A136514 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of radius n.

0, 2, 8, 16, 30, 44, 60, 82, 108, 138, 166, 196, 238, 278, 324, 366, 416, 468, 526, 588, 644, 714, 780, 848, 930, 1008, 1090, 1170, 1256, 1350, 1438, 1540, 1638, 1744, 1856, 1954, 2072, 2180, 2310, 2432, 2548, 2678, 2808, 2950, 3090, 3220, 3366, 3510, 3664

Offset

1,2

Links

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

Formula

Lim_{n -> oo} a(n)/(n^2) -> Pi/8.

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

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

a(n) = 2*A001182(n). - R. J. Mathar, Jan 10 2008

Example

a(2) = 2 because a circle centered at the origin and of radius 2 encloses (-1,1) and (1,1) in the upper half plane.

Mathematica

Table[2*Sum[Floor[Sqrt[n^2 -k^2]], {k, n-1}], {n, 100}]

Prog

(Magma)
A136514:= func< n | n eq 1 select 0 else 2*(&+[Floor(Sqrt(n^2-j^2)): j in [1..n-1]]) >;
[A136514(n): n in [1..100]]; // G. C. Greubel, Jul 27 2023
(SageMath)
def A136514(n): return 2*sum(isqrt(n^2-k^2) for k in range(1, n))
[A136514(n) for n in range(1, 101)] # G. C. Greubel, Jul 27 2023
(PARI) a(n) = 2*sum(k=1, n-1, sqrtint(n^2-k^2)); \\ Michel Marcus, Jul 27 2023

Crossrefs

Cf. A001182, A136513.

Sequence in context: A137882 A194643 A327329 * A077071 A187216 A210729

Adjacent sequences: A136511 A136512 A136513 * A136515 A136516 A136517

Keyword

easy,nonn

Author

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

Status

approved