A017901 Expansion of 1/(1 - x^7 - x^8 - ...).

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851, 31200, 39173

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 >= 7. - Milan Janjic, Jun 28 2010

a(n+7) equals the number of n-length binary words such that 0 appears only in a run length that is a multiple of 7. - Milan Janjic, Feb 17 2015

A017847(n) = |a(-n)| for n>=0. - Michael Somos, Oct 28 2018

Examples

			G.f. = 1 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + 2*x^14 + ... - _Michael Somos_, Oct 28 2018
		

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

Crossrefs

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898, A017899, A017900, A017901, A017902, A017903, A017904.

Cf. A005709, A017847.

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(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$5, 1][i] else 0 fi)^n)[7,7]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

Mathematica

LinearRecurrence[{1,0,0,0,0,0,1}, {1,0,0,0,0,0,0}, 60] (* Jean-François Alcover, Mar 28 2017 *)

PARI

Vec((x-1)/(x-1+x^7)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

PARI

{a(n) = if( n < 0, polcoeff( 1 / (1 + x^6 - x^7) + x * O(x^-n), -n), polcoeff( (1 - x) / (1 - x - x^7) + x * O(x^n), n))}; /* Michael Somos, Oct 28 2018 */

Formula

G.f.: (x-1)/(x-1+x^7). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+7) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

a(n) = A005709(n) - A005709(n-1). - R. J. Mathar, Sep 07 2016

0 == a(n) + a(n+6) - a(n+7) for all n in Z. - Michael Somos, Oct 28 2018