login
A165217
Count of interior bounded regions in a regular 2n-sided polygon dissected by all diagonals parallel to sides.
3
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
3,1
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!
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Chintan (timtamboy63(AT)gmail.com), Sep 08 2009
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
STATUS
approved