Grimaldi, Ternary strings ..., Congressus Numerantium, 205 (2011), 129-149. (The sequences w_n and z_n)
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LINKS
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Table of n, a(n) for n=0..28.
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FORMULA
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a(n)= sum(b(k)*b(n-k), k=0..n) with b(k) := A000129(k).
a(n)= sum(2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, k=0..floor((n-2)/2)), n>=2.
a(n)= ((n-1)*P(n)+n*P(n-1))/4, P(n)=A000129(n). G.f.: (x/(1-2*x-x^2))^2 - Wolfdieter Lang, Apr 11 2000
a(n)=F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006
G.f.: sage: taylor( mul(x/(1 - 2*x - x^2) for i in xrange(1,3)),x,0,28)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]
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EXAMPLE
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sage: taylor( mul(x/(1 - 2*x - x^2) for i in xrange(1,3)),x,0,28)# solution>> x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + 2888*x^9 + 7813*x^10 + 20892*x^11 + 55338*x^12 +...+ 4474555476*x^24 + 11266301976*x^25 + 28318897549*x^26+ 71070913036*x^27 + 178106093666*x^28+etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]
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MAPLE
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a:= n-> (Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -2, -4, -1][i] else 0 fi)^n) [1, 3]: seq (a(n), n=0..40); [From Alois P. Heinz, Oct 28 2008]
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MATHEMATICA
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pell[n_] := Simplify[ ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2])]; a[n_] := First[ ListConvolve[ pp = Array[pell, n+1, 0], pp]]; Table[a[n], {n, 0, 28}] (* From Jean-François Alcover, Oct 21 2011 *)
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CROSSREFS
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a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.
Sequence in context: A007466 A062109 A118042 * A094309 A000300 A005323
Adjacent sequences: A006642 A006643 A006644 * A006646 A006647 A006648
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Sum formulae and cross-references added.- Wolfdieter Lang, Aug 07 2002
More terms from Alois P. Heinz, Oct 28 2008
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STATUS
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approved
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