A017900 Expansion of 1/(1 - x^6 - x^7 - x^8 - ...).

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17887, 22990, 29548, 37975, 48804, 62721, 80608

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A Lamé sequence of higher order.

Number of compositions of n into parts >= 6. - Milan Janjic, Jun 28 2010

a(n+6) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 6. - Milan Janjic, Feb 17 2015

Same as sequence A005708 with 1, 0, 0, 0, 0, 0 prepended. - Linas Vepstas, Feb 06 2024

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• I. M. Gessel and 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 (1,0,0,0,0,1).

Crossrefs

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

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(6, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [1, 0$4, 1][i], 0)))^n)[6, 6]: seq(a(n), n=0..80); # Alois P. Heinz, Aug 04 2008

Mathematica

f[n_] := If[n < 1, 1, Sum[ Binomial[ n - 5 k - 5, k], {k, 0, (n - 5)/6}]]; Array[f, 49, 0] (* Adi Dani, Robert G. Wilson v, Jul 04 2011 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, 60] (* Jean-François Alcover, Feb 13 2016 *)

PARI

Vec((1-x)/(1-x-x^6)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

G.f.: 1/(1-Sum_{k>=6} x^k).

G.f.: (1-x)/(1-x-x^6). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 6*k, and 5 divides n-k, define c(n,k) = binomial(k,(n-k)/5), and c(n,k) = 0 otherwise. Then, for n >= 1, a(n+6) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011