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

Author

Philipp O. Tsvetkov, Aug 26 2013

Keywords

• nonn
• easy

Comments

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

Examples

			For n=6 we have xxoxxo, oxxxxo and oxxoxx, so a(6) = number of o's = 6. - _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

• 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).

Crossrefs

Cf. A228361, A228364.

Programs

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

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]

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