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G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.
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%I #62 Dec 16 2023 20:09:48

%S 0,0,1,2,3,6,9,14,22,32,48,70,101,146,208,296,419,590,829,1160,1619,

%T 2254,3130,4338,6000,8284,11419,15716,21600,29648,40645,55658,76135,

%U 104042,142045,193758,264078,359636,489408,665538,904449,1228342,1667216,2261592

%N G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.

%C 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 Comments from _Rigoberto Florez_, Oct 13 2019 (Start)

%C 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 k=2: Intervals covering or almost covering interval [0,2] = {[0,2]}. So a(2)=1.

%C k=3: Interval [0,3] = {[0,2]},{[1,3]}

%C k=4: Interval [0,4] = {[0,2],[2,4]},{[1,3]}

%C k=5: Interval [0,5] = {[0,2],[2,4]},{[1,3],[3,5]},{[0,2],[3,5]}

%C k=6: Interval [0,6] = {[0,2],[2,4],[4,6]},{[1,3],[3,5]},{[1,3],[4,6]},{[0,2],[3,5]}

%C (End)

%D 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-2,-1).

%F G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.

%F 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 a(n) = A228677(n-3) + 2*A228677(n-2) + A228677(n-1). - _R. J. Mathar_, Sep 02 2013

%F a(n) = Sum_{i=1..n} P(i+4)*P(n-i+4), where P(n) = A000931(n). - _Rigoberto Florez_, Oct 13 2019

%t 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 LinearRecurrence[{0,2,2,-1,-2,-1},{0,0,1,2,3,6},50] (* _Harvey P. Dale_, Dec 31 2018 *)

%t 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 Cf. A228677, A228361.

%K nonn,easy

%O 0,4

%A _Philipp O. Tsvetkov_, Aug 21 2013

%E 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.