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

0,4

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

Table of n, a(n) for n=0..43.

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

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

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 *)

Crossrefs

Cf. A228677, A228361.

Sequence in context: A022567 A134004 A123631 * A018060 A115856 A147007

Adjacent sequences: A228361 A228362 A228363 * A228365 A228366 A228367

Keyword

nonn,easy

Author

Philipp O. Tsvetkov, Aug 21 2013

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.

Status

approved