A165217 Count of interior bounded regions in a regular 2n-sided polygon dissected by all diagonals parallel to sides.

6, 25, 50, 145, 224, 497, 630, 1281, 1606, 2761, 3302, 5265, 5940, 9185, 10472, 14977, 16834, 23161, 25284, 34321, 37720, 49105, 53500, 68225, 73278, 92457, 99470, 122641, 131316, 159681, 169158, 204545, 217210, 258265, 273282, 321937, 338208, 396721, 417380, 483841, 507830

Offset: 3

Author

Chintan (timtamboy63(AT)gmail.com), Sep 08 2009

Keywords

• nonn

Comments

The rule is: get a regular polygon with at least 6 sides and an even number of sides (hexagon, octagon, etc.) and pick a point, then pick the point directly clockwise it, draw a line then draw lines parallel to it going through the other points. Then do the same with all the other points. a(n) is the count of interior bounded regions.

Please email me if you can find a pattern!

Links

• R. J. Mathar, Tile Count in the Interior of Regular 2n-Gons Dissected by Diagonals Parallel to Sides, arxiv:0911.3434 [math.CO], 2009.
• Index to sequences on drawing diagonals in regular polygons

Crossrefs

Cf. A003454, A320422

Formula

Conjecture: a(2n) = (2*n-1)*(4*n^3-4*n^2+6*n-3)/3. - Thomas Young (tyoung(AT)district16.org), Dec 23 2018

Extensions

a(6)-a(8) corrected and a(9)-a(10) added by R. J. Mathar, Oct 09 2009

a(11)-a(22) from R. J. Mathar, Nov 19 2009

Typo in a(14) corrected by Thomas Young (tyoung(AT)district16.org), Dec 23 2018

a(23)-a(43) from Christopher Scussel, Jun 25 2023