A001182 Number of cells of square lattice of edge 1/n inside quadrant of unit circle centered at 0.

0, 1, 4, 8, 15, 22, 30, 41, 54, 69, 83, 98, 119, 139, 162, 183, 208, 234, 263, 294, 322, 357, 390, 424, 465, 504, 545, 585, 628, 675, 719, 770, 819, 872, 928, 977, 1036, 1090, 1155, 1216, 1274, 1339, 1404, 1475, 1545, 1610, 1683, 1755, 1832, 1911, 1992, 2072

Offset: 1

Author

Tihamer von Ghyczy (ghyczy(AT)esinet.net)

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A261849, A242118, A281795.

Programs

Mathematica

Table[Sum[Floor@ Sqrt[n^2 - k^2], {k, n - 1}], {n, 52}] (* Michael De Vlieger, Jan 30 2017 *)

Python

from math import isqrt
def A001182(n): return sum(isqrt(k*((n<<1)-k)) for k in range(1,n)) # Chai Wah Wu, Jul 18 2024

Formula

a(n) = Sum_{k=1..n-1} floor(sqrt(n^2-k^2)). - Horst Kraemer (horst.kraemer(AT)epost.de) Apr 07 2004

a(n) = A261849(2*n)/4 = (A281795(n)-A242118(n))/4. - Andrey Zabolotskiy, Jan 30 2017

a(n) = [x^(n^2)] (theta_3(x) - 1)^2/(4*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018

Extensions

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 19 2000