A008616 Expansion of 1/((1-x^2)(1-x^5)).

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 9, 9, 9, 9, 9, 10, 9, 10, 9, 10, 10, 10, 10, 10, 10

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts of size two and five.

It appears that, for n >= 2, a(n-2) is also (1) the number of partitions of 3n that are 6-term arithmetic progressions and (2) floor(n/2) - floor(2n/5). - John W. Layman, Jun 29 2009

References

• G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 30, Exercise 48.
• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 213
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 1, 0, -1).

Crossrefs

Cf. A000217, A025810.

Cf. A008615. - John W. Layman, Jun 29 2009

Programs

Magma

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x^2)*(1-x^5)))); // Wesley Ivan Hurt, Dec 27 2021

Mathematica

CoefficientList[Series[1 / ((1 - x^2) (1 - x^5)),{x, 0, 100}], x] (* Vincenzo Librandi, Jun 21 2013 *)

PARI

{a(n) = if( n<-6, -a(-7 - n), polcoeff( 1 / (1 - x^2) / (1 - x^5) + x * O(x^n), n))} /* Michael Somos, Jan 25 2005 */

PARI

a(n) = floor(n/10+(3+(-1)^n)/4) \\ Charles R Greathouse IV, Jun 19 2013

Formula

G.f.: 1/((1-x^2)(1-x^5)) = 1/((x-1)^2*(1+x)*(1+x+x^2+x^3+x^4)).

Euler transform of finite sequence [0, 1, 0, 0, 1].

a(n) = -a(-7 - n) = a(n - 10) + 1 = a(n - 2) + a(n - 5) - a(n - 7). - Michael Somos, Jan 25 2005

A000217(a(n)) = A025810(n). - Michael Somos, Dec 15 2002

a(n) = 7/20 + n/10 + (-1)^n/4 + (A105384(n) + 2*(A010891(n) + A105384(n+4)))/5. - R. J. Mathar, Jun 28 2009

a(n) = floor(n/10 + (3 + (-1)^n)/4). - Tani Akinari, Jun 20 2013