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

1,1

Comments

See A374338 for further details.

Links

Table of n, a(n) for n=1..51.

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.

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.

Crossrefs

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

Sequence in context: A164845 A024194 A011941 * A219621 A033960 A147173

Adjacent sequences: A374329 A374331 A374333 * A374338 A374339 A374340

Keyword

nonn,new

Author

Scott R. Shannon, Jul 05 2024

Status

approved