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

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

Cf. A194106, A183143.

Sequence in context: A303795 A175704 A350602 * A111371 A078514 A324519

Adjacent sequences: A362870 A362871 A362872 * A362874 A362875 A362876

Keyword

nonn

Author

Nicolay Avilov, May 07 2023

Status

approved