cp's OEIS Frontend

A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

%I A108914 #20 Jul 22 2022 10:36:57
%S A108914 4,32,96,188,332,460,712,916,1204,1488,1904,2108,2716,3080,3532,4068,
%T A108914 4772,5140,6016,6392,7188,7992,8936,9260,10484,11312,12208,12968,
%U A108914 14396,14660,16504,17220,18436,19680,20756,21548,23692,24728,25992,26868,29204,29704,32176,33068,34444,36552,38552
%N A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.
%H A108914 Scott R. Shannon, <a href="/A108914/b108914.txt">Table of n, a(n) for n = 1..100</a>
%H A108914 Scott R. Shannon, <a href="/A108914/a108914.jpg">Image for n = 2</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_1.jpg">Image for n = 3</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_2.jpg">Image for n = 4</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_3.jpg">Image for n = 5</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_4.jpg">Image for n = 6</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_5.jpg">Image for n = 11</a>.
%H A108914 Scott R. Shannon, <a href="/A108914/a108914_6.jpg">Image for n = 30</a>.
%H A108914 L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/square6b.jpg">The case n=6</a>. Note 3- and 4-fold off-diagonal concurrencies.
%H A108914 L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/square7b.jpg">The case n=7</a>. Note there are no off-diagonal concurrencies.
%F A108914 If n=1 or n is prime, a(n)=18*n^2-26*n+12.
%F A108914 If n is composite, vanishing regions from 3- and 4-fold concurrency must be subtracted.
%F A108914 a(n) = A355948(n) - A355949(n) + 1 by Euler's formula.
%Y A108914 A092098 is the corresponding count for triangles.
%Y A108914 A355949 (vertices), A355948 (edges), A355992 (k-gons), A355838, A355798.
%K A108914 nonn
%O A108914 1,1
%A A108914 _Len Smiley_ and Brian Wick ( mathclub(AT)math.uaa.alaska.edu ), Jul 19 2005
%E A108914 a(23), a(33) corrected, a(41) and above by _Scott R. Shannon_, Jul 22 2022