

A253663


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


11



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
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OFFSET

0,4


COMMENTS

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


LINKS



FORMULA

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


EXAMPLE

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).


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



