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%I #47 Jan 23 2019 18:50:39
%S 1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,11,14,18,23,29,36,44,
%T 53,64,78,96,119,148,184,228,281,345,423,519,638,786,970,1198,1479,
%U 1824,2247,2766,3404,4190,5160,6358,7837,9661,11908,14674,18078
%N Expansion of 1/(1 - x^8 - x^9 - ...).
%C A Lamé sequence of higher order.
%C a(n) = number of compositions of n in which each part is >=8. - _Milan Janjic_, Jun 28 2010
%C a(n+8) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 8. - _Milan Janjic_, Feb 17 2015
%H Alois P. Heinz, <a href="/A017902/b017902.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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_08">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 1).
%F G.f.: (x-1)/(x-1+x^8). - _Alois P. Heinz_, Aug 04 2008
%F For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+8) = sum(c(n,k), k=1..n). - _Milan Janjic_, Dec 09 2011
%F a(n) = A005710(n)-A005710(n-1). - _R. J. Mathar_, Sep 07 2016
%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(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$6, 1][i] else 0 fi)^n)[8,8]: seq(a(n), n=0..53); # _Alois P. Heinz_, Aug 04 2008
%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 60] (* _Jean-François Alcover_, Feb 13 2016 *)
%o (PARI) a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,0,0,0,0,0,0,1]^n*[1;0;0;0;0;0;0;0])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.
%K nonn,easy
%O 0,17
%A _N. J. A. Sloane_
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