A017902 Expansion of 1/(1 - x^8 - x^9 - ...).

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078

Offset

0,17

Comments

A Lamé sequence of higher order.

a(n) = number of compositions of n in which each part is >=8. - Milan Janjic, Jun 28 2010

a(n+8) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 8. - 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, 1).

Formula

G.f.: (x-1)/(x-1+x^8). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+8) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

a(n) = A005710(n)-A005710(n-1). - R. J. Mathar, Sep 07 2016

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(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 0$6, 1][i] else 0 fi)^n)[8, 8]: seq(a(n), n=0..53); # Alois P. Heinz, Aug 04 2008

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 60] (* Jean-François Alcover, Feb 13 2016 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; 1, 0, 0, 0, 0, 0, 0, 1]^n*[1; 0; 0; 0; 0; 0; 0; 0])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

Crossrefs

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

Sequence in context: A219955 A079064 A123176 * A005710 A367800 A291146

Adjacent sequences: A017899 A017900 A017901 * A017903 A017904 A017905

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved