A051132 Number of ordered pairs of integers (x,y) with x^2+y^2 < n^2.

0, 1, 9, 25, 45, 69, 109, 145, 193, 249, 305, 373, 437, 517, 609, 697, 793, 889, 1005, 1125, 1245, 1369, 1513, 1649, 1789, 1941, 2109, 2285, 2449, 2617, 2809, 2997, 3205, 3405, 3613, 3841, 4049, 4281, 4509, 4765, 5013, 5249, 5521, 5785, 6073, 6349, 6621

Offset: 0

Author

Jostein Trondal (jostein.trondal(AT)protech.no)

Keywords

• nonn
• nice
• easy

Examples

			a(3)=25 from the points of shapes 00 (1), 10 (4), 11 (4), 20 (4), 21 (8), 22 (4).
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. See p. 18.

Crossrefs

Changing "<" to "<=" in the definition gives A000328.

Programs

Haskell

a051132 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 < n^2]
-- Reinhard Zumkeller, Jan 23 2012

Mathematica

Table[Sum[SquaresR[2, k], {k, 0, n^2 - 1}], {n, 0, 46}]
a[0]=0;a[n_]:=4*n-3+4Sum[Ceiling[Sqrt[n^2-i^2]]-1,{i,n-1}];Array[a,47,0] (* Giorgos Kalogeropoulos, May 20 2025 *)

Python

from math import isqrt
def A051132(n): return 1+(sum(isqrt(k*((n<<1)-k)-1) for k in range(1,n+1))<<2) if n else 0 # Chai Wah Wu, Feb 12 2025

Formula

a(n) = A000328(n) - A046109(n). - Reinhard Zumkeller, Jan 23 2012

Limit_{n->oo} a(n)/n^2 = Pi. - Chai Wah Wu, Feb 12 2025

a(n) = 4*n - 3 + 4 Sum_{i=1..n-1} ceiling(sqrt(n^2 - i^2)) - 1, for n > 0 (see Zhao). - Giorgos Kalogeropoulos, May 20 2025

Extensions

More terms from James Sellers