A017899 Expansion of 1/(1 -x^5 -x^6 -x^7 - ...).

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions of n into parts >=5. - Joerg Arndt, Jun 22 2011

a(n+5) equals the number of binary words such that 0 appears only in runs whose lengths are a multiple of 5. - Milan Janjic, Feb 17 2015

a(n-5) equals the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 2 or 3. - Ethan Patrick White, Jun 24 2020

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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.
• J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97.
• Ethan P. White, Richard K. Guy, Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1).

Crossrefs

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

Apart from initial terms, same as A003520.

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(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$3, 1][i] else 0 fi)^n)[5,5]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

Mathematica

CoefficientList[ Series[(1 - x)/(1 - x - x^5), {x, 0, 50}], x] (* Adi Dani, Jun 25 2011 *)
LinearRecurrence[{1,0,0,0,1},{1,0,0,0,0},60] (* Harvey P. Dale, Jun 07 2015 *)

PARI

Vec((1-x)/(1-x-x^5)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

Formula

G.f.: (1-x)/(1-x-x^5) = 1/(1-Sum_{k>=5} x^k).

For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+5) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011