A008722 Molien series for 3-dimensional group [2,9] = *229.

1, 0, 2, 0, 3, 0, 4, 0, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 11, 21, 13, 23, 15, 25, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 30, 45, 33, 48, 36, 51, 39, 54, 42, 58, 45, 62, 48, 66, 51, 70, 54, 74, 58, 78, 62, 82, 66, 86, 70, 90, 74, 95, 78, 100, 82

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

It appears that a(n) is the number of (n+11)-digit fixed points under the base-7 Kaprekar map A165071 (see A165075 for the list of fixed points). - Joseph Myers, Sep 04 2009

a(n) is the number of partitions of n into parts 2 and 9 where there are two kinds of parts 2. - Hoang Xuan Thanh, Jun 20 2025

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 225
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,0,1,0,-2,0,1).

Programs

GAP

a:=[1,0,2,0,3,0,4,0,5,1,6,2,7];; for n in [14..80] do a[n]:= 2*a[n-2] -a[n-4]+a[n-9]-2*a[n-11]+a[n-13]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)^2*(1-x^9)) )); // G. C. Greubel, Sep 09 2019

Maple

1/((1-x^2)^2*(1-x^9)); seq(coeff(series(%, x, n+1), x, n), n = 0..80); # modified by G. C. Greubel, Sep 09 2019

Mathematica

LinearRecurrence[{0,2,0,-1,0,0,0,0,1,0,-2,0,1}, {1,0,2,0,3,0,4,0,5,1,6, 2,7}, 80] (* Ray Chandler, Jul 15 2015 *)

PARI

my(x='x+O('x^80)); Vec(1/((1-x^2)^2*(1-x^9))) \\ G. C. Greubel, Sep 09 2019

Sage

def A008722_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)^2*(1-x^9)) ).list()
A008722_list(80) # G. C. Greubel, Sep 09 2019

Formula

G.f.: 1/((1-x^2)^2*(1-x^9)).

a(n) = 2*a(n-2) - a(n-4) + a(n-9) - 2*a(n-11) + a(n-13). - R. J. Mathar, Dec 18 2014

a(n) = floor((n^2 + n*(13+9*(-1)^n) + 62*(-1)^n + 75)/72) - [(n mod 9)=7], where [] is Iverson bracket. - Hoang Xuan Thanh, Jun 20 2025