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A295344 Maximum number of lattice points inside and on a circle of radius n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Maximum number of lattice points (i.e., points with integer coordinates) in the plane that can be covered by a circle of radius n.

a(n) >= A000328(n).

Conjecture: sequence contains infinitely many terms that are divisible by 4.

REFERENCES

B. R. Srinivasan, Lattice Points in a Circle, Proc. Nat. Inst. Sci. India, Part A, 29 (1963), pp. 332-346.

LINKS

Table of n, a(n) for n=0..47.

FORMULA

a(n) = Pi*n^2 + O(n), as n goes to infinity.

a(n) = A123690(2*n) for n >= 1.

EXAMPLE

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.

.

.     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

PROG

(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));

CROSSREFS

Cf. A000328, A123690, A291259.

Sequence in context: A101648 A265100 A070134 * A219902 A064412 A211803

Adjacent sequences:  A295341 A295342 A295343 * A295345 A295346 A295347

KEYWORD

nonn

AUTHOR

Arkadiusz Wesolowski, Nov 20 2017

EXTENSIONS

a(10) corrected by Giovanni Resta, Nov 24 2017

STATUS

approved

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Last modified May 19 08:53 EDT 2019. Contains 323389 sequences. (Running on oeis4.)