A089800 Expansion of Jacobi theta function theta_2(q)/q^(1/4).

2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Eric W. Weisstein, Nov 12 2003

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Programs

Magma

[2*(Floor(Sqrt(n+1)+1/2) - Floor(Sqrt(n)+1/2)): n in [0..50]]; // G. C. Greubel, Nov 20 2017

Maple

A089800 := proc(n)
    if issqr(1+4*n) then
        if type( sqrt(1+4*n)-1,'even') then
            2;
        else
            0;
        end if;
    else
        0;
    end if;
end proc:
seq( A089800(n),n=0..40) ; # R. J. Mathar, Feb 22 2021

Mathematica

a[n_] := SeriesCoefficient[ EllipticTheta[2, 0, q]/q^(1/4), {q, 0, n}]; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Nov 12 2012 *)
Table[2*(Floor[Sqrt[n+1]+1/2] - Floor[Sqrt[n]+1/2]), {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)

PARI

for(n=0,50, print1(2*(floor(sqrt(n+1)+1/2) - floor(sqrt(n)+1/2)), ", ")) \\ G. C. Greubel, Nov 20 2017

Formula

For n > 0, a(n) = 2*(floor(sqrt(n+1/4)-1/2) - floor(sqrt(n-1+1/4)-1/2)). - Mikael Aaltonen, Jan 18 2015

a(n) = 2*(floor(sqrt(n+1)+1/2)-floor(sqrt(n)+1/2)). - Mikael Aaltonen, Jan 20 2015

a(n) = 2*A005369(n). - Michel Marcus, Jan 20 2015