A136486 -id:A136486 - cp's OEIS frontend

A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

0, 1, 3, 6, 13, 19, 28, 37, 48, 64, 77, 94, 110, 131, 152, 172, 199, 226, 253, 281, 308, 343, 377, 412, 447, 488, 528, 567, 612, 654, 703, 750, 796, 847, 902, 957, 1013, 1068, 1129, 1187, 1252, 1313, 1378, 1446, 1511, 1582, 1650, 1725, 1800, 1877, 1955, 2034

Offset: 0

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

Number of unit square lattice cells inside quadrant of origin centered circle of radius n+1/2.

			a(2) = 3 because a circle of radius 2+1/2 in the first quadrant encloses (2,1), (1,1), (1,2).

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

Cf. A019683, A136483, A136486, A136515.

A136484:= func< n | n eq 0 select 0 else (&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
[A136484(n): n in [0..100]]; // G. C. Greubel, Jul 29 2023

Table[Sum[Floor[Sqrt[(n+1/2)^2 - k^2]], {k,n}], {n,0,100}]

def A136484(n): return sum(floor(sqrt((n+1/2)^2-k^2)) for k in range(1, n+1))
[A136484(n) for n in range(101)] # G. C. Greubel, Jul 29 2023

a(n) = Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
a(n) = (1/2) * A136515(n).
a(n) = (1/4) * A136486(n).
a(n) = A136483(2*n+1).
Lim_{n -> oo} a(n)/(n^2) -> Pi/16 (A019683).

A136515 Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter 2n+1.

Original entry on oeis.org

0, 2, 6, 12, 26, 38, 56, 74, 96, 128, 154, 188, 220, 262, 304, 344, 398, 452, 506, 562, 616, 686, 754, 824, 894, 976, 1056, 1134, 1224, 1308, 1406, 1500, 1592, 1694, 1804, 1914, 2026, 2136, 2258, 2374, 2504, 2626, 2756, 2892, 3022, 3164, 3300, 3450, 3600

Offset: 0

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

Number of unit square lattice cells inside two adjacent quadrants of origin centered circle of radius n+1/2.

			a(2) = 6 because a circle centered at the origin and of radius 2.5 encloses (-2,1),(-1,1),(-1,2),(2,1),(1,1),(1,2) in the upper half plane.

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

Cf. A136484, A136513, A136514.

A136515:= func< n | n eq 0 select 0 else 2*(&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
[A136515(n): n in [0..100]]; // G. C. Greubel, Jul 27 2023

Table[2*Sum[Floor[Sqrt[(n +1/2)^2 -k^2]], {k,n}], {n,0,100}]

a(n) = 2*sum(k=1, n, sqrtint((n+1/2)^2-k^2)); \\ Michel Marcus, Jul 27 2023

def A136515(n): return 2*sum(isqrt((n+1/2)^2-k^2) for k in range(1,n+1))
[A136515(n) for n in range(101)] # G. C. Greubel, Jul 27 2023

a(n) = 2*Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
a(n) = 2 * A136484(n).
a(n) = (1/2)*A136486(2*n+1).

cp's OEIS Frontend

A136484 Number of unit square lattice cells inside quadrant of origin centered circle of diameter 2n+1.

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