A228577 The number of 1-length gaps in all possible covers of n-length line by 2-length segments.

0, 1, 0, 2, 2, 3, 6, 7, 12, 17, 24, 36, 50, 72, 102, 143, 202, 282, 394, 549, 762, 1057, 1462, 2019, 2784, 3832, 5268, 7232, 9916, 13581, 18580, 25394, 34674, 47303, 64478, 87819, 119520, 162549, 220920, 300060, 407302, 552552, 749186, 1015259, 1375134

Offset

0,4

Comments

2-gaps must be filled, so, for example, xxoo doesn't count for n=4. - Jon Perry, Nov 18 2014

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 Eqn. (3.13). - N. J. A. Sloane, Jan 11 2022

Links

Table of n, a(n) for n=0..44.

D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014  and J. Int. Seq. 18 (2015) # 15.1.2.

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

Formula

For n>1, a(n) = n * A228361(n) - 2 * A228364(n).

G.f.: x/(x^3 + x^2 - 1)^2, convolution of A182097 by itself.

a(n) = 2*a(n-2) +2*a(n-3) -a(n-4) -2*a(n-5) -a(n-6) for n>5.

(n-1)*a(n) - (n+1)*a(n-2) - (n+2)*a(n-3) = 0 for n>2. - Michael D. Weiner, Nov 18 2014

Example

For n=6 we have xxoxxo, oxxxxo and oxxoxx, so a(6) = number of o's = 6. - Jon Perry, Nov 18 2014

Maple

A228577 := proc(n) coeftayl(x/(x^3+x^2-1)^2, x=0, n); end proc: seq(A228577(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2014

Mathematica

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

Prog

(Magma) I:=[0, 1, 0, 2, 2, 3]; [n le 6 select I[n] else 2*Self(n-2)+2*Self(n-3)-Self(n-4)-2*Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Nov 18 2014

Crossrefs

Cf. A228361, A228364.

Sequence in context: A039866 A106369 A298436 * A032062 A303028 A011141

Adjacent sequences: A228574 A228575 A228576 * A228578 A228579 A228580

Keyword

nonn,easy

Author

Philipp O. Tsvetkov, Aug 26 2013

Status

approved