A008679 Expansion of 1/((1-x^3)*(1-x^4)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts 3 and 4. - Reinhard Zumkeller, Feb 09 2009

Convolution of A112689 (shifted left once) by A033999. - R. J. Mathar, Feb 13 2009

With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose largest three parts are equal. - Wesley Ivan Hurt, Jan 06 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 216
• A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).

Crossrefs

Cf. A005044, A033999, A112689.

Programs

GAP

a:=[1,0,0,1,1,0,1,1];; for n in [8..90] do a[n]:=a[n-3]+a[n-4]-a[n-7]; od; a; # G. C. Greubel, Sep 09 2019

Magma

R:=PowerSeriesRing(Integers(), 90); Coefficients(R!( 1/((1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 09 2019

Maple

seq(coeff(series(1/((1-x^3)*(1-x^4)), x, n+1), x, n), n = 0..90); # G. C. Greubel, Sep 09 2019

Mathematica

LinearRecurrence[{0,0,1,1,0,0,-1}, {1,0,0,1,1,0,1}, 90] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
CoefficientList[Series[1/((1-x)^2(1+x)(1+x+x^2)(1+x^2)), {x,0,90}], x] (* Vincenzo Librandi, Jun 11 2013 *)

PARI

my(x='x+O('x^90)); Vec(1/((1-x^3)*(1-x^4))) \\ G. C. Greubel, Sep 09 2019

Sage

def A008679_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^3)*(1-x^4))).list()
A008679_list(90) # G. C. Greubel, Sep 09 2019

Formula

a(n+12) = a(n) + 1. - Reinhard Zumkeller, Feb 09 2009

G.f.: 1/((1-x)^2*(1+x)*(1+x+x^2)*(1+x^2)). - R. J. Mathar, Feb 13 2009

a(n) = 1 + floor(n/3) + floor(-n/4). - Tani Akinari, Sep 02 2013

E.g.f.: (1/72)*(9*exp(-x)+21*exp(x)+6*exp(x)*x+18*cos(x)+24*exp(-x/2)*cos(sqrt(3)*x/2)-18*sin(x)+8*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Sep 09 2019

a(n) = A005044(n+3) - A005044(n+1). - Yuchun Ji, Oct 10 2020

From Wesley Ivan Hurt, Jan 17 2021: (Start)

a(n) = a(n-3) + a(n-4) - a(n-7).

a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i = n+4-i-k-j], where [ ] is the Iverson bracket. (End)