This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A073380 #25 May 07 2025 09:12:55
%S A073380 1,8,44,200,810,3032,10716,36248,118435,376240,1167720,3553840,
%T A073380 10636180,31375440,91392040,263266512,750922021,2123059448,5955034740,
%U A073380 16584106040,45884989054,126202397032
%N A073380 Third convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.
%H A073380 G. C. Greubel, <a href="/A073380/b073380.txt">Table of n, a(n) for n = 0..1000</a>
%H A073380 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
%H A073380 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,8,26,-8,-20,-8,-1).
%F A073380 a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A054457(k).
%F A073380 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+3, 3) * binomial(n-k, k).
%F A073380 a(n) = ((147 +94*n +14*n^2)*(n+1)*U(n+1) + 3*(15 +12*n +2*n^2)*(n+2)*U(n))/ (3*2^7), with U(n) = A000129(n+1), n >= 0.
%F A073380 G.f.: 1/(1-(2+x)*x)^4.
%F A073380 a(n) = F'''(n+4, 2)/6, that is, 1/6 times the 3rd derivative of the (n+4)th Fibonacci polynomial evaluated at x=2. - _T. D. Noe_, Jan 19 2006
%t A073380 CoefficientList[Series[1/(1-2*x-x^2)^4, {x,0,40}], x] (* _G. C. Greubel_, Oct 02 2022 *)
%t A073380 LinearRecurrence[{8,-20,8,26,-8,-20,-8,-1},{1,8,44,200,810,3032,10716,36248},30] (* _Harvey P. Dale_, Feb 18 2023 *)
%o A073380 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^4 )); // _G. C. Greubel_, Oct 02 2022
%o A073380 (SageMath)
%o A073380 def A073380_list(prec):
%o A073380 P.<x> = PowerSeriesRing(ZZ, prec)
%o A073380 return P( 1/(1-2*x-x^2)^4 ).list()
%o A073380 A073380_list(30) # _G. C. Greubel_, Oct 02 2022
%Y A073380 Fourth (m=3) column of triangle A054456, A054457 (m=2).
%Y A073380 Cf. A000129.
%K A073380 nonn,easy
%O A073380 0,2
%A A073380 _Wolfdieter Lang_, Aug 02 2002