%I #27 May 25 2021 07:58:18

%S 0,5,14,29,42,65,94,123,154,187,234,289,328,383,436,507,572,645,716,

%T 789,884,961,1058,1159,1244,1347,1454,1573,1692,1805,1940,2057,2194,

%U 2325,2454,2621,2758,2927,3060,3221,3404,3571,3746,3909,4086,4293,4478,4677,4868,5061,5256,5465,5698,5915

%N a(n) is the number of lattice points in a Cartesian grid between an equilateral triangle and an inscribed circle of radius n; one of the side of triangle is perpendicular to the X-axis; the circle's center is at the origin.

%H Kirill Ustyantsev, <a href="https://www.desmos.com/calculator/j3ojy7gvmb">Geometric illustration</a>

%e For n = 2 we have 5 lattice points: (-1, 2); (-1, -2); (2, -1); (2, 1); (3, 0).

%o (Python)

%o import math

%o tan=math.sqrt(3)/3

%o for n in range (1,71):

%o count=0

%o for x in range (-n, 2*n):

%o for y in range (-2*n, 2*n):

%o if (x*x+y*y>n*n and y<-tan*x+2*tan*n and y>tan*x-2*tan*n and x>-n):

%o count=count+1

%o print(count)

%o (PARI) a(n) = sum(x=-n+1, 2*n, sum(y=-2*n, 2*n, ((x^2+y^2) > n^2) && (3*y^2 < (x-2*n)^2))); \\ _Michel Marcus_, May 22 2018

%Y Cf. A303644, A303646.

%K nonn

%O 1,2

%A _Kirill Ustyantsev_, Apr 29 2018