A303669 a(n) is the number of lattice points in a Cartesian grid between a circle of radius n, centered at the origin, and an inscribed equilateral triangle; one of the sides of triangle is perpendicular to X-axis.

0, 2, 13, 21, 36, 56, 81, 103, 144, 166, 215, 239, 298, 342, 405, 447, 514, 568, 655, 707, 796, 864, 961, 1019, 1128, 1208, 1337, 1405, 1524, 1614, 1749, 1847, 1990, 2082, 2249, 2333, 2502, 2600, 2789, 2899, 3064, 3192, 3383, 3519, 3718, 3832, 4047, 4175

Offset

1,2

Links

Kirill Ustyantsev, Table of n, a(n) for n = 1..1000

Kirill Ustyantsev, Illustration of this configuration for n=1..10

Example

For n = 2 we have two lattice points between the defined circle and its inscribed equilateral triangle: (1, 1) and (1, -1).

Prog

(Python)
import math
tan=math.sqrt(3)/3
for n in range (1, 70):
.count=0
.count1=0
.for x in range (-n, n):
..for y in range (-n, n):
...if (x*x+y*y<n*n and y>-tan*x+tan*n):
....count=count+1
...if (x*x+y*y<n*n and y<-n/2):
....count1=count1+1
.print(2*count+count1)
(PARI) a(n) = sum(x=-n, +n, sum(y=-n, +n, ((x^2+y^2) < n^2) && ((2*x < - n) || (3*y^2 > (n-x)^2)))); \\ Michel Marcus, May 22 2018

Crossrefs

Cf. A302829, A303642, A303706.

Sequence in context: A061871 A182459 A333216 * A084651 A285087 A085509

Adjacent sequences: A303666 A303667 A303668 * A303670 A303671 A303672

Keyword

nonn

Author

Kirill Ustyantsev, Apr 28 2018

Status

approved