cp's OEIS Frontend

A017909 Expansion of 1/(1 - x^15 - x^16 - ...).

%I A017909 #42 Jul 08 2025 06:19:30
%S A017909 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,
%T A017909 6,7,8,9,10,11,12,13,14,15,16,18,21,25,30,36,43,51,60,70,81,93,106,
%U A017909 120,135,151,169,190,215,245,281,324,375,435,505,586,679
%N A017909 Expansion of 1/(1 - x^15 - x^16 - ...).
%C A017909 a(n+29) equals the number of binary words of length n having at least 14 zeros between every two successive ones. - _Milan Janjic_, Feb 09 2015
%C A017909 Number of compositions of n into parts >= 15. - _Ilya Gutkovskiy_, May 23 2017
%H A017909 Alois P. Heinz, <a href="/A017909/b017909.txt">Table of n, a(n) for n = 0..1000</a>
%H A017909 I. M. Gessel, Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5
%H A017909 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%F A017909 G.f.: (x-1)/(x-1+x^15). - _Alois P. Heinz_, Aug 04 2008
%F A017909 For positive integers n and k such that k <= n <= 15*k, and 14 divides n-k, define c(n,k) = binomial(k,(n-k)/14), and c(n,k) = 0, otherwise. Then, for n>=1,  a(n+15) = sum(c(n,k), k=1..n). - _Milan Janjic_, Dec 09 2011
%p A017909 a:= n -> (Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$13, 1][i] else 0 fi)^n)[15,15]: seq(a(n), n=0..80); # _Alois P. Heinz_, Aug 04 2008
%t A017909 LinearRecurrence[{1, 0, 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, 0}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2012 *)
%t A017909 CoefficientList[Series[(x-1)/(x-1+x^15),{x,0,100}],x] (* _Harvey P. Dale_, Sep 04 2020 *)
%o A017909 (PARI) Vec((x-1)/(x-1+x^15)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%K A017909 nonn,easy
%O A017909 0,31
%A A017909 _N. J. A. Sloane_