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%I #47 Jan 23 2019 19:24:19
%S 1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,11,13,16,
%T 20,25,31,38,46,55,65,76,89,105,125,150,181,219,265,320,385,461,550,
%U 655,780,930,1111,1330,1595,1915,2300,2761,3311,3966,4746,5676
%N Expansion of 1/(1 - x^10 - x^11 - ...).
%C A Lamé sequence of higher order.
%C a(n) = number of compositions of n in which each part is >=10. - _Milan Janjic_, Jun 28 2010
%C a(n+19) equals the number of binary words of length n having at least 9 zeros between every two successive ones. - _Milan Janjic_, Feb 09 2015
%H Alois P. Heinz, <a href="/A017904/b017904.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Hermes, <a href="http://dx.doi.org/10.1007/BF01446684">Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden</a>, Math. Ann., 45 (1894), 371-380.
%H Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%F G.f.: (x-1)/(x-1+x^10). - _Alois P. Heinz_, Aug 04 2008
%F For positive integers n and k such that k <= n <= 10*k, and 9 divides n-k, define c(n,k) = binomial(k,(n-k)/9), and c(n,k) = 0, otherwise. Then, for n>= 1, a(n+10) = sum(c(n,k), k=1..n). - _Milan Janjic_, Dec 09 2011
%p f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1),0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order
%p a:= n-> (Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$8, 1][i] else 0 fi)^n)[10,10]: seq(a(n), n=0..80); # _Alois P. Heinz_, Aug 04 2008
%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2012 *)
%o (PARI) a(n)=([0,1,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,1; 1,0,0,0,0,0,0,0,0,1]^n)[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017903, and this one.
%K nonn,easy
%O 0,21
%A _N. J. A. Sloane_
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