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A237526
a(n) = number of unit squares in the first quadrant of a disk of radius sqrt(n) centered at the origin of a square lattice.
3
0, 0, 1, 1, 1, 3, 3, 3, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 11, 11, 13, 13, 13, 13, 13, 15, 17, 17, 17, 19, 19, 19, 20, 20, 22, 22, 22, 24, 24, 24, 26, 28, 28, 28, 28, 30, 30, 30, 30, 30, 33, 33, 35, 37, 37, 37, 37, 37, 39, 39, 39, 41, 41, 41, 41, 45, 45, 45, 47, 47
OFFSET
0,6
LINKS
FORMULA
a(A000404(n)) = A232499(n).
a(n) = Sum_{k=1..floor(sqrt(n))} floor(sqrt(n-k^2)). - M. F. Hasler, Feb 09 2014
G.f.: (theta_3(x) - 1)^2/(4*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
For n > 1, Pi*(n+2-sqrt(8n)) < a(n) < Pi*n. (This is trivial and can probably be improved by methods like Euler-Maclaurin and perhaps even a modification of the Dirichlet hyperbola method.) - Charles R Greathouse IV, Jul 17 2024
MATHEMATICA
a[n_]:=Sum[Floor[Sqrt[n-k^2]], {k, Floor[Sqrt[n]]}]; Array[a, 70, 0] (* Stefano Spezia, Jul 19 2024 *)
PROG
(PARI) A237526(n)=sum(k=1, sqrtint(n), sqrtint(n-k^2)) \\ M. F. Hasler, Feb 09 2014
(Python)
from math import isqrt
def A237526(n): return sum(isqrt(n-k**2) for k in range(1, isqrt(n)+1)) # Chai Wah Wu, Jul 18 2024
CROSSREFS
Partial sums of A063725.
Sequence in context: A120204 A177018 A156349 * A240117 A333536 A069941
KEYWORD
nonn
AUTHOR
L. Edson Jeffery, Feb 09 2014
STATUS
approved