A008725 Molien series for 3-dimensional group [2,n] = *22n.

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 24, 27, 30, 33, 36, 39, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 111, 117, 123, 129, 135, 141, 147, 154, 161, 168, 175, 182, 189, 196, 204, 212, 220, 228, 236, 244, 252, 261, 270, 279, 288, 297, 306

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of partitions of n into parts 1 and 7, where there are two kinds of part 1. - Joerg Arndt, Sep 27 2020

Define a general Somos-4 sequence by b(n) = (p1*b(n-1)*b(n-3) + p2*b(n-2)^2)/b(n-4) with b(0) = b0, b(1) = b1, b(2) = b2, b(3) = b3 and where p1 = (b1^3*b2 - b0^3*b3) / (b0*(b1^3 + b0^2*b2)), p2 = -b1*(b2^2 + b0*b3) / (b1^3 + b0^2*b2). Then b(n) = -b(-1-n) for all n in Z. The denominator of b(n) is a power of b0 times (b1^3 + b0^2*b2)^a(n-4). - Michael Somos, Nov 23 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 190
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-2,1).

Crossrefs

Cf. A011867, A174738.

Programs

GAP

a:=[1,2,3,4,5,6,7,9,11];; for n in [10..80] do a[n]:=2*a[n-1] -a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x)^2*(1-x^7)) )); // G. C. Greubel, Sep 09 2019

Maple

1/((1-x)^2*(1-x^7)): seq(coeff(series(%, x, n+1), x, n), n=0..80);

Mathematica

CoefficientList[Series[1/((1-x)^2*(1-x^7)), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,9,11}, 80] (* Harvey P. Dale, Sep 27 2014 *)
a[ n_] := Floor[(n+4)*(n+5)/14]; (* Michael Somos, Nov 23 2023 *)

PARI

my(x='x+O('x^80)); Vec(1/((1-x)^2*(1-x^7))) \\ G. C. Greubel, Sep 09 2019

PARI

{a(n) = (n+4)*(n+5)\14}; /* Michael Somos, Nov 23 2023 */

Sage

def A008725_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x)^2*(1-x^7))).list()
A008725_list(80) # G. C. Greubel, Sep 09 2019

Formula

G.f.: 1/((1-x)^2*(1-x^7)).

From Mitch Harris, Sep 08 2008: (Start)

a(n) = Sum_{j=0..n+7} floor(j/7).

a(n-7) = (1/2)*floor(n/7)*(2*n - 5 - 7*floor(n/7)). (End)

a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9). - R. J. Mathar, Apr 20 2010

a(n) = A011867(n+5). - Pontus von Brömssen, Sep 27 2020

a(n) = a(-9-n) = A174738(n+7) = floor((n+4)*(n+5)/14) for all n in Z. - Michael Somos, Nov 23 2023

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010