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%I A303706 #27 May 25 2021 07:58:18
%S A303706 0,5,14,29,42,65,94,123,154,187,234,289,328,383,436,507,572,645,716,
%T A303706 789,884,961,1058,1159,1244,1347,1454,1573,1692,1805,1940,2057,2194,
%U A303706 2325,2454,2621,2758,2927,3060,3221,3404,3571,3746,3909,4086,4293,4478,4677,4868,5061,5256,5465,5698,5915
%N 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.
%H A303706 Kirill Ustyantsev, <a href="https://www.desmos.com/calculator/j3ojy7gvmb">Geometric illustration</a>
%e A303706 For n = 2 we have 5 lattice points: (-1, 2); (-1, -2); (2, -1); (2, 1); (3, 0).
%o A303706 (Python)
%o A303706 import math
%o A303706 tan=math.sqrt(3)/3
%o A303706 for n in range (1,71):
%o A303706   count=0
%o A303706   for x in range (-n, 2*n):
%o A303706    for y in range (-2*n, 2*n):
%o A303706     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 A303706      count=count+1
%o A303706   print(count)
%o A303706 (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 A303706 Cf. A303644, A303646.
%K A303706 nonn
%O A303706 1,2
%A A303706 _Kirill Ustyantsev_, Apr 29 2018

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