cp's OEIS Frontend

A331766 Number of regions formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

%I A331766 #37 Jan 14 2023 09:55:22
%S A331766 16,56,142,296,544,892,1436,2136,3066,4272,5840,7688,10094,12884,
%T A331766 16182,20192,24918,30200,36614,43692,51756,61008,71544,83040,96202,
%U A331766 110692,126702,144372,164144,185200,209192,234928,262706,293244,326002,361240,400170,441516
%N A331766 Number of regions formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).
%C A331766 The grid consists of a rectangular array of 3 X (n+1) dots. If we instead count squares, the dimensions are 2 X n.
%C A331766 Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.
%C A331766 See A331452 for other illustrations.
%C A331766 For n<=100, 7-gons: 4 for n=9, 4 for n=18; 8-gons: 2 for n=9; no 9-gons or 10-gons. _Lars Blomberg_, Apr 28 2020
%H A331766 Lars Blomberg, <a href="/A331766/b331766.txt">Table of n, a(n) for n = 1..100</a>
%H A331766 Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2020). Also arXiv:2009.07918.
%H A331766 Scott R. Shannon, <a href="/A331452/a331452_13.png">Colored illustration for a(3) = 142.</a>
%H A331766 Scott R. Shannon, <a href="/A331452/a331452_1.txt">Data specifically for nX2 (or 2Xn) rectangles</a>
%H A331766 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%H A331766 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 20.
%Y A331766 Column 2 of A331452.
%Y A331766 Cf. A331763, A331765.
%K A331766 nonn
%O A331766 1,1
%A A331766 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 05 2020
%E A331766 More terms from _Scott R. Shannon_, Mar 11 2020
%E A331766 a(21) and beyond from _Lars Blomberg_, Apr 28 2020