A281795 -id:A281795 - cp's OEIS frontend

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

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

nonn

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

Cf. A261849, A242118, A281795.

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

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

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

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

A302829 a(n) is the number of lattice points in a Cartesian grid between a circle of radius n and an inscribed square whose vertices lie on the coordinate axes.

Original entry on oeis.org

0, 0, 4, 8, 12, 28, 36, 52, 72, 88, 112, 128, 156, 192, 220, 252, 280, 324, 368, 408, 448, 504, 548, 592, 644, 708, 776, 828, 880, 952, 1016, 1096, 1164, 1236, 1324, 1388, 1472, 1548, 1648, 1736, 1808, 1912, 2004, 2116, 2212, 2300, 2408, 2508, 2624, 2728, 2860, 2976, 3076

Offset: 1

Kirill Ustyantsev, Apr 27 2018

nonn

Points are not lying on the borders of the circle and the square.

Note that if the square is rotated so that its sides are parallel to the coordinate axes, the resulting sequence is A303642 instead.

Kirill Ustyantsev, Desmos graph calculator

Cf. A034856, A281795, A303642, A303644, A303646.

a(n) = sum(x=-n, +n, sum(y=-n, +n, ((x^2+y^2) < n^2) && ((abs(x)+abs(y))^2 > n^2))); \\ Michel Marcus, May 22 2018

for n in range (1, 100):
    count=0
    for x in range (0, n):
        for y in range (0, n):
            if (x*x+y*yn):
                count=count+1
    print(4*count)

a(n) = A281795(n) - 4*A034856(n). - Andrey Zabolotskiy, Apr 29 2018

cp's OEIS Frontend

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

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