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In theory this should be easier to analyze than A335703, since the construction follows a cycle of four steps.

This sequence completes a set of four. (1) The original Conant warp and woof construction used dissection lines that alternated between the base and left side of a square (see A328078).

(2) Robert Fathauer observed that if the construction starts with an equilateral triangle, and the dissection lines start from each of the three sides in rotation, the resulting structure in generation 3n converges to the Sierpinski Gasket fractal (see A329774).

(3) If the construction is applied to a square, and the dissection lines start from each of the four sides in rotation, we obtain the structures shown in A335703. To our surprise, these is no apparent fractal structure.

(4) The remaining case, an equilateral triangle with the dissection lines alternating between the base and the left side, is the subject of the present sequence.

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.

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.

This is a variation of A335703 where the edge where the dissections begin starts at the bottom edge of the square and then proceeds clockwise around the square after each generation. However unlike A335703, where the dissection halves in size after every two generations, here it halves in size after every single generation. The resulting pattern, which resembles images of a printed circuit for large n, stays fairly constant while the internal regions are cut into smaller rectangles and squares after each generation.

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

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.

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.

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.