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.

1, 6, 24, 54, 124, 214, 382, 598, 950, 1334, 1912, 2622, 3624, 4690, 6096, 7686, 9764, 12010, 14866, 18026, 21904, 25918, 30818, 36246, 42654, 49246, 57006, 65334, 75098, 85414, 97384, 110138, 124726, 139642, 156286, 174018, 194106, 214570, 237534, 261666, 288686, 316770, 348048, 380798, 416524, 452794, 492830

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, May 30 2020

Keywords

• nonn

Comments

It would be nice to have a formula or recurrence. - N. J. A. Sloane, Jun 22 2020

Links

• Lars Blomberg, Table of n, a(n) for n = 1..500
• Lars Blomberg, 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.
• Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
• Scott R. Shannon, Colored illustration showing regions for n=1
• Scott R. Shannon, Images of vertices for n=1.
• Scott R. Shannon, Colored illustration showing regions for n=2
• Scott R. Shannon, Images of vertices for n=2.
• Scott R. Shannon, Colored illustration showing regions for n=3
• Scott R. Shannon, Images of vertices for n=3.
• Scott R. Shannon, Colored illustration showing regions for n=4
• Scott R. Shannon, Images of vertices for n=4.
• Scott R. Shannon, Colored illustration showing regions for n=5
• Scott R. Shannon, Images of vertices for n=5
• Scott R. Shannon, Colored illustration showing regions for n=6
• Scott R. Shannon, Images of vertices for n=6
• Scott R. Shannon, Images of vertices for n=7
• Scott R. Shannon, Images of vertices for n=8
• Scott R. Shannon, Images of vertices for n=9.
• Scott R. Shannon, Images of vertices for n=11.
• Scott R. Shannon, Images of vertices for n=14.
• Index entries for sequences related to stained glass windows

Crossrefs

Column 4 of array in A333275.

Cf. A306302, A331755, A290131, A333274.

See also A115004, A331761.

Formula

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

Extensions

More terms from Lars Blomberg, Jun 17 2020