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A006645
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Self-convolution of Pell numbers (A000129).
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16
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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, 445740656420, 1114147888655
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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 with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequences w_n and z_n)
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} b(k)*b(n-k) with b(k) := A000129(k).
a(n) = Sum_{k=0..floor((n-2)/2)} 2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, 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. (End)
a(n) = F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006
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EXAMPLE
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G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...
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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); # 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}] (* Jean-François Alcover, Oct 21 2011 *)
Table[(n Fibonacci[n - 1, 2] + (n - 1) Fibonacci[n, 2])/4, {n, 0, 30}] (* Vladimir Reshetnikov, May 08 2016 *)
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PROG
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(Sage) taylor( mul(x/(1 - 2*x - x^2) for i in range(1, 3)), x, 0, 28) # Zerinvary Lajos, Jun 03 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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