A228364 G.f.: x^2*(x+1)^2/(x^3+x^2-1)^2.

0, 0, 1, 2, 3, 6, 9, 14, 22, 32, 48, 70, 101, 146, 208, 296, 419, 590, 829, 1160, 1619, 2254, 3130, 4338, 6000, 8284, 11419, 15716, 21600, 29648, 40645, 55658, 76135, 104042, 142045, 193758, 264078, 359636, 489408, 665538, 904449, 1228342, 1667216, 2261592

Offset: 0

Author

Philipp O. Tsvetkov, Aug 21 2013

Keywords

• nonn
• easy

Comments

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).

Comments from Rigoberto Florez, Oct 13 2019 (Start)

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.

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

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

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

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

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

(End)

References

• 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

Links

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

Crossrefs

Cf. A228677, A228361.

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, 2], {j, 0, 30}] (* or *) CoefficientList[Series[x^2 (x + 1)^2/(x^3 + x^2 - 1)^2, {x, 0, 100}], x]
LinearRecurrence[{0,2,2,-1,-2,-1},{0,0,1,2,3,6},50] (* Harvey P. Dale, Dec 31 2018 *)
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 *)

Formula

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

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).

a(n) = A228677(n-3) + 2*A228677(n-2) + A228677(n-1). - R. J. Mathar, Sep 02 2013

a(n) = Sum_{i=1..n} P(i+4)*P(n-i+4), where P(n) = A000931(n). - Rigoberto Florez, Oct 13 2019

Extensions

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.