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A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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

Cf. A000604.

Sequence in context: A273947 A140121 A102770 * A298369 A213789 A058273

Adjacent sequences:  A253660 A253661 A253662 * A253664 A253665 A253666

KEYWORD

nonn

AUTHOR

R. J. Mathar, Jan 07 2015

STATUS

approved

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Last modified September 18 16:10 EDT 2021. Contains 347527 sequences. (Running on oeis4.)