A119677 a(n) is the number of complete squares that fit inside the circle with radius n, drawn on squared paper at (0, 0).

0, 0, 4, 16, 32, 60, 88, 120, 164, 216, 276, 332, 392, 476, 556, 648, 732, 832, 936, 1052, 1176, 1288, 1428, 1560, 1696, 1860, 2016, 2180, 2340, 2512, 2700, 2876, 3080, 3276, 3488, 3712, 3908, 4144, 4360, 4620, 4864, 5096, 5356, 5616, 5900

Offset: 0

Author

Tomas Kyselica (Tomas.Kyselica(AT)gmail.com), Jul 29 2006

Keywords

• nonn

Links

• David Dewan, Table of n, a(n) for n = 0..10000
• David Dewan, Drawings for n=1..12

Crossrefs

Cf. A001182.

Programs

Mathematica

a[n_] := 4 Sum[Floor[Sqrt[n^2 - k^2]], {k, n - 1}];
Array[a, 45, 0]  (* David Dewan, May 27 2024 *)

Python

from math import isqrt
def A119677(n): return sum(isqrt(k*((n<<1)-k)) for k in range(1,n))<<2 # Chai Wah Wu, Jul 18 2024

Formula

a(n) = 4 * A001182(n).

Limit_{n->oo} a(n)/(n^2) = Pi.

Extensions

More terms from Don Reble, Aug 01 2006

Name clarified by Kirill Zolotuskiy, Apr 15 2023