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 A113301 #20 Sep 08 2022 08:45:23
%S A113301 0,1,5,18,62,211,715,2420,8188,27701,93713,317030,1072506,3628263,
%T A113301 12274327,41523752,140473848,475219625,1607656477,5438662906,
%U A113301 18398864822,62242913851,210566269283,712340586524,2409830942708,8152399683933,27579370581033,93300342369742
%N A113301 Sum of odd-indexed terms of tribonacci numbers.
%C A113301 A000073 is the tribonacci numbers. A113300 is the sum of even-indexed terms of tribonacci numbers. A099463 is the bisection of the tribonacci numbers. A113300(n) + A113301(n) = cumulative sum of tribonacci numbers = A008937(n). Primes in A113300 include a(2) = 5, a(5) = 211, a(9) = 27701, .... A113300 is semiprime for n = 4, 10, 14, ...
%H A113301 Harvey P. Dale, <a href="/A113301/b113301.txt">Table of n, a(n) for n = 0..1000</a>
%H A113301 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,0,-1).
%F A113301 a(n) = Sum_{j=0..n} A000073(2*j+1).
%F A113301 a(n) + A113300(n) = A008937(n).
%F A113301 a(n) = 4*a(n-1) - 2*a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=5, a(3)=18. - _Harvey P. Dale_, Apr 12 2013
%F A113301 G.f.: x*(1+x) / ((1-x)*(1-3*x-x^2-x^3)). - _Colin Barker_, May 06 2013
%e A113301 a(0) = 0 = A000073(1);
%e A113301 a(1) = 0+1 = A000073(1) + A000073(3) = 1;
%e A113301 a(2) = 0+1+4 = A000073(1) + A000073(3) + A000073(5) = 5, prime;
%e A113301 a(3) = 0+1+4+13 = A000073(1) + A000073(3) + A000073(5) + A000073(7) = 18;
%e A113301 a(4) = 0+1+4+13+44 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) = 62 = 2 * 31, semiprime;
%e A113301 a(5) = 0+1+4+13+44+149 = A000073(1) + A000073(3) + A000073(5) + A000073(7) + A000073(9) + A000073(11) = 211, prime.
%t A113301 Accumulate[Take[LinearRecurrence[{1,1,1},{0,1,1},40],{1,-1,2}]] (* or *) LinearRecurrence[{4,-2,0,-1},{0,1,5,18},30] (* _Harvey P. Dale_, Apr 12 2013 *)
%o A113301 (Magma) I:=[0,1,5,18]; [n le 4 select I[n] else 4*Self(n-1) - 2*Self(n-2) -Self(n-4): n in [1..41]]; // _G. C. Greubel_, Nov 20 2021
%o A113301 (Sage)
%o A113301 @CachedFunction
%o A113301 def T(n): # A000073
%o A113301 if (n<2): return 0
%o A113301 elif (n==2): return 1
%o A113301 else: return T(n-1) +T(n-2) +T(n-3)
%o A113301 def A113301(n): return sum(T(2*j+1) for j in (0..n))
%o A113301 [A113301(n) for n in (0..40)] # _G. C. Greubel_, Nov 20 2021
%Y A113301 Cf. A000073, A008937, A099463, A113300.
%K A113301 easy,nonn
%O A113301 0,3
%A A113301 _Jonathan Vos Post_, Oct 24 2005
%E A113301 More terms from _Colin Barker_, May 06 2013