0,5

Related with the number of all possible covers of L-length line segment by 3-length line segments with allowed gaps < 3 (A228362).

Index entries for linear recurrences with constant coefficients, signature (0,0,2,2,2,-1,-2,-3,-2,-1)

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

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}]

CoefficientList[Series[x^3(x^2+x+1)^2/(x^5+x^4+x^3-1)^2, {x, 0, 100}], x]

LinearRecurrence[{0, 0, 2, 2, 2, -1, -2, -3, -2, -1}, {0, 0, 0, 1, 2, 3, 4, 7, 12, 17}, 50] (* Harvey P. Dale, May 21 2025 *)

(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

Cf. A228362, A228364.

Sequence in context: A325244 A376987 A217786 * A292324 A289919 A293411

Adjacent sequences: A228491 A228492 A228493 * A228495 A228496 A228497

nonn,easy

Philipp O. Tsvetkov, Aug 23 2013

approved