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 A228364 #62 Dec 16 2023 20:09:48
%S A228364 0,0,1,2,3,6,9,14,22,32,48,70,101,146,208,296,419,590,829,1160,1619,
%T A228364 2254,3130,4338,6000,8284,11419,15716,21600,29648,40645,55658,76135,
%U A228364 104042,142045,193758,264078,359636,489408,665538,904449,1228342,1667216,2261592
%N A228364 G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.
%C A228364 a(n) = number of segments of length 2 in all possible covers of a line of length k+1 by segments of length 2 with allowed gaps < 2 (cf. A228361).
%C A228364 Comments from _Rigoberto Florez_, Oct 13 2019 (Start)
%C A228364 Consider the interval [0,k] on the real line, where k an integer. We are looking for all length 2 subintervals covering or almost covering [0,k] where their ends are integers and the distance between two consecutive subintervals is at most 1. Examples.
%C A228364 k=2: Intervals covering or almost covering interval [0,2] = {[0,2]}. So a(2)=1.
%C A228364 k=3: Interval [0,3] = {[0,2]},{[1,3]}
%C A228364 k=4: Interval [0,4] = {[0,2],[2,4]},{[1,3]}
%C A228364 k=5: Interval [0,5] = {[0,2],[2,4]},{[1,3],[3,5]},{[0,2],[3,5]}
%C A228364 k=6: Interval [0,6] = {[0,2],[2,4],[4,6]},{[1,3],[3,5]},{[1,3],[4,6]},{[0,2],[3,5]}
%C A228364 (End)
%D A228364 A. G. Shannon, P. G. Anderson and A. F. Horadam, Properties of Cordonnier, Perrin and Van der Laan numbers, International Journal of Mathematical Education in Science and Technology, Volume 37:7 (2006), 825-831. See Equation (3.11). - _N. J. A. Sloane_, Jan 11 2022
%H A228364 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-2,-1).
%F A228364 G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.
%F A228364 a(0)=a(1)=0, a(2)=1, a(3)=2, a(4)=3, a(5)=6; for n>5, a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) - 2*a(n-5) - a(n-6).
%F A228364 a(n) = A228677(n-3) + 2*A228677(n-2) + A228677(n-1). - _R. J. Mathar_, Sep 02 2013
%F A228364 a(n) = Sum_{i=1..n} P(i+4)*P(n-i+4), where P(n) = A000931(n). - _Rigoberto Florez_, Oct 13 2019
%t A228364 c[k_, l_, m_] := Sum[(-1)^i Binomial[k - 1 - i*l, m - 1] Binomial[m, i], {i, 0, Floor[(k - m)/l]}]; a[L_, l_, m_] := Sum[Binomial[m + 1, m + 1 - j]*c[L - l*m, l - 1, j], {j, 0, m + 1}]; sa[L_, l_] := Sum[j*a[L, l, j], {j, 1, Ceiling[L/l]}]; Table[sa[j, 2], {j, 0, 30}] (* or *) CoefficientList[Series[x^2 (x + 1)^2/(x^3 + x^2 - 1)^2, {x, 0, 100}], x]
%t A228364 LinearRecurrence[{0,2,2,-1,-2,-1},{0,0,1,2,3,6},50] (* _Harvey P. Dale_, Dec 31 2018 *)
%t A228364 P[0] = 1; P[1] = 0; P[2] = 0; P[n_] := P[n] = P[n - 2] + P[n - 3]; Table[Sum[P[i + 4]*P[n -i + 4], {i, 1, n}], {n, 0, 20}] (* _Rigoberto Florez_, Oct 13 2019 *)
%Y A228364 Cf. A228677, A228361.
%K A228364 nonn,easy
%O A228364 0,4
%A A228364 _Philipp O. Tsvetkov_, Aug 21 2013
%E A228364 Edited by _N. J. A. Sloane_, Nov 06 2019, replacing not very clear original definition by simple generating function, rewriting original definition using comments from _Rigoberto Florez_, and moving it to comments.