A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

6, 12, 30, 60, 102, 156, 222, 300, 390, 468, 606, 708, 870, 1020, 1152, 1356, 1542, 1740, 1950, 2112, 2406, 2652, 2910, 3072, 3462, 3756, 4062, 4350, 4710, 4974, 5406, 5772, 6126, 6540, 6918, 7260, 7782, 8220, 8646, 8946, 9606, 10032, 10590, 11052, 11568, 12156, 12702, 13116, 13830, 14388

Offset

0,1

Comments

Unlike similar dissections of the triangle and square, see A357007 and A357060, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .

Links

Table of n, a(n) for n=0..49.

Scott R. Shannon, Image for n = 1.

Scott R. Shannon, Image for n = 2.

Scott R. Shannon, Image for n = 5.

Scott R. Shannon, Image for n = 9. This is the first term that forms hexagons with non-simple intersections.

Scott R. Shannon, Image for n = 50.

Scott R. Shannon, Image for n = 150.

Formula

a(n) = A357198(n) - A357196(n) + 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 6 for hexagons that only contain simple intersections when cut by n internal hexagons.

Crossrefs

Cf. A357196 (regions), A357198 (edges), A330846, A357007 (triangle), A357060 (square).

Sequence in context: A036690 A229746 A256579 * A322374 A351694 A014131

Adjacent sequences: A357194 A357195 A357196 * A357198 A357199 A357200

Keyword

nonn

Author

Scott R. Shannon, Sep 17 2022

Status

approved