A337270 - OEIS

%I #24 Aug 29 2020 14:51:42

%S 1,2,3,5,7,13,20,36,57,108,185,355,637,1246,2344,4595,8895,17532,

%T 34592,68287,136053,269046,539516,1068111,2147477,4254870,8567392,

%U 16982215,34213477,67850054,136710948,271162515,546323617,1083843471

%N Number of regions formed at generation n when the Conant "warp and woof" construction is applied to the base and left side of an equilateral triangle.

%C 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).

%C (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).

%C (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.

%C (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.

%H Rémy Sigrist, <a href="/A337270/a337270_0.png">Illustration for a(0)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_1.png">Illustration for a(1)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_2.png">Illustration for a(2)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_3.png">Illustration for a(3)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_4.png">Illustration for a(4)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_5.png">Illustration for a(5)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_6.png">Illustration for a(6)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_7.png">Illustration for a(7)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_8.png">Illustration for a(8)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_9.png">Illustration for a(9)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_10.png">Illustration for a(10)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_11.png">Illustration for a(11)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_12.png">Illustration for a(12)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_13.png">Illustration for a(13)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_14.png">Illustration for a(14)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_15.png">Illustration for a(15)</a>

%H Rémy Sigrist, <a href="/A337270/a337270_16.png">Illustration for a(16)</a>

%H Rémy Sigrist, <a href="/A337270/a337270.gif">Illustration of generations 0 through 16</a> (animated gif)

%H Rémy Sigrist, <a href="/A337270/a337270.txt">C++ program for A337270</a>

%H N. J. A. Sloane, <a href="/A337270/a337270.png">Illustration of generations 0 through 7.</a> (The colored tick marks on the edges indicate the generation.)

%o (C++) See Links section.

%Y Cf. A328078, A329774, A335703.

%K nonn

%O 0,2

%A _Rémy Sigrist_ and _N. J. A. Sloane_, Aug 27 2020