A006645 Self-convolution of Pell numbers (A000129).
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
Offset: 0
Examples
G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...
References
- 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)
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2605
- Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
- Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
- Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
- Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
- Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).
Programs
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Maple
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
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 *) -
Sage
taylor( mul(x/(1 - 2*x - x^2) for i in range(1,3)),x,0,28) # Zerinvary Lajos, Jun 03 2009
Formula
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.
From Wolfdieter Lang, Apr 11 2000: (Start)
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
Extensions
Sum formulas and cross-references added by Wolfdieter Lang, Aug 07 2002