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A000328
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Number of points of norm <= n^2 in square lattice.
(Formerly M3829 N1570)
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42
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1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441, 529, 613, 709, 797, 901, 1009, 1129, 1257, 1373, 1517, 1653, 1793, 1961, 2121, 2289, 2453, 2629, 2821, 3001, 3209, 3409, 3625, 3853, 4053, 4293, 4513, 4777, 5025, 5261, 5525, 5789, 6077, 6361, 6625
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OFFSET
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0,2
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COMMENTS
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Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = 1 + 4 * Sum_{j>=0} floor(n^2/(4*j+1)) - floor(n^2/(4*j+3)). Also a(n) = A057655(n^2). - Max Alekseyev, Nov 18 2007
a(n) = [x^(n^2)] theta_3(x)^2/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018
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MATHEMATICA
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Table[Sum[SquaresR[2, k], {k, 0, n^2}], {n, 0, 46}]
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PROG
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(PARI) { a(n) = 1 + 4 * sum(j=0, n^2\4, n^2\(4*j+1) - n^2\(4*j+3) ) } /* Max Alekseyev, Nov 18 2007 */
(Haskell)
a000328 n = length [(x, y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n^2]
(Python)
return (sum([int((n**2 - y**2)**0.5) for y in range(1, n)]) * 4 + 4*n + 1)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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