A335703 - OEIS

%I #53 Sep 14 2020 19:55:29

%S 1,2,3,5,9,16,30,52,95,174,326,618,1195,2300,4478,8764,17251,34038,

%T 67326,133858,266331,530876,1058146,2112904,4216075,8417830,16808282,

%U 33592910,67130923,134211556,268281390,536403988,1072424939,2144322638,4287655814,8574721850

%N Number of regions after n generations of symmetrized Conant Gasket.

%C Generation 0. Start with a square of side 1

%C Generation 1. Go to bottom edge of the square, and at the point 1/2 draw a vertical line from bottom to top.

%C Generation 2. Go to the left edge, and at the point 1/2 draw a line from left to right. Stop when you reach the vertical line.

%C Generation 3. Go to the top edge and draw downward vertical lines at the 1/4 and 3/4 points. If you reach a horizontal line, lift the pen, skip to the next horizontal line and lower the pen. Repeat until reaching the bottom edge.

%C Generation 4. Go to the right edge and draw horizontal lines at the 1/4 and 3/4 points. If you reach a vertical line, lift the pen, skip to the next vertical line and lower the pen. Repeat until reaching the left edge.

%C Generation 5. Return to the bottom edge and draw vertical lines at the 1/8, 3/8, 5/8, and 7/8 points. If you reach a horizontal line, lift the pen, skip to the next horizontal line and lower the pen. Repeat until reaching the top edge.

%C Generation 6. Return to the left edge and draw horizontal lines at the points 1/8, 3/8, 5/8, 7/8, following the same rules.

%C Continue in this way, visiting each side in turn.

%C At generations 2k-1 and 2k, the lines start at the points 1/2^k, 3/2^k, 5/2^k, ..., (2^k-1)/2^k.

%C The tick marks on the edges of the illustrations indicate which side of the square the lines start from.

%C This is similar to Conant's Gasket as described in A328078, except there lines were drawn alternately from the bottom and left edges of the square, which led to a figure with a strong bias to the North-East (see Robert Fathauer's colored illustration in A328078 of the gasket after 16 generations).

%C The present definition produces a more homogeneous design, although now there is no obvious fractal structure.

%H Rémy Sigrist, <a href="/A335703/a335703_0.png">Illustration for a(0)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_1.png">Illustration for a(1)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_2.png">Illustration for a(2)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_3.png">Illustration for a(3)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_4.png">Illustration for a(4)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_5.png">Illustration for a(5)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_6.png">Illustration for a(6)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_7.png">Illustration for a(7)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_8.png">Illustration for a(8)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_9.png">Illustration for a(9)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_10.png">Illustration for a(10)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_11.png">Illustration for a(11)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_12.png">Illustration for a(12)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_13.png">Illustration for a(13)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_14.png">Illustration for a(14)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_15.png">Illustration for a(15)</a>

%H Rémy Sigrist, <a href="/A335703/a335703_16.png">Illustration for a(16)</a>

%H Rémy Sigrist, <a href="/A335703/a335703.png">Illustration for a(18)</a>

%H Rémy Sigrist, <a href="/A335703/a335703.gif">Illustration of generations 0 to 18</a> (Animated gif)

%H Rémy Sigrist, <a href="/A335703/a335703.txt">C++ program for A335703</a>

%H N. J. A. Sloane, <a href="/A335703/a335703.pdf">Illustration of generations 0 through 6.</a>

%H N. J. A. Sloane, <a href="/A335703/a335703_1.pdf">Illustration of generations 7 (black lines) and 8 (both black and red lines).</a>

%H N. J. A. Sloane, <a href="https://vimeo.com/457349959">Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows</a>, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)

%o (C++) See Links section.

%Y Cf. A328078, A328080, A139250, A160124.

%Y For bisections see A337267 and A337268. See also A337269.

%K nonn

%O 0,2

%A _Rémy Sigrist_ and _N. J. A. Sloane_, Aug 25 2020

%E a(34)-a(35) from _N. J. A. Sloane_, Sep 06 2020 using _Rémy Sigrist_'s C++ program.