A017904 Expansion of 1/(1 - x^10 - x^11 - ...).

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, 20, 25, 31, 38, 46, 55, 65, 76, 89, 105, 125, 150, 181, 219, 265, 320, 385, 461, 550, 655, 780, 930, 1111, 1330, 1595, 1915, 2300, 2761, 3311, 3966, 4746, 5676

Offset

0,21

Comments

A Lamé sequence of higher order.

a(n) = number of compositions of n in which each part is >=10. - Milan Janjic, Jun 28 2010

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Formula

G.f.: (x-1)/(x-1+x^10). - Alois P. Heinz, Aug 04 2008

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

Maple

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
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

Mathematica

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 *)

Prog

(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

Crossrefs

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017903, and this one.

Sequence in context: A107062 A178538 A332166 * A368902 A143290 A272038

Adjacent sequences: A017901 A017902 A017903 * A017905 A017906 A017907

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved