A017903 Expansion of 1/(1 - x^9 - x^10 - ...).

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, 30, 37, 45, 54, 64, 76, 91, 110, 134, 164, 201, 246, 300, 364, 440, 531, 641, 775, 939, 1140, 1386, 1686, 2050, 2490, 3021, 3662, 4437, 5376, 6516, 7902, 9588

Offset

0,19

Comments

A Lamé sequence of higher order.

a(n) = number of compositions of n in which each part is >=9. - Milan Janjic, Jun 28 2010

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

Links

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

I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5

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 (1,0,0,0,0,0,0,0,1)

Formula

G.f.: (x-1)/(x-1+x^9). - Alois P. Heinz, Aug 04 2008

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

a(n) = A005711(n-10) for n > 9. - Alois P. Heinz, May 21 2018

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(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

Mathematica

CoefficientList[(1-x)/(1-x-x^9) + O[x]^70, x] (* Jean-François Alcover, Jun 08 2015 *)

Prog

(PARI) Vec((x-1)/(x-1+x^9)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Crossrefs

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.

Cf. A005711.

Sequence in context: A246084 A260768 A130224 * A005711 A322856 A280863

Adjacent sequences: A017900 A017901 A017902 * A017904 A017905 A017906

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved