0,2

For the points that form the Pythagorean triple (for example see illustration n = 5, on the first quadrant at coordinate (4,3) and (3,4)), the transit of circumference occurs exactly at the corners, therefore there are no additional intersecting squares on the upper or lower rows (diagonally NE & SW directions) of such points.

If the center of the circle is instead chosen at the middle of a square grid centered at (1/2,0), the sequence will be 2*A004767(n-1).

Kival Ngaokrajang, Illustration of initial terms

a(n) = 4*Sum{k=1..n} ceiling(sqrt(n^2 - (k-1)^2)) - floor(sqrt(n^2 - k^2)). - Orson R. L. Peters, Jan 30 2017

a(n) = 8*n - A046109(n) for n > 0. - conjectured by Orson R. L. Peters, Jan 30 2017, proved by Andrey Zabolotskiy, Jan 31 2017

(Python)

a = lambda n: sum(4 for x in range(n) for y in range(n)

if x**2 + y**2 < n**2 and (x+1)**2 + (y+1)**2 > n**2)

(Python)

from sympy import factorint

def a(n):

r = 1

for p, e in factorint(n).items():

if p%4 == 1: r *= 2*e + 1

return 8*n - 4*r if n > 0 else 0

Cf. A009003, A004767.

Sequence in context: A285526 A321466 A227226 * A030387 A269931 A043437

Adjacent sequences: A242115 A242116 A242117 * A242119 A242120 A242121

Kival Ngaokrajang, May 05 2014

Terms corrected by Orson R. L. Peters, Jan 30 2017

approved