A228494 The number of 3-length segments in all possible covers of L-length line by these segments with allowed gaps < 3.

0, 0, 0, 1, 2, 3, 4, 7, 12, 17, 24, 36, 54, 77, 108, 155, 222, 312, 436, 612, 858, 1194, 1656, 2298, 3184, 4397, 6060, 8346, 11480, 15762, 21612, 29607, 40518, 55385, 75632, 103197, 140692, 191647, 260856, 354814, 482290, 655131, 889364, 1206649, 1636218

Offset: 0

Author

Philipp O. Tsvetkov, Aug 23 2013

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A228362, A228364.

Programs

Mathematica

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

Formula

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