A357196 - OEIS

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

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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