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A334701 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly two lines cross.

%I A334701 #51 Jan 01 2024 22:46:53
%S A334701 1,6,24,54,124,214,382,598,950,1334,1912,2622,3624,4690,6096,7686,
%T A334701 9764,12010,14866,18026,21904,25918,30818,36246,42654,49246,57006,
%U A334701 65334,75098,85414,97384,110138,124726,139642,156286,174018,194106,214570,237534,261666,288686,316770,348048,380798,416524,452794,492830
%N A334701 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly two lines cross.
%C A334701 It would be nice to have a formula or recurrence. - _N. J. A. Sloane_, Jun 22 2020
%H A334701 Lars Blomberg, <a href="/A334701/b334701.txt">Table of n, a(n) for n = 1..500</a>
%H A334701 Lars Blomberg, <a href="/A334701/a334701.txt">Array (s,n) of the number of internal vertices where exactly s=2..501 lines cross in a figure made up of a row of n=1..500 adjacent congruent rectangles, with diagonals of all possible rectangles drawn. Rows are stored comma-separated.</a>
%H A334701 Lars Blomberg, Scott R. Shannon, 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 A334701 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration showing regions for n=1</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755.png">Images of vertices for n=1</a>.
%H A334701 Scott R. Shannon, <a href="/A331452/a331452_7.png">Colored illustration showing regions for n=2</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_1.png">Images of vertices for n=2</a>.
%H A334701 Scott R. Shannon, <a href="/A331452/a331452_8.png">Colored illustration showing regions for n=3</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_2.png">Images of vertices for n=3</a>.
%H A334701 Scott R. Shannon, <a href="/A331452/a331452_9.png">Colored illustration showing regions for n=4</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_3.png">Images of vertices for n=4</a>.
%H A334701 Scott R. Shannon, <a href="/A331452/a331452_10.png">Colored illustration showing regions for n=5</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_7.png">Images of vertices for n=5</a>
%H A334701 Scott R. Shannon, <a href="/A331452/a331452_11.png">Colored illustration showing regions for n=6</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_8.png">Images of vertices for n=6</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_11.png">Images of vertices for n=7</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_10.png">Images of vertices for n=8</a>
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_4.png">Images of vertices for n=9</a>.
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_5.png">Images of vertices for n=11</a>.
%H A334701 Scott R. Shannon, <a href="/A331755/a331755_6.png">Images of vertices for n=14</a>.
%H A334701 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%F A334701 Conjecture: As n -> oo, a(n) ~ C*n^4/Pi^2, where C is about 0.95 (compare A115004, A331761). - _N. J. A. Sloane_, Jul 03 2020
%Y A334701 Column 4 of array in A333275.
%Y A334701 Cf. A306302, A331755, A290131, A333274.
%Y A334701 See also A115004, A331761.
%K A334701 nonn
%O A334701 1,2
%A A334701 _Scott R. Shannon_ and _N. J. A. Sloane_, May 30 2020
%E A334701 More terms from _Lars Blomberg_, Jun 17 2020