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A253663
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Number of positive solutions to x^2+y^2+z^2 <= n^2.
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11
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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
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0,4
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COMMENTS
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Whereas A000604 counts solutions where x>=0, y>=0, z>=0, this sequence counts solutions where x>0, y>0, z>0.
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LINKS
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FORMULA
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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
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).
Raising it to the power 3 counts points with x^2+y^2+z^2 = k.
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)
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EXAMPLE
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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).
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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