A333597 The number of unit cells intersected by the circumference of a circle centered on the origin with radius squared equal to the norm of the Gaussian integers A001481(n).

0, 4, 8, 12, 12, 16, 20, 20, 20, 28, 28, 32, 28, 28, 36, 36, 40, 36, 44, 44, 44, 44, 44, 52, 48, 52, 52, 52, 52, 60, 52, 60, 64, 60, 60, 60, 68, 68, 60, 68, 68, 68, 72, 68, 76, 76, 76, 76, 76, 76, 76, 84, 84, 76, 88, 76, 84, 84, 92, 84, 92

Offset: 1

Author

Scott R. Shannon, Mar 28 2020

Keywords

• nonn

Comments

Draw a circle on a 2D square grid centered at the origin with a radius squared equal to the norm of the Gaussian integers A001481(n). See the images in the links. This sequence gives the number of unit cells intersected by the circumference of the circle. Equivalently this is the number of intersections of the circumference with the x and y integer grid lines.

Links

• Scott R. Shannon, Illustration for n = 3. The circle has a radius squared of 2, resulting in 8 unit cells intersected/intersection points.
• Scott R. Shannon, Illustration for n = 4. The circle has a radius squared of 4, resulting in 12 unit cells intersected/intersection points.
• Scott R. Shannon, Illustration for n = 8. The circle has a radius squared of 10, resulting in 20 unit cells intersected/intersection points.
• Scott R. Shannon, Illustration for n = 12. The circle has a radius squared of 18, resulting in 32 unit cells intersected/intersection points.
• Scott R. Shannon, Illustration for n = 13. The circle has a radius squared of 20, resulting in 28 unit cells intersected/intersection points. This is the first term where the number of intersection points decreases relative to the previous term.
• Scott R. Shannon, Illustration for n = 26. The circle has a radius squared of 52, resulting in 52 unit cells intersected/intersection points.
• Wikipedia, Gaussian integer.

Crossrefs

Cf. A001481, A055025, A057655, A119439, A242118 (a subsequence of this sequence), A234300.

Formula

a(n) = 4*A234300(2*(n-1)). - Andrey Zabolotskiy, Feb 22 2025