A008624 Expansion of g.f. (1 + x^3)/((1 - x^2)*(1 - x^4)) = (1 - x + x^2)/((1 + x)*(1 - x)^2*(1 + x^2)).

1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Molien series of 2-dimensional representation of group of order 16 over GF(3).

References

• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 107.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Essentially the same as A059169.

Cf. A028242, A110654, A110659.

Programs

Maple

f := x -> (1+x^3)/((1-x^2)*(1-x^4)): seq(coeff(series(f(x), x, n+1), x, n), n=0..64);
a := n -> floor(n/4) + ((n mod 2 + 1 - floor((n mod 4)/3)) mod 2): seq(a(n), n=0..64); # Johannes W. Meijer, Oct 08 2013

Mathematica

CoefficientList[Series[(1 + x^3) / (1 - x^2) / (1 - x^4), {x, 0, 70}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{1,0,0,1,-1},{1,0,1,1,2},70] (* Harvey P. Dale, Sep 27 2024 *)

PARI

a(n) = (3 + 3*(-1)^n + (1-I)*(-I)^n + (1+I)*I^n + 2*n) / 8 \\ Colin Barker, Oct 15 2015

PARI

my(x='x+O('x^100)); Vec((1+x^3)/((1-x^2)*(1-x^4))) \\ Altug Alkan, Dec 24 2015

Formula

From Reinhard Zumkeller, Aug 05 2005: (Start)

a(n) = floor(n/4) + ((n mod 2 + 1 - floor((n mod 4)/3)) mod 2).

a(n) = A110654(A028242(n)). (End)

a(n) = (3 + 3*(-1)^n + (1-i)*(-i)^n + (1+i)*i^n + 2*n) / 8 where i = sqrt(-1). - Colin Barker, Oct 15 2015

a(n) = (2*n+3+2*cos(n*Pi/2)+3*cos(n*Pi)-2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017

E.g.f.: (cos(x) + (3 + x)*cosh(x) - sin(x) + x*sinh(x))/4. - Stefano Spezia, Jan 03 2023

Extensions

Replaced x^2 three times with x in the generating function (un-aerated). - R. J. Mathar, Oct 23 2008