A335093 Number of regions after generation n of Conant's dissection of a square when dissected with diagonal lines and where the starting pair of edges rotates counterclockwise around the square.

1, 2, 3, 5, 7, 12, 18, 33, 57, 105, 186, 348, 648, 1226, 2347, 4535, 8840, 17386, 34229, 67708, 134210, 266779, 531277, 1058724, 2112238, 4215738, 8421830, 16822772, 33624420, 67204921, 134363536, 268636845, 537171420, 1074115099

Offset

0,2

Comments

This is a variation of A334630 where the pair of edges where the dissection begins starts at the left and bottom edges of the square and then proceeds counterclockwise around the square, the dissection halving in size after every two steps.

For the first generation a single dissection line is drawn from the bottom left to the top right corner of the square. For the second generation a single line is drawn from the bottom right corner toward to top left corner. For the third generation the dissection size halves and moves counterclockwise so now two lines are drawn from the top edge toward the left edge and also two lines from the right edge toward the bottom edge. For the fourth generation two lines are drawn from the top edge toward the right edge and two lines from the left edge toward the bottom edge. The dissection again moves counterclockwise and halves in size so now starts at the left and bottom edge again, and for the fifth generation draws four lines on each edge. The sequence gives the number of regions in the resulting dissection after generation n.

Like the standard orthogonal lines dissection of A328078 no obvious repetitive pattern appears as the generations increases. But unlike A334630 there appears to be no high density horizontal lines appearing in the resulting dissection, see the linked images.

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

Links

Table of n, a(n) for n=0..33.

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. A334630, A328078, A335703, A337270.

Sequence in context: A326490 A191385 A345669 * A143642 A192685 A293543

Adjacent sequences: A335090 A335091 A335092 * A335094 A335095 A335096

Keyword

nonn,more

Author

Scott R. Shannon, Sep 12 2020

Status

approved