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

1,2

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

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

Crossrefs

Column 4 of array in A333275.

Cf. A306302, A331755, A290131, A333274.

See also A115004, A331761.

Sequence in context: A277014 A033581 A213393 * A274205 A009943 A028595

Adjacent sequences: A334698 A334699 A334700 * A334702 A334703 A334704

Keyword

nonn

Author

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

Extensions

More terms from Lars Blomberg, Jun 17 2020

Status

approved