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 A099568 #17 Jul 26 2022 01:35:53
%S A099568 1,2,4,9,19,39,80,163,330,666,1341,2695,5409,10846,21733,43526,87140,
%T A099568 174409,349007,698291,1396988,2794571,5590014,11181306,22364485,
%U A099568 44731715,89467453,178940802,357890245,715793154,1431604868,2863236937
%N A099568 Expansion of (1-x)/((1-2*x)*(1-x-x^3)).
%C A099568 Row sums of number triangle A099567.
%H A099568 G. C. Greubel, <a href="/A099568/b099568.txt">Table of n, a(n) for n = 0..1000</a>
%H A099568 Denis Neiter and Amsha Proag, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Proag/proag3.html">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
%H A099568 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1,-2).
%F A099568 a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - 2*a(n-4).
%F A099568 a(n) = Sum_{k=0..n} Sum_{j=0..floor(n/3)} binomial(n-2*j, k+j).
%t A099568 LinearRecurrence[{3,-2,1,-2}, {1,2,4,9}, 40] (* _G. C. Greubel_, Jul 26 2022 *)
%o A099568 (PARI) Vec((1-x)/((1-2*x)*(1-x-x^3)) + O(x^40)) \\ _Michel Marcus_, Oct 18 2016
%o A099568 (Magma) [n le 4 select Round(9^((n-1)/3)) else 3*Self(n-1) -2*Self(n-2) +Self(n-3) -2*Self(n-4): n in [1..41]]; // _G. C. Greubel_, Jul 26 2022
%o A099568 (SageMath)
%o A099568 @CachedFunction
%o A099568 def a(n): # a = A099568
%o A099568 if (n<4): return round(9^(n/3))
%o A099568 else: return 3*a(n-1) -2*a(n-2) + a(n-3) - 2*a(n-4)
%o A099568 [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 26 2022
%Y A099568 Cf. A099567.
%K A099568 easy,nonn
%O A099568 0,2
%A A099568 _Paul Barry_, Oct 22 2004