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The number of 1-length gaps in all possible covers of n-length line by 2-length segments.
3

%I #55 Sep 08 2022 08:46:05

%S 0,1,0,2,2,3,6,7,12,17,24,36,50,72,102,143,202,282,394,549,762,1057,

%T 1462,2019,2784,3832,5268,7232,9916,13581,18580,25394,34674,47303,

%U 64478,87819,119520,162549,220920,300060,407302,552552,749186,1015259,1375134

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

%C 2-gaps must be filled, so, for example, xxoo doesn't count for n=4. - _Jon Perry_, Nov 18 2014

%D 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

%H D. Birmajer, J. Gil and M. Weiner, <a href="http://arxiv.org/abs/1405.7727">Linear recurrence sequences and their convolutions via Bell polynomials</a>, arXiv:1405.7727 [math.CO], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Gil/gil3.html">J. Int. Seq. 18 (2015) # 15.1.2</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-2,-1).

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

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

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

%F (n-1)*a(n) - (n+1)*a(n-2) - (n+2)*a(n-3) = 0 for n>2. - _Michael D. Weiner_, Nov 18 2014

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

%p 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

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

%o (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

%Y Cf. A228361, A228364.

%K nonn,easy

%O 0,4

%A _Philipp O. Tsvetkov_, Aug 26 2013