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 A094388 #29 Sep 27 2024 02:02:13
%S A094388 1,2,4,10,28,82,244,730,2188,6562,19684,59050,177148,531442,1594324,
%T A094388 4782970,14348908,43046722,129140164,387420490,1162261468,3486784402,
%U A094388 10460353204,31381059610,94143178828,282429536482,847288609444
%N A094388 Expansion of (1- 2*x - x^2)/((1-x)*(1-3*x)).
%C A094388 Binomial transform of 0^n + A001045(n).
%C A094388 From _J. M. Bergot_, Nov 10 2012: (Start)
%C A094388 Form an array with the first row and column containing all 1's: m(n,1) = m(1,n) = 1 for n=1,2,3,... An interior term m(i,j) is the sum of all preceding terms in row(i) and all preceding terms in column(j): m(i,j) = Sum_{k=1..j-1} m(i,k) + Sum_{l=1..i-1} m(l,j). The sum of the terms in each antidiagonal will reproduce the terms in this sequence beginning at a(0).
%C A094388 The upper left corner of the array begins
%C A094388 1 1 1 1 1 ...
%C A094388 1 2 4 8 16 ...
%C A094388 1 4 10 24 56 ...
%C A094388 1 8 24 66 172 ...
%C A094388 1 16 56 172 490 ...
%C A094388 ...
%C A094388 (End) [edited by _Jon E. Schoenfield_, Sep 08 2018]
%H A094388 Vincenzo Librandi, <a href="/A094388/b094388.txt">Table of n, a(n) for n = 0..600</a>
%H A094388 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A094388 a(n) = 3^n/3 - 0^n/3 + 1.
%F A094388 a(n+1) = 2*A007051(n).
%F A094388 a(n) = A034472(n-1), n > 0. - _R. J. Mathar_, Sep 05 2008
%F A094388 G.f.: G(0), where G(k)= 1 + 3^k*x/(1 - x/(x + 3^k*x/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 26 2013
%F A094388 E.g.f.: (1/3)*(-1 + 3*exp(x) + exp(3*x)). - _G. C. Greubel_, Sep 27 2024
%t A094388 CoefficientList[Series[(1-2x-x^2)/((1-x)(1-3x)),{x,0,30}],x] (* _Harvey P. Dale_, May 20 2011 *)
%o A094388 (Magma) [3^n/3-0^n/3+1: n in [0..30]]; // _Vincenzo Librandi_, May 21 2011
%o A094388 (PARI) a(n)=3^n/3-0^n/3+1 \\ _Charles R Greathouse IV_, Nov 27 2012
%o A094388 (SageMath) [(3^n +3 -int(n==0))//3 for n in range(41)] # _G. C. Greubel_, Sep 27 2024
%Y A094388 Cf. A001045, A007051, A034472.
%K A094388 easy,nonn
%O A094388 0,2
%A A094388 _Paul Barry_, Apr 28 2004