cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

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

Keywords

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

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

A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

Original entry on oeis.org

1, 1, 31, 13, 71, 25, 127, 41, 199, 61, 287, 85, 391, 113, 511, 145, 647, 181, 799, 221, 967, 265, 1151, 313, 1351, 365, 1567, 421, 1799, 481, 2047, 545, 2311, 613, 2591, 685, 2887, 761, 3199, 841, 3527, 925, 3871, 1013, 4231, 1105, 4607, 1201, 4999, 1301, 5407, 1405, 5831, 1513, 6271, 1625, 6727, 1741
Offset: 3

Views

Author

R. J. Mathar, Jan 08 2019

Keywords

Comments

Sequence proposed by Thomas Young: draw the regular n-gon and construct 2*n lines that run from both ends of the n edges perpendicular into the n-gon until they hit an opposite edge. (For n even the lines actually hit another vertex, so there are only n additional lines). a(n) is the number of non-overlapping tiles inside the n-gon with edges that are sections of the lines or n-gon edges.

Links

Crossrefs

Cf. A165217, A320422

Formula

a(2n) = 2*n^2+2*n+1 = A001844(n), n>1. a(2n+1) = 8*n^2-1 = A157914(n), n>1. - Thomas Young (tyoung(AT)district16.org), Nov 11 2017
G.f.: x^3 +x^4 -x^5*(31+13*x-22*x^2-14*x^3+7*x^4+5*x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Jan 21 2019
a(n) = 1+n*A064680(n-2), n>=5. - R. J. Mathar, Jan 21 2019
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