A303706 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.

0, 5, 14, 29, 42, 65, 94, 123, 154, 187, 234, 289, 328, 383, 436, 507, 572, 645, 716, 789, 884, 961, 1058, 1159, 1244, 1347, 1454, 1573, 1692, 1805, 1940, 2057, 2194, 2325, 2454, 2621, 2758, 2927, 3060, 3221, 3404, 3571, 3746, 3909, 4086, 4293, 4478, 4677, 4868, 5061, 5256, 5465, 5698, 5915

Offset: 1

Author

Kirill Ustyantsev, Apr 29 2018

Keywords

• nonn

Examples

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

Links

• Kirill Ustyantsev, Geometric illustration

Crossrefs

Cf. A303644, A303646.

Programs

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

Python

import math
tan=math.sqrt(3)/3
for n in range (1,71):
  count=0
  for x in range (-n, 2*n):
   for y in range (-2*n, 2*n):
    if (x*x+y*y>n*n and y<-tan*x+2*tan*n and y>tan*x-2*tan*n and x>-n):
     count=count+1
  print(count)