A374337 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of regions constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections.

3, 11, 27, 55, 99, 145, 203, 277, 353, 441, 545, 651, 769, 903, 1039, 1187, 1351, 1517, 1695, 1889, 2085, 2293, 2517, 2743, 2981, 3235, 3491, 3759, 4043, 4329, 4627, 4941, 5257, 5585, 5929, 6275, 6633, 7007, 7383, 7771, 8175, 8581, 8999, 9433, 9869, 10317, 10781, 11247, 11725, 12219, 12715

Offset: 1

Author

Scott R. Shannon, Jul 05 2024

Keywords

• nonn

Comments

See A374338 for further details.

Links

• Scott R. Shannon, Image for n = 1. In this and other images the initial vertices that form the circles' centers are shown as white dots.
• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 16.

Crossrefs

Cf. A374338 (vertices), A374339 (edges), A359570, A371374, A371253.

Formula

a(n) = A374339(n) - A374338(n) + 1, by Euler's formula.

Conjectured:

If n = 3*k + 1, k >= 0, a(n) = |(15*n^2 - 17*n - 7)/3|.

If n = 3*k, k >= 1, a(n) = (15*n^2 - 17*n - 3)/3.

If n = 3*k - 1, k >= 1, a(n) = (15*n^2 - 17*n + 7)/3.