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%I #32 Dec 27 2017 02:02:26
%S 1,5,14,32,52,81,116,157,208,258,319,384,457,540,623,716,812,914,1025,
%T 1142,1268,1396,1528,1669,1816,1976,2131,2300,2472,2650,2836,3028,
%U 3228,3436,3644,3859,4080,4314,4548,4792,5038,5289,5555,5818,6092,6376,6668,6952
%N Maximum number of lattice points inside and on a circle of radius n.
%C Maximum number of lattice points (i.e., points with integer coordinates) in the plane that can be covered by a circle of radius n.
%C a(n) >= A000328(n).
%C Conjecture: sequence contains infinitely many terms that are divisible by 4.
%D B. R. Srinivasan, Lattice Points in a Circle, Proc. Nat. Inst. Sci. India, Part A, 29 (1963), pp. 332-346.
%F a(n) = Pi*n^2 + O(n), as n goes to infinity.
%F a(n) = A123690(2*n) for n >= 1.
%e For a circle centered at the point (x, y) = (1/2, 1/4) with radius 2, there are 14 lattice points inside and on the circle.
%e .
%e . Center # Pts in/
%e . x y Radius on circle
%e . ----- ----- ------ ---------
%e . 0 0 1 5
%e . 1/2 1/4 2 14
%e . 1/2 1/2 3 32
%e . 1/2 1/2 4 52
%e . 0 0 5 81
%e . 1/2 1/3 6 116
%e . 2/5 1/5 7 157
%e . 1/2 1/2 8 208
%e . 1/2 2/9 9 258
%e . 20/47 19/56 10 319
%e . 1/2 1/2 11 384
%e . 11/23 7/20 12 457
%e . 1/2 1/2 13 540
%e . 10/21 3/13 14 623
%e . 1/2 1/2 15 716
%e . 1/2 1/2 16 812
%e . 2/5 2/5 17 914
%e . 3/8 5/14 18 1025
%e . 1/2 1/6 19 1142
%e . 9/19 8/17 20 1268
%o (PARI) L=List([]); for(n=0, 47, if(n>0, j=5, j=1); g=0; h=0; f=ceil(Pi*n^2); for(d=2, floor(f/2), for(c=1, floor(d/2), if(gcd(c, d)==1, for(e=d, d+1, if(e/f<=1/2, a=c/d; b=e/f; if(a+b>=1/2, t=0; for(x=-n, n+1, for(y=-n, n+1, z=(a-x)^2+(b-y)^2; if(z<=n^2,t++))); if(t>j, j=t; if(a>=b, g=a; h=b, g=b; h=a)))))))); print("a("n") = "j", the center of the circle is at point ("g", "h")."); listput(L, j)); print(); print(Vec(L));
%Y Cf. A000328, A123690, A291259.
%K nonn
%O 0,2
%A _Arkadiusz Wesolowski_, Nov 20 2017
%E a(10) corrected by _Giovanni Resta_, Nov 24 2017