

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.


5



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 nonsimple 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 nonsimple 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).
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



