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A295344
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Maximum number of lattice points inside and on a circle of radius n.
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3
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1, 5, 14, 32, 52, 81, 116, 157, 208, 258, 319, 384, 457, 540, 623, 716, 812, 914, 1025, 1142, 1268, 1396, 1528, 1669, 1816, 1976, 2131, 2300, 2472, 2650, 2836, 3028, 3228, 3436, 3644, 3859, 4080, 4314, 4548, 4792, 5038, 5289, 5555, 5818, 6092, 6376, 6668, 6952
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OFFSET
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0,2
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COMMENTS
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Maximum number of lattice points (i.e., points with integer coordinates) in the plane that can be covered by a circle of radius n.
Conjecture: sequence contains infinitely many terms that are divisible by 4.
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REFERENCES
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B. R. Srinivasan, Lattice Points in a Circle, Proc. Nat. Inst. Sci. India, Part A, 29 (1963), pp. 332-346.
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LINKS
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FORMULA
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a(n) = Pi*n^2 + O(n), as n goes to infinity.
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EXAMPLE
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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.
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. Center # Pts in/
. x y Radius on circle
. ----- ----- ------ ---------
. 0 0 1 5
. 1/2 1/4 2 14
. 1/2 1/2 3 32
. 1/2 1/2 4 52
. 0 0 5 81
. 1/2 1/3 6 116
. 2/5 1/5 7 157
. 1/2 1/2 8 208
. 1/2 2/9 9 258
. 20/47 19/56 10 319
. 1/2 1/2 11 384
. 11/23 7/20 12 457
. 1/2 1/2 13 540
. 10/21 3/13 14 623
. 1/2 1/2 15 716
. 1/2 1/2 16 812
. 2/5 2/5 17 914
. 3/8 5/14 18 1025
. 1/2 1/6 19 1142
. 9/19 8/17 20 1268
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PROG
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(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));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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