A092867 Number of regions in an equilateral triangular figure formed by the straight line segments connecting all vertices and all points that divide the sides into n equal parts.

1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, 83286, 108384, 140152, 181710, 225565, 282978, 342792, 415614, 502318, 606642, 708505, 839874, 983007, 1141416, 1315102, 1529526, 1733476, 1994550, 2259420, 2559990, 2878053, 3237414, 3593521, 4047906, 4510590, 5002350, 5506918, 6128100, 6704800, 7414518, 8113992, 8858622, 9682927, 10626774, 11478142, 12519492

Offset

1,2

Links

Table of n, a(n) for n=1..50.

Hugo Pfoertner, Intersections of diagonals in polygons of triangular shape.

Cynthia Miaina Rasamimanananivo and Max Alekseyev, Sage program for this sequence

Scott R. Shannon, Triangle regions for n = 2.

Scott R. Shannon, Triangle regions for n = 3.

Scott R. Shannon, Triangle regions for n = 4.

Scott R. Shannon, Triangle regions for n = 5.

Scott R. Shannon, Triangle regions for n = 6.

Scott R. Shannon, Triangle regions for n = 7.

Scott R. Shannon, Triangle regions for n = 8.

Scott R. Shannon, Triangle regions for n = 9.

Scott R. Shannon, Triangle regions for n = 10.

Scott R. Shannon, Triangle regions for n = 11.

Scott R. Shannon, Triangle regions for n = 12.

Scott R. Shannon, Triangle regions for n = 13.

Scott R. Shannon, Triangle regions for n = 14.

Scott R. Shannon, Triangle regions for n = 9, random distance-based coloring.

Scott R. Shannon, Triangle regions for n = 12, random distance-based coloring

Index entries for sequences related to stained glass windows

Sequences formed by drawing all diagonals in regular polygon

Formula

By the Euler characteristic, a(n) = A274586(n) - A274585(n) + 1 = A274586(n) - A092866(n) - 3n - 1.

Example

a(2)=12 because the 6 line segments mutually connecting the vertices and the mid-side nodes form 12 congruent right triangles of two different sizes.
a(3)=75: 48 triangles, 24 quadrilaterals and 3 pentagons are formed. See pictures at Pfoertner link.

Crossrefs

Cf. A092866 (number of intersections), A274585 (number of points both inside and on the triangle sides), A274586 (number of edges), A331911 (number of n-gons).

Cf. A092098 (regions in triangle cut by line segments connecting vertices with subdivision points on opposite side), A006533 (regions formed by all diagonals in regular n-gon), A002717 (triangles in triangular matchstick arrangement).

If the boundary points are in general position, we get A367117, A213827, A367118, A367119. - N. J. A. Sloane, Nov 09 2023

Sequence in context: A003368 A246767 A328526 * A292532 A053310 A368530

Adjacent sequences: A092864 A092865 A092866 * A092868 A092869 A092870

Keyword

more,nonn

Author

Hugo Pfoertner, Mar 15 2004

Extensions

a(1)=1 prepended by Max Alekseyev, Jun 29 2016

a(6)-a(50) from Cynthia Miaina Rasamimanananivo, Jun 28 2016, Jul 01 2016, Aug 05 2016, Aug 15 2016

Definition edited by N. J. A. Sloane, May 13 2020

Status

approved