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 A113300 #39 Mar 16 2025 02:36:16
%S A113300 0,1,3,10,34,115,389,1316,4452,15061,50951,172366,583110,1972647,
%T A113300 6673417,22576008,76374088,258371689,874065163,2956941266,10003260650,
%U A113300 33840788379,114482567053,387291750188,1310198605996,4432370135229,14994600761871,50726371026838
%N A113300 Sum of even-indexed terms of tribonacci numbers.
%C A113300 Partial sums of A099463. a(n+1) gives row sums of Riordan array (1/(1-x)^2,(1+x)^2/(1-x)^2). Congruent to 0,1,1,0,0,1,1,0,0,... modulo 2. - _Paul Barry_, Feb 07 2006
%H A113300 G. C. Greubel, <a href="/A113300/b113300.txt">Table of n, a(n) for n = 0..1000</a>
%H A113300 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,1).
%F A113300 a(n) = Sum_{i=0..n} A000073(2*n).
%F A113300 a(n) = Sum_{i=0..n} A099463(n).
%F A113300 a(n) + A113301(n) = A008937(n).
%F A113300 From _Paul Barry_, Feb 07 2006: (Start)
%F A113300 G.f.: x/(1 - 3*x - x^2 - x^3).
%F A113300 a(n) = 3*a(n-1) + a(n-2) + a(n-3). (End)
%t A113300 Accumulate[Take[LinearRecurrence[{1,1,1},{0,0,1},60],{1,-1,2}]] (* _Harvey P. Dale_, Nov 06 2011 *)
%t A113300 LinearRecurrence[{3,1,1},{0,1,3},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)
%t A113300 a[ n_] := Sum[ SeriesCoefficient[ SeriesCoefficient[ x / (1 - x - y - x y) , {x, 0, n - k}]^2 , {y, 0, k}], {k, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)
%o A113300 (Magma) I:=[0,1,3]; [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..61]]; // _G. C. Greubel_, Nov 19 2021
%o A113300 (Sage)
%o A113300 @CachedFunction
%o A113300 def T(n): # T(n) = A000073(n)
%o A113300 if (n<2): return 0
%o A113300 elif (n==2): return 1
%o A113300 else: return T(n-1) +T(n-2) +T(n-3)
%o A113300 def a(n): return sum( T(2*j) for j in (0..n) )
%o A113300 [a(n) for n in (0..60)] # _G. C. Greubel_, Nov 19 2021
%Y A113300 Cf. A000073, A008937, A099463, A113301.
%K A113300 easy,nonn
%O A113300 0,3
%A A113300 _Jonathan Vos Post_, Oct 24 2005