A387220 Arithmetic mean of number of lattice points strictly inside circle of radius n centered on origin, and number of points not outside that circle.

3, 11, 27, 47, 75, 111, 147, 195, 251, 311, 375, 439, 523, 611, 703, 795, 895, 1007, 1127, 1251, 1371, 1515, 1651, 1791, 1951, 2115, 2287, 2451, 2623, 2815, 2999, 3207, 3407, 3619, 3847, 4051, 4287, 4511, 4771, 5019, 5255, 5523, 5787, 6075, 6355, 6623, 6919, 7211

Offset

1,1

Comments

a(n) is the number of integer values (x, y) strictly inside the circle x^2+y^2=n^2, plus half the number of such lattice points that are part of the perimeter of that circle.

All terms are odd. - Chai Wah Wu, Aug 23 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A051132(n) + A000328(n))/2.

a(n) = A256465(n^2). - R. J. Mathar, Aug 26 2025

Example

The unit circle has 1 lattice point strictly inside it and 5 lattice points not outside it. Halfway between 1 and 5 is 3, so a(1) = 3.

Prog

(PARI) a(n) = {4*n -1 + 2*sum(k=1, n-1, my(t=n^2-k^2); 2*sqrtint(t)-issquare(t))} \\ Andrew Howroyd, Aug 22 2025
(Python)
from math import isqrt
def A387220(n): return 1+(sum(isqrt(m:=k*((n<<1)-k))+isqrt(m-1) for k in range(1, n+1))<<1) # Chai Wah Wu, Aug 23 2025

Crossrefs

Average of A051132 and A000328.

Sequence in context: A123928 A186301 A170945 * A164897 A322595 A212982

Adjacent sequences: A387217 A387218 A387219 * A387221 A387222 A387223

Keyword

nonn

Author

Lorraine Lee, Aug 22 2025

Status

approved