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%I #17 Jun 13 2015 00:54:44
%S 0,0,0,1,2,3,4,7,12,17,24,36,54,77,108,155,222,312,436,612,858,1194,
%T 1656,2298,3184,4397,6060,8346,11480,15762,21612,29607,40518,55385,
%U 75632,103197,140692,191647,260856,354814,482290,655131,889364,1206649,1636218
%N The number of 3-length segments in all possible covers of L-length line by these segments with allowed gaps < 3.
%C Related with the number of all possible covers of L-length line segment by 3-length line segments with allowed gaps < 3 (A228362).
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,2,2,-1,-2,-3,-2,-1)
%F G.f.: x^3*(x^2+x+1)^2/((x^2+1)*(x^3+x^2-1))^2.
%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, 3], {j, 0, 100}]
%t CoefficientList[Series[x^3(x^2+x+1)^2/(x^5+x^4+x^3-1)^2,{x, 0, 100}], x]
%o (PARI) concat([0,0,0], Vec(x^3*(x^2+x+1)^2/((x^2+1)*(x^3+x^2-1))^2+O(x^66))) \\ _Joerg Arndt_, Aug 23 2013
%Y Cf. A228362, A228364.
%K nonn,easy
%O 0,5
%A _Philipp O. Tsvetkov_, Aug 23 2013
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