A337675 -id:A337675 - cp's OEIS frontend

A335628 Number of regions after generation n of Conant's dissection of a square when dissected with both orthogonal and diagonal lines and where the starting edges rotate clockwise around the square and the dissection halves in size every second generation.

1, 2, 3, 6, 11, 20, 37, 68, 123, 232, 457, 879, 1679, 3269, 6478, 12799, 25272, 50127, 99888, 198867, 396267, 791069, 1580460, 3156095, 6305694, 12606152, 25205005, 50388077

Offset: 0

Scott R. Shannon, Oct 02 2020

This is a variation of A328078 and A334630 where the square is dissected with both orthogonal and diagonal lines.
For the first generation, a single orthogonal dissection line is drawn from the bottom to the top edge of the square. For the second generation, a single diagonal line is drawn from the bottom left corner toward to top right corner. The edge where the dissections start now rotates clockwise around the square and the dissection size halves. For the third generation, two orthogonal dissection lines are drawn from the left edge toward the right edge. For the fourth generation, four diagonal lines, two from the left edge and two from the top edge, are drawn from the top-left corner toward the bottom right corner. The edge now rotates clockwise again and the dissection size halves. The sequences gives the number of regions in the resulting dissection after generation n.
The author thanks Rémy Sigrist whose code given in A328078 was modified to generate the larger values of this sequence.

Scott R. Shannon, Illustration for n=2.
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.

Cf. A328078, A334630, A335703, A337270, A335093, A337675.

A337780 a(n)/A163403(n) = the total length of the lines drawn at generation n for Conant's dissection of a square with size 1.

Original entry on oeis.org

1, 1, 3, 5, 9, 17, 37, 73, 141, 273, 541, 1065, 2085, 4081, 8013, 15737, 30869, 60545, 118781, 233097, 457317, 897233, 1760269, 3453785, 6776181, 13294881, 26083869, 51176745, 100407301, 196998513, 386505517, 758320121, 1487807381, 4137567061

Offset: 1

Scott R. Shannon, Oct 07 2020

This sequence is the numerator of the fraction which gives the total length of the lines drawn at generation n for Conant's dissection of a square, assuming the square has a size of 1 unit. The corresponding denominator is given by A163403(n).

See A328078 for a description of the iterative dissection and images of the dissection after generation n.

			a(1) = 1 as the first dissection line is from the bottom to the top of the square giving a total drawn length of 1/1 = 1/A163403(1) = 1.
a(2) = 1 as the second dissection line is from the left edge to halfway across the square toward the right edge, giving a total drawn length of 1/2 = 1/A163403(2) = 0.5.
a(3) = 3 as the third dissection draws two lines from the bottom edge toward the top edge, one of length 1/2 the other of length 1, giving a total drawn length of 1/2 + 1 = 3/2 = 3/A163403(3) = 1.5.
a(4) = 5 as the fourth dissection draws two lines from the left edge toward the right edge, each line being broken into two smaller lines. The total drawn length of the four smaller lines is 1/2 + 1/4 + 1/4 + 1/4 = 5/4 = 5/A163403(4) = 1.25.
a(5) = 9 as the fifth dissection draws four lines from the bottom edge toward the top edge, two lines being broken into two smaller lines. The total drawn length of the six lines is 1/4 + 1/4 + 1/2 + 1/4 + 1/4 + 3/4 = 9/4 = 9/A163403(5) = 2.25.

Cf. A163403, A328078, A328080, A337642, A328081, A337675.

Empirical g.f.: -(4*x^6 + 4*x^3 - 2*x^2 + 2*x + 3)/(4*x^7 - 4*x^6 + 4*x^4 - 2*x^3 + 2*x^2 + x - 1).

A335632 Number of regions after generation n of Conant's dissection of a square when dissected with both diagonal and orthogonal lines and where the starting edges rotate clockwise around the square and the dissection halves in size every second generation.

Original entry on oeis.org

1, 2, 3, 8, 11, 25, 37, 97, 142, 316, 463, 1150, 1747, 4298, 6599, 16641, 25800, 65025, 101027, 256245, 399972, 1017939, 1589375, 4048559, 6328766, 16148228, 25252615, 53243252

Offset: 0

Scott R. Shannon, Oct 03 2020

This is a variation of A335628 where the same rules are applied but the first dissection is with a diagonal line from the lower left corner to the upper right corner, followed by dissection with an orthogonal line from the left edge. For further details see A335628.

Scott R. Shannon, Illustration for n=2.
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.

Cf. A335628, A328078, A334630, A335703, A337270, A335093, A337675.

A337693 Number of regions after generation n of Conant's dissection of a square when dissected with diagonal lines and where the starting edges rotate counterclockwise around the square and the dissection halves in size after every generation.

Original entry on oeis.org

1, 2, 4, 9, 25, 61, 197, 597, 2165, 7861, 30549, 118869, 471765, 1873621, 7479637, 29864277, 119397205

Offset: 0

Scott R. Shannon, Sep 15 2020

This is a variation of A335093 in which the edges where the diagonal dissections begin start at the left and bottom edge of the square and then proceeds counterclockwise around the square after each generation. However unlike A335093, where the dissection halves in size after every two generations, here it halves in size after every single generation. The resulting pattern for larger n shows vertical and horizontal lines of higher density crossings similar to those seen in A334630.

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

Scott R. Shannon, Illustration for n=2.
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.

Cf. A335093, A337675, A335703, A328078, A337270, A334630.

cp's OEIS Frontend

A335628 Number of regions after generation n of Conant's dissection of a square when dissected with both orthogonal and diagonal lines and where the starting edges rotate clockwise around the square and the dissection halves in size every second generation.

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A337780 a(n)/A163403(n) = the total length of the lines drawn at generation n for Conant's dissection of a square with size 1.

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A335632 Number of regions after generation n of Conant's dissection of a square when dissected with both diagonal and orthogonal lines and where the starting edges rotate clockwise around the square and the dissection halves in size every second generation.

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A337693 Number of regions after generation n of Conant's dissection of a square when dissected with diagonal lines and where the starting edges rotate counterclockwise around the square and the dissection halves in size after every generation.

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