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Search: id:A000328
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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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+0
16
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.
Or, numerator of N(r)/r^2, where N(r) is the number of lattice points inside a circle of radius r.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
W. Fraser and C. C. Gotlieb, A calculation of the number of lattice points in the circle and sphere, Math. Comp., 16 (1962), 282-290.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Eric Weisstein's World of Mathematics, Gauss's Circle Problem
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FORMULA
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a(n) = 1 + 4 * Sum[j=0..oo] [n^2/(4*j+1)] - [n^2/(4*j+3)]. Also a(n) = A057655(n^2). - Max Alekseyev (maxale(AT)gmail.com), Nov 18 2007
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MATHEMATICA
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Needs["NumberTheory`NumberTheoryFunctions`"]; Table[Sum[SumOfSquaresR[2, k], {k, 0, n^2}], {n, 0, 46}]
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PROGRAM
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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 (maxale(AT)gmail.com), Nov 18 2007
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CROSSREFS
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Equals A051132 + A046109. For another version see A057655.
Cf. A093832, A093837.
Sequence in context: A130066 A095085 A093836 this_sequence A100438 A129371 A130230
Adjacent sequences: A000325 A000326 A000327 this_sequence A000329 A000330 A000331
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net), May 22, 2000
Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 18 2007, at the suggestion of Max Alekseyev.
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