A092867 - OEIS

%I #57 Nov 13 2023 11:27:43

%S 1,12,75,252,715,1572,3109,5676,9291,14556,22081,32502,44935,62868,

%T 83286,108384,140152,181710,225565,282978,342792,415614,502318,606642,

%U 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

%N 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.

%H Hugo Pfoertner, <a href="/A092866/a092866.pdf">Intersections of diagonals in polygons of triangular shape.</a>

%H Cynthia Miaina Rasamimanananivo and Max Alekseyev, <a href="/A092867/a092867.py.txt">Sage program for this sequence</a>

%H Scott R. Shannon, <a href="/A331911/a331911.png">Triangle regions for n = 2</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_1.png">Triangle regions for n = 3</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_13.png">Triangle regions for n = 4</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_3.png">Triangle regions for n = 5</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_4.png">Triangle regions for n = 6</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_5.png">Triangle regions for n = 7</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_6.png">Triangle regions for n = 8</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_7.png">Triangle regions for n = 9</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_8.png">Triangle regions for n = 10</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_10.png">Triangle regions for n = 11</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_9.png">Triangle regions for n = 12</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_11.png">Triangle regions for n = 13</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_12.png">Triangle regions for n = 14</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_14.png">Triangle regions for n = 9, random distance-based coloring</a>.

%H Scott R. Shannon, <a href="/A331911/a331911_15.png">Triangle regions for n = 12, random distance-based coloring</a>

%H <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>

%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>

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

%e 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.

%e a(3)=75: 48 triangles, 24 quadrilaterals and 3 pentagons are formed. See pictures at Pfoertner link.

%Y 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).

%Y 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).

%Y If the boundary points are in general position, we get A367117, A213827, A367118, A367119. - _N. J. A. Sloane_, Nov 09 2023

%K more,nonn

%O 1,2

%A _Hugo Pfoertner_, Mar 15 2004

%E a(1)=1 prepended by _Max Alekseyev_, Jun 29 2016

%E a(6)-a(50) from _Cynthia Miaina Rasamimanananivo_, Jun 28 2016, Jul 01 2016, Aug 05 2016, Aug 15 2016

%E Definition edited by _N. J. A. Sloane_, May 13 2020