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

0,11

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000217, A025810.

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

Sequence in context: A083023 A084359 A143935 * A331973 A097471 A025868

Adjacent sequences: A008613 A008614 A008615 * A008617 A008618 A008619

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved