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A006645 Self-convolution of Pell numbers (A000129).

%I A006645 #61 May 07 2025 09:03:14
%S A006645 0,0,1,4,14,44,131,376,1052,2888,7813,20892,55338,145428,379655,
%T A006645 985520,2545720,6547792,16777993,42847988,109099078,277040572,
%U A006645 701794187,1773851304,4474555476,11266301976,28318897549,71070913036,178106093666,445740656420,1114147888655
%N A006645 Self-convolution of Pell numbers (A000129).
%D A006645 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)
%H A006645 Michael De Vlieger, <a href="/A006645/b006645.txt">Table of n, a(n) for n = 0..2605</a>
%H A006645 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A006645 Sergio Falcon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.96-106">Half self-convolution of the k-Fibonacci sequence</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
%H A006645 Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264">The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.
%H A006645 Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H A006645 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H A006645 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,-1).
%F A006645 a(n) = Sum_{k=0..n} b(k)*b(n-k) with b(k) := A000129(k).
%F A006645 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.
%F A006645 From _Wolfdieter Lang_, Apr 11 2000: (Start)
%F A006645 a(n) = ((n-1)*P(n) + n*P(n-1))/4, P(n)=A000129(n).
%F A006645 G.f.: (x/(1 - 2*x - x^2))^2. (End)
%F A006645 a(n) = F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%e A006645 G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...
%p A006645 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
%t A006645 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 *)
%t A006645 Table[(n Fibonacci[n - 1, 2] + (n - 1) Fibonacci[n, 2])/4, {n, 0, 30}] (* _Vladimir Reshetnikov_, May 08 2016 *)
%o A006645 (Sage) taylor( mul(x/(1 - 2*x - x^2) for i in range(1,3)),x,0,28) # _Zerinvary Lajos_, Jun 03 2009
%Y A006645 a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.
%K A006645 nonn,easy
%O A006645 0,4
%A A006645 _N. J. A. Sloane_
%E A006645 Sum formulas and cross-references added by _Wolfdieter Lang_, Aug 07 2002