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%I #57 Jan 23 2019 18:48:36
%S 1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,12,15,19,24,
%T 30,37,45,54,64,76,91,110,134,164,201,246,300,364,440,531,641,775,939,
%U 1140,1386,1686,2050,2490,3021,3662,4437,5376,6516,7902,9588
%N Expansion of 1/(1 - x^9 - x^10 - ...).
%C A Lamé sequence of higher order.
%C a(n) = number of compositions of n in which each part is >=9. - _Milan Janjic_, Jun 28 2010
%C a(n+9) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 9. - _Milan Janjic_, Feb 17 2015
%H Alois P. Heinz, <a href="/A017903/b017903.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_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,1)
%F G.f.: (x-1)/(x-1+x^9). - _Alois P. Heinz_, Aug 04 2008
%F For positive integers n and k such that k <= n <= 9*k, and 8 divides n-k, define c(n,k) = binomial(k,(n-k)/8), and c(n,k) = 0, otherwise. Then, for n>= 1, a(n+9) = sum(c(n,k), k=1..n). - _Milan Janjic_, Dec 09 2011
%F a(n) = A005711(n-10) for n > 9. - _Alois P. Heinz_, May 21 2018
%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(9, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$7, 1][i] else 0 fi)^n)[9,9]: seq(a(n), n=0..55); # _Alois P. Heinz_, Aug 04 2008
%t CoefficientList[(1-x)/(1-x-x^9) + O[x]^70, x] (* _Jean-François Alcover_, Jun 08 2015 *)
%o (PARI) Vec((x-1)/(x-1+x^9)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.
%Y Cf. A005711.
%K nonn,easy
%O 0,19
%A _N. J. A. Sloane_