A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

0, 0, 4, 4, 12, 16, 24, 32, 52, 60, 76, 88, 112, 120, 148, 164, 192, 216, 256, 276, 308, 332, 376, 392, 440, 476, 524, 556, 608, 648, 688, 732, 796, 832, 904, 936, 1012, 1052, 1124, 1176, 1232, 1288, 1372, 1428, 1508, 1560, 1648, 1696, 1788, 1860, 1952, 2016

Offset: 1

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 diameter n, drawn on squared paper.

Examples

			a(3) = 4 because a circle centered at the origin and of radius 3/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003881, A136483, A136513.

Alternating merge of A119677 of A136485.

Programs

Magma

A136485:= func< n | n le 1 select 0 else 4*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136485(n): n in [1..100]]; // G. C. Greubel, Jul 29 2023

Mathematica

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

SageMath

def A136485(n): return 4*sum(floor(sqrt((n/2)^2-k^2)) for k in range(1,(n//2)+1))
[A136485(n) for n in range(1,101)] # G. C. Greubel, Jul 29 2023

Formula

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

a(n) = 4 * A136483(n).

a(n) = 2 * A136513(n).

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

a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (1 - x). - Ilya Gutkovskiy, Nov 24 2021