A036693 Number of Gaussian integers z = a + bi satisfying n-1 < |z| <= n.

1, 4, 8, 16, 20, 32, 32, 36, 48, 56, 64, 60, 64, 88, 84, 96, 88, 104, 108, 120, 128, 116, 144, 136, 140, 168, 160, 168, 164, 176, 192, 180, 208, 200, 216, 228, 200, 240, 220, 264, 248, 236, 264, 264, 288, 284, 264, 296, 292, 312

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(10^2) = 660, a(10^3) = 6392, a(10^4) = 62952, a(10^5) = 628520, a(10^6) = 6281404. - _Reinhard Zumkeller_, Jan 13 2002
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A000328.

Programs

Magma

[#[: x in [-n..n], y in [-n..n]| n-1 lt r and r  le n where r is Sqrt(x^2+ y^2)]: n in [0..50]]; // Marius A. Burtea, Feb 18 2020

Sage

def A036693(n):
    if n == 0: return 1
    Range = lambda n: ((i, j) for i in (-n..n) for j in (-n..n))
    return sum(1 for (j, k) in Range(n) if (n-1)^2 < j^2 + k^2 <= n^2)
print([A036693(n) for n in range(20)]) # Peter Luschny, Mar 27 2020

Formula

From Reinhard Zumkeller, Jan 13 2002: (Start)

a(n)/n ~ 2*Pi.

a(n) = A000328(n)-A000328(n-1). (End)