A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2.

0, 0, 1, 7, 17, 38, 78, 127, 196, 296, 410, 564, 738, 958, 1220, 1514, 1848, 2235, 2686, 3175, 3719, 4365, 5007, 5758, 6568, 7442, 8415, 9477, 10597, 11779, 13100, 14459, 15954, 17566, 19231, 21029, 22916, 24930, 27030, 29293, 31616, 34103, 36732, 39459

Offset: 0

Author

R. J. Mathar, Jan 07 2015

Keywords

• nonn

Comments

Whereas A000604 counts solutions where x>=0, y>=0, z>=0, this sequence counts solutions where x>0, y>0, z>0.

Examples

			a(4)=17 counts the following solutions (x,y,z): (1,1,1), (2,2,2), three permutations of (1,1,2), three permutations of (1,1,3), three permutations of (1,2,2), and six permutations of (1,2,3).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000604.

Programs

Sage

[len([(x,y,z) for x in [1..n] for y in [1..n] for z in [1..n] if x^2+y^2+z^2<=n^2]) for n in [0..43]] # Tom Edgar, Jan 07 2015

Formula

a(n) = A211639(n^2).

a(n) = [x^(n^2)] (theta_3(x) - 1)^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018

Comment from N. J. A. Sloane, Jun 02 2024 (Start)

The one-dimensional lattice {n: n an integer} , which graphically looks like

...o o o o o o ...

has theta series 1 + 2 q + 2 q^4 + 2 q^9 + 2 q^16 + ... = sum {n=-oo..oo} q^(n^2),

and that power series is called theta_3(q), A000122.

Raising it to the power 3 counts points with x^2+y^2+z^2 = k, A005875.

Dividing it by 1-x gives the partial sums, which basically is what this sequence is.

So a first approximation to a theta series for the sequence is theta_3(q)^8/(1-q).

Subtracting 1 and dividing by 8 is because here we only want positive solutions.

(End)