A362873 - OEIS

A362873

a(n) is the number of points with integer coordinates that are inside an equilateral triangle inscribed in a circle of radius n, the location of the triangle in the Oxy coordinate plane is described in the comments.

1, 4, 12, 17, 33, 42, 64, 77, 105, 122, 158, 177, 219, 242, 292, 319, 375, 406, 470, 503, 573, 610, 688, 729, 813, 856, 948, 995, 1093, 1144, 1248, 1303, 1415, 1472, 1592, 1653, 1779, 1844, 1976, 2045, 2185, 2256, 2402, 2477, 2631, 2710, 2870, 2951, 3119, 3204, 3378, 3467, 3649

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OFFSET

1,2

COMMENTS

An equilateral triangle is located in the coordinate plane Oxy so that its center coincides with the origin O, one of the vertices lies on the Oy axis.

LINKS

Table of n, a(n) for n=1..53.

Nicolay Avilov, Illustration of initial terms

FORMULA

a(n) = (3*n - 2)/2 + 2*Sum_{k=1..floor(sqrt(3)*n/2)} floor(-sqrt(3)*k + 3*n/2) if n is even;

a(n) = (3*n - 1)/2 + 2*Sum_{k=1..floor(sqrt(3)*n/2)} floor(-sqrt(3)*k + (3*n + 1)/2) if n is odd.

EXAMPLE

a(3) = 4 + 2*4 = 12;

a(4) = 5 + 2*6 = 17.

MATHEMATICA

a[n_]:=(3n-2+Mod[n, 2])/2+2Sum[Floor[(3n+Mod[n, 2])/2-Sqrt[3]k], {k, Floor[Sqrt[3]n/2]}]; Array[a, 53] (* Stefano Spezia, May 08 2023 *)

CROSSREFS