A053415 Circle numbers (version 3): a(n) = number of points (i,j), i,j integers, contained in a circle of diameter n, centered at (1/2, 1/2).

0, 0, 4, 4, 12, 16, 32, 32, 52, 60, 80, 88, 112, 124, 156, 172, 208, 216, 256, 276, 316, 332, 384, 408, 448, 484, 540, 560, 616, 648, 716, 740, 812, 848, 912, 952, 1020, 1060, 1124, 1184, 1264, 1304, 1396, 1436, 1528, 1576, 1664, 1716, 1804, 1876, 1976

Offset: 0

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A053411, A053414, A053415, A053416, A053417.

Programs

Mathematica

cx = 1/2; cy = 1/2; a[n_] := Sum[ dj = 1/2*Sqrt[ Abs[n^2 - 4*cx^2 + 8*cx*i - 4*i^2]]; j1 = cy - dj // Floor; j2 = cy + dj // Ceiling; Sum[Boole[ (i - cx)^2 + (j - cy)^2 <= n^2/4], {j, j1, j2}], {i, cx - n/2 // Floor, cx + n/2 // Ceiling}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

Formula

a(n)/(n/2)^2 -> Pi.

a(n) = [x^(n^2)] theta_2(x^4)^2 / (1 - x). - Ilya Gutkovskiy, Nov 23 2021