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 A014334 #73 Jan 05 2025 19:51:34
%S A014334 0,0,2,6,22,70,230,742,2406,7782,25190,81510,263782,853606,2762342,
%T A014334 8939110,28927590,93611622,302933606,980313702,3172361830,10265978470,
%U A014334 33221404262,107506722406,347899061862,1125825013350,3643246274150,11789792601702
%N A014334 Exponential convolution of Fibonacci numbers with themselves.
%H A014334 Vincenzo Librandi, <a href="/A014334/b014334.txt">Table of n, a(n) for n = 0..1000</a>
%H A014334 C. A. Church and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Issues/11-3.pdf">Exponential generating functions for Fibonacci identities</a>, Fibonacci Quarterly 11, no. 3 (1973), 275-281.
%H A014334 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, Vol. 26, No. 3 (2020), pp. 96-106.
%H A014334 Helmut Prodinger, <a href="https://arxiv.org/abs/1910.08323">Convolution identities for Tribonacci numbers via the diagonal of a bivariate generating function</a>, arXiv:1910.08323 [math.NT], 2019.
%H A014334 Charles R. Wall, <a href="https://fq.math.ca/Scanned/24-2/elementary24-2.pdf">Problem B-573</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 24, No. 2 (1986), p. 181; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-2/elementary25-2.pdf">Solution to Problem B-573</a> by Bob Prielipp, ibid., Vol. 25, No. 2 (1987), p. 184.
%H A014334 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-4).
%F A014334 From _Benoit Cloitre_, May 29 2003: (Start)
%F A014334 a(n) = 3*a(n-1) + 2*a(n-2) - 4*a(n-3), a(0)=0, a(1)=0, a(2)=2.
%F A014334 a(n) = Sum_{k=0..n-1} 2^k*Fibonacci(k) for n > 0.
%F A014334 a(n) = (-2 + ((1+sqrt(5))^n + (1-sqrt(5))^n))/5. (End)
%F A014334 a(n) = Sum_{k=0..n} Fibonacci(k)*Fibonacci(n-k)*binomial(n, k). - _Benoit Cloitre_, May 11 2005
%F A014334 From _R. J. Mathar_, Sep 29 2010: (Start)
%F A014334 a(n) = 2*A014335(n).
%F A014334 G.f.: 2*x^2/((1-x)*(1-2*x-4*x^2)).
%F A014334 a(n) = Sum_{k=1..n-1} A103435(k). (End)
%F A014334 a(n) = (2^n*A000032(n) - 2)/5. - _Vladimir Reshetnikov_, May 18 2016
%F A014334 E.g.f.: 2*(cosh(sqrt(5)*x)-1)*exp(x)/5. - _Ilya Gutkovskiy_, May 18 2016
%F A014334 a(n) = ((Sum_{k=0..n} Lucas(k)*Lucas(n-k)*binomial(n, k)) - 4)/5 (Wall, 1986). - _Amiram Eldar_, Jan 27 2022
%t A014334 LinearRecurrence[{3,2,-4}, {0,0,2}, 30] (* _Harvey P. Dale_, Oct 24 2015 *)
%t A014334 Table[(2^n LucasL[n] -2)/5, {n,0,100}] (* _Vladimir Reshetnikov_, May 18 2016 *)
%o A014334 (PARI) a(n)=if(n<1,0,sum(k=0,n-1,fibonacci(k)*2^k))
%o A014334 (Magma) [(2^n*Lucas(n) -2)/5: n in [0..40]]; // _Vincenzo Librandi_, Jul 15 2018
%o A014334 (SageMath) [(2^n*lucas_number2(n,1,-1) -2)/5 for n in range(41)] # _G. C. Greubel_, Jan 06 2023
%Y A014334 Cf. A000032, A000045, A014335, A103435.
%K A014334 nonn,easy
%O A014334 0,3
%A A014334 _N. J. A. Sloane_