Search: id:A057655
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%I A057655
%S A057655 1,5,9,9,13,21,21,21,25,29,37,37,37,45,45,45,49,57,61,61,69,69,
%T A057655 69,69,69,81,89,89,89,97,97,97,101,101,109,109,113,121,121,121,
%U A057655 129,137,137,137,137,145,145,145,145,149,161,161,169,177,177,177
%N A057655 The circle problem: number of points (x,y) in square lattice with x^2+y^2 
               <= n.
%D A057655 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 106.
%D A057655 F. Fricker, Einfuehrung in die Gitterpunktlehre, Birkhaeuser, Boston, 
               1982.
%D A057655 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 
               2000, p. 5.
%D A057655 E. Kraetzel, lattice Points, Kluwer, Dordrecht, 1988.
%D A057655 C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. 
               Assoc. Amer., 2000, p. 51.
%D A057655 W. Sierpinski, Elementary Theory of Numbers, Elsevier, North-Holland, 
               1988.
%H A057655 T. D. Noe, <a href="b057655.txt">Table of n, a(n) for n = 0..1000</a>
%H A057655 F. Richman, <a href="http://www.math.fau.edu/Richman/gausdisk.htm">Count 
               Gaussian integers of norm less than and equal to n</a>
%H A057655 W. Sierpi\'{n}ski, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&wyd=10">
               Elementary Theory of Numbers</a>, Warszawa 1964.
%H A057655 F. Richman, <a href="http://www.math.fau.edu/Richman/gausdisk.htm">Counting 
               Gaussian integers in a disk</a>
%F A057655 a(n) = 1 + 4*{ [n/1] - [n/3] + [n/5] - [n/7] + ... }. - Gauss
%F A057655 a(n) = 1 + 4*Sum_{ k = 0 .. [sqrt(n)] } [ sqrt(n-k^2) ]. - Liouville 
               (?)
%F A057655 a(n) - Pi*n = O(sqrt(n)) (Gauss). a(n) - Pi*n = O(n^c), c = 23/73 + epsilon 
               ~ 0.3151 (Huxley). If a(n) - Pi*n = O(n^c) then c > 1/4 (Landau, 
               Hardy). It is conjectured that a(n) - Pi*n = O(n^(1/4 + epsilon)) 
               for all epsilon >0.
%t A057655 f[n_] := 1 + 4Sum[ Floor@ Sqrt[n - k^2], {k, 0, Sqrt[n]}]; Table[ f[n], 
               {n, 0, 60}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jun 16 2006)
%o A057655 (PARI) a(n)=sum(x=-n,n,sum(y=-n,n,if((sign(x^2+y^2-n)+1)*sign(x^2+y^2-n),
               0,1)))
%Y A057655 Partial sums of A004018. Cf. A057656, A057961, A057962. For another version 
               see A000328.
%Y A057655 A014198(n) + 1.
%Y A057655 Sequence in context: A105643 A073168 A127500 this_sequence A141124 A046255 
               A068388
%Y A057655 Adjacent sequences: A057652 A057653 A057654 this_sequence A057656 A057657 
               A057658
%K A057655 nonn,easy,nice
%O A057655 0,2
%A A057655 N. J. A. Sloane (njas(AT)research.att.com), Oct 15 2000

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