A334630 Number of regions after generation n of Conant's dissection of a square when dissected with diagonal lines.

1, 2, 3, 5, 7, 12, 20, 35, 61, 106, 187, 347, 642, 1223, 2348, 4554, 8949, 17486, 34736, 68220, 135861, 268434, 535977, 1063240, 2125421, 4228727, 8457226, 16860125, 33723361, 67309114, 134632684, 268894671, 537869772, 1074864547

Offset: 0

Author

Scott R. Shannon, Sep 10 2020

Keywords

• nonn
• more

Comments

See A328078 for details of the iterative dissection. Here a similar procedure is performed on a square except that diagonal lines are used. The first step cuts from the lower left corner to the upper right corner, forming two regions for generation 1. The next generation cuts from the lower right corner to the upper left corner, creating three regions in all. From then on each generation alternates from cutting from the left and bottom edge (towards the top and right edge), to cutting from the bottom and right edge (towards the left and top edge).

Like the standard orthogonal lines dissection of A328078 no obvious repetitive pattern appears as the generations increases. However from about n=17 the images, see attached links, show lines of higher density crossings, the first about one quarter of the way up from the bottom. Underneath this are three similar smaller separate lines, and underneath those additional lines. Interestingly the largest top line appears to be slightly off-center and shifted to the left.

The author thanks Rémy Sigrist whose code given in A328078 was modified to generate the larger values of this sequence.

Links

• Scott R. Shannon, Hand drawn image for n=12. Used as a confirmation of the computer generated images and data.
• Scott R. Shannon, Illustration for n=3.
• Scott R. Shannon, Illustration for n=4.
• Scott R. Shannon, Illustration for n=5.
• Scott R. Shannon, Illustration for n=6.
• Scott R. Shannon, Illustration for n=7.
• Scott R. Shannon, Illustration for n=8.
• Scott R. Shannon, Illustration for n=9.
• Scott R. Shannon, Illustration for n=10.
• Scott R. Shannon, Illustration for n=11.
• Scott R. Shannon, Illustration for n=12.
• Scott R. Shannon, Illustration for n=13.
• Scott R. Shannon, Illustration for n=14.
• Scott R. Shannon, Illustration for n=15.
• Scott R. Shannon, Illustration for n=16.
• Scott R. Shannon, Illustration for n=17.
• Scott R. Shannon, Illustration for n=18.
• Scott R. Shannon, Illustration for n=19.
• Scott R. Shannon, Illustration for n=20.

Crossrefs

Cf. A328078, A335703, A337270.