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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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)

Crossrefs

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

Cf. A005711.

Programs

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 *)

PARI

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

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