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

%I A334630 #32 Sep 11 2020 04:41:02
%S A334630 1,2,3,5,7,12,20,35,61,106,187,347,642,1223,2348,4554,8949,17486,
%T A334630 34736,68220,135861,268434,535977,1063240,2125421,4228727,8457226,
%U A334630 16860125,33723361,67309114,134632684,268894671,537869772,1074864547
%N A334630 Number of regions after generation n of Conant's dissection of a square when dissected with diagonal lines.
%C A334630 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).
%C A334630 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.
%C A334630 The author thanks _Rémy Sigrist_ whose code given in A328078 was modified to generate the larger values of this sequence.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_1.jpg">Hand drawn image for n=12</a>. Used as a confirmation of the computer generated images and data.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630.png">Illustration for n=3</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_1.png">Illustration for n=4</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_2.png">Illustration for n=5</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_3.png">Illustration for n=6</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_4.png">Illustration for n=7</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_5.png">Illustration for n=8</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_6.png">Illustration for n=9</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_7.png">Illustration for n=10</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_8.png">Illustration for n=11</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_9.png">Illustration for n=12</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_10.png">Illustration for n=13</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_11.png">Illustration for n=14</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_12.png">Illustration for n=15</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_13.png">Illustration for n=16</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_14.png">Illustration for n=17</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_15.png">Illustration for n=18</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_16.png">Illustration for n=19</a>.
%H A334630 Scott R. Shannon, <a href="/A334630/a334630_17.png">Illustration for n=20</a>.
%Y A334630 Cf. A328078, A335703, A337270.
%K A334630 nonn,more
%O A334630 0,2
%A A334630 _Scott R. Shannon_, Sep 10 2020