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 A114199 #19 Oct 25 2024 07:14:23
%S A114199 1,2,4,8,17,38,87,200,458,1044,2373,5388,12233,27782,63112,143392,
%T A114199 325805,740266,1681935,3821412,8682310,19726316,44818473,101828344,
%U A114199 231355953,525645354,1194276812,2713420728,6164945513,14006877390
%N A114199 Row sums of a Pascal-Fibonacci triangle.
%C A114199 Binomial transform of double Fibonacci sequence A103609(n+2). Row sums of A114197.
%H A114199 Michael De Vlieger, <a href="/A114199/b114199.txt">Table of n, a(n) for n = 0..2806</a>
%H A114199 Sergio Falcón, <a href="https://www.rgnpublications.com/journals/index.php/cma/article/viewFile/1221/950">Binomial Transform of the Generalized k-Fibonacci Numbers</a>, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
%H A114199 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,1).
%F A114199 G.f.: (1-x)^2/(1-4*x+5*x^2-2*x^3-x^4).
%F A114199 a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*Fibonacci(j).
%F A114199 a(n) = Sum_{k=0..n} C(n, k)*Fibonacci(floor((k+2)/2)).
%t A114199 LinearRecurrence[{4,-5,2,1},{1,2,4,8},30] (* _Harvey P. Dale_, Dec 07 2015 *)
%o A114199 (Magma) [n le 4 select 2^(n-1) else 4*Self(n-1) -5*Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..30]]; // _G. C. Greubel_, Oct 23 2024
%o A114199 (SageMath)
%o A114199 @CachedFunction # a = A114199
%o A114199 def a(n): return 2^n if n<4 else 4*a(n-1) -5*a(n-2) +2*a(n-3) +a(n-4)
%o A114199 [a(n) for n in range(71)] # _G. C. Greubel_, Oct 23 2024
%Y A114199 Cf. A000045, A103609, A114197.
%K A114199 easy,nonn
%O A114199 0,2
%A A114199 _Paul Barry_, Nov 16 2005