A357196 Number of regions in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

1, 7, 25, 55, 97, 151, 217, 295, 385, 475, 601, 715, 865, 1015, 1159, 1351, 1537, 1735, 1945, 2131, 2401, 2647, 2905, 3115, 3457, 3751, 4057, 4357, 4705, 5005, 5401, 5767, 6133, 6535, 6925, 7303, 7777, 8215, 8653, 9025, 9601, 10051, 10585, 11071, 11587, 12151, 12697, 13171, 13825, 14395, 14989

Offset: 0

Author

Scott R. Shannon, Sep 17 2022

Keywords

• nonn

Comments

Unlike similar dissections of the triangle and square, see A356984 and A357058, 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

• 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.

Crossrefs

Cf. A357197 (vertices), A357198 (edges), A331931, A356984 (triangle), A357058 (square).

Cf. A227776 (6*n^2 + 1).

Formula

a(n) = A357198(n) - A357197(n) + 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 1 for hexagons that only contain simple intersections when cut by n internal hexagons.