|
|
A136515
|
|
Number of unit square lattice cells inside half-plane (two adjacent quadrants) of origin centered circle of diameter 2n+1.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Number of unit square lattice cells inside two adjacent quadrants of origin centered circle of radius n+1/2.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 2*Sum_{k=1..n} floor(sqrt((n+1/2)^2 - k^2)).
Lim_{n -> oo} a(n)/(n^2) -> Pi/8.
|
|
EXAMPLE
|
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.
|
|
MATHEMATICA
|
Table[2*Sum[Floor[Sqrt[(n +1/2)^2 -k^2]], {k, n}], {n, 0, 100}]
|
|
PROG
|
(Magma)
A136515:= func< n | n eq 0 select 0 else 2*(&+[Floor(Sqrt((n+1/2)^2-j^2)): j in [1..n]]) >;
(SageMath)
def A136515(n): return 2*sum(isqrt((n+1/2)^2-k^2) for k in range(1, n+1))
(PARI) a(n) = 2*sum(k=1, n, sqrtint((n+1/2)^2-k^2)); \\ Michel Marcus, Jul 27 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008
|
|
STATUS
|
approved
|
|
|
|