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A017908 Expansion of 1/(1 - x^14 - x^15 - ...).

%I A017908 #39 Jul 08 2025 06:19:24
%S A017908 1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,
%T A017908 8,9,10,11,12,13,14,15,17,20,24,29,35,42,50,59,69,80,92,105,119,134,
%U A017908 151,171,195,224,259,301,351,410,479,559,651,756,875,1009
%N A017908 Expansion of 1/(1 - x^14 - x^15 - ...).
%C A017908 a(n) = number of compositions of n in which each part is >=14. - _Milan Janjic_, Jun 28 2010
%C A017908 a(n+27) equals the number of binary words of length n having at least 13 zeros between every two successive ones. - _Milan Janjic_, Feb 09 2015
%H A017908 Alois P. Heinz, <a href="/A017908/b017908.txt">Table of n, a(n) for n = 0..1000</a>
%H A017908 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%F A017908 G.f.: (x-1)/(x-1+x^14). - _Alois P. Heinz_, Aug 04 2008
%F A017908 For positive integers n and k such that k <= n <= 14*k, and 13 divides n-k, define c(n,k) = binomial(k,(n-k)/13), and c(n,k) = 0, otherwise. Then, for n>=1,  a(n+14) = sum(c(n,k), k=1..n). - _Milan Janjic_, Dec 09 2011
%p A017908 a:= n-> (Matrix(14, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$12, 1][i] else 0 fi)^n)[14, 14]: seq(a(n), n=0..62); # _Alois P. Heinz_, Aug 04 2008
%t A017908 LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2012 *)
%o A017908 (PARI) Vec((x-1)/(x-1+x^14)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%K A017908 nonn,easy
%O A017908 0,29
%A A017908 _N. J. A. Sloane_