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

Author

Lorraine Lee, Aug 22 2025

Keywords

• nonn
• new

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

Examples

			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.
		

Crossrefs

Average of A051132 and A000328.

Programs

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

Formula

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

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