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 A215448 #38 Jul 27 2025 08:57:26
%S A215448 1,0,2,5,15,43,124,357,1028,2960,8523,24541,70663,203466,585857,
%T A215448 1686908,4857258,13985917,40270843,115955271,333879896,961368845,
%U A215448 2768151264,7970573896,22950352843,66082907265,190278147899,547884090854,1577569365297,4542429947992
%N A215448 a(0)=1, a(1)=0, a(n) = a(n-1) + a(n-2) + Sum_{i=0...n-1} a(i).
%C A215448 For the general recurrence X(n) = 3*X(n-1) - X(n-3) we get Sum_{k=3..n} X(k) = 3*Sum_{k=2..n-1} X(k) - Sum_{k=0..n-3} X(k), which implies the following summation formula: X(n) - X(n-1) - X(n-2) - X(2) + X(1) + X(0) = Sum_{k=2..n-1} X(k). Similarly from the formula X(n) + X(n-3) = 3*X(n-1) we deduce the following relations: Sum_{k=0..2*n-1} X(3*k) = 3*Sum_{k=0..n-1} X(6*k+2), Sum_{k=0..2*n-1} X(3*k+1) = 3*Sum_{k=1..n} X(6*k), and Sum_{k=0..2*n-1} X(3*k+2) = 3*Sum_{k=1..n} X(6*k-2). Lastly from the formula X(n)-X(n-1)=(X(n-1)-X(n-3))+X(n-1) we obtain the relations: Sum_{k=2..2*n+1} (-1)^(k-1)*X(k) = X(2*n) - X(0) + Sum_{k=1..n} X(2*k) and Sum_{k=3..2n} (-1)^k*X(k) = X(2*n-1) - X(1) + Sum_{k=2..n} X(2*k-1). - _Roman Witula_, Aug 27 2012
%H A215448 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-1).
%F A215448 a(0)=1, a(1)=0, for n>=2, a(n) = a(n-1) + a(n-2) + (a(0)+...+a(n-1)).
%F A215448 Conjecture: a(n) = +3*a(n-1) -a(n-3) = A076264(n) -3 *A076264(n-1) +2*A076264(n-2). G.f. (2*x-1)*(x-1) / ( 1-3*x+x^3 ). - _R. J. Mathar_, Aug 11 2012
%F A215448 Proof of the above conjecture: we have a(n) - a(n-1) = a(n-1) + a(n-2) + (a(0) + ... + a(n-1)) - a(n-2) - a(n-3) - (a(0) + ... + a(n-2)), which after simple algebra implies a(n) - a(n-1) = 2*a(n-1) - a(n-3), so the Mathar's formula holds true (see also Witula's comment above). - _Roman Witula_, Aug 27 2012
%t A215448 LinearRecurrence[{3,0,-1},{1,0,2},30] (* _Harvey P. Dale_, Jan 26 2017 *)
%o A215448 (Python)
%o A215448 a = [1]*33
%o A215448 a[1]=0
%o A215448 sum = a[0]+a[1]
%o A215448 for n in range(2,33):
%o A215448 print(a[n-2], end=', ')
%o A215448 a[n] = a[n-1] + a[n-2] + sum
%o A215448 sum += a[n]
%Y A215448 Cf. A052536: same formula, seed {0, 1}, first term removed.
%Y A215448 Cf. A122100: same formula, seed {0,-1}, first two terms removed.
%Y A215448 Cf. A052545: same formula, seed {1, 1}.
%K A215448 nonn,easy
%O A215448 0,3
%A A215448 _Alex Ratushnyak_, Aug 10 2012