A136486 Number of unit square lattice cells enclosed by origin centered circle of diameter 2n+1.

0, 4, 12, 24, 52, 76, 112, 148, 192, 256, 308, 376, 440, 524, 608, 688, 796, 904, 1012, 1124, 1232, 1372, 1508, 1648, 1788, 1952, 2112, 2268, 2448, 2616, 2812, 3000, 3184, 3388, 3608, 3828, 4052, 4272, 4516, 4748, 5008, 5252, 5512, 5784, 6044, 6328, 6600

Offset: 0

Author

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

Keywords

• easy
• nonn

Comments

a(n) is the number of complete squares that fit inside the circle with radius n+1/2, drawn on squared paper.

Examples

			a(1) = 4 because a circle centered at the origin and of radius 1+1/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A003881, A136484, A136485, A136515.

Programs

Magma

A136486:= func< n | n eq 0 select 0 else 4*(&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
[A136486(n): n in [0..100]]; // G. C. Greubel, Jul 30 2023

Mathematica

Table[4*Sum[Floor[Sqrt[(n + 1/2)^2 - k^2]], {k,n}], {n, 0, 100}]

SageMath

def A136486(n): return 4*sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
[A136486(n) for n in range(101)] # G. C. Greubel, Jul 30 2023

Formula

a(n) = 4*Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).

a(n) = 4 * A136484(n).

a(n) = 2 * A136515(n).

a(n) = A136485(2*n+1).

Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).