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

0,2

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

Table of n, a(n) for n=0..50.

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) - 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.

Crossrefs

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

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

Sequence in context: A179436 A254963 A304075 * A227776 A155286 A155313

Adjacent sequences: A357193 A357194 A357195 * A357197 A357198 A357199

Keyword

nonn

Author

Scott R. Shannon, Sep 17 2022

Status

approved