This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334576 #23 May 09 2020 02:31:55
%S A334576 0,1,2,2,2,3,3,3,4,5,6,6,5,4,4,4,4,5,6,6,6,7,7,7,7,6,5,5,6,7,7,7,8,9,
%T A334576 10,10,10,11,11,11,12,13,14,14,13,12,12,12,11,10,9,9,9,8,8,8,8,9,10,
%U A334576 10,9,8,8,8,8,9,10,10,10,11,11,11,12,13,14,14,13
%N A334576 a(n) is the X-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334577 gives Y-coordinates.
%C A334576 The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:
%C A334576 - we define the family {P_k, k > 0}:
%C A334576 - P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:
%C A334576 +
%C A334576 |
%C A334576 |
%C A334576 +----+----+
%C A334576 O
%C A334576 - for any k > 0, P_{n+1} is built from four copies of P_n as follows:
%C A334576 +
%C A334576 |A
%C A334576 + |
%C A334576 C| +----+ |
%C A334576 A B| ---> |C B| |B C
%C A334576 +-------+ + | +----+-+
%C A334576 O C| | C|
%C A334576 A B| A| A B|
%C A334576 +-------+ +-+-------+
%C A334576 O
%C A334576 - the space filling curve P is the limit of P_k as k tends to infinity.
%C A334576 We can also describe the space filling curve P by mean of an L-system (see Links section).
%H A334576 Rémy Sigrist, <a href="/A334576/b334576.txt">Table of n, a(n) for n = 0..4095</a>
%H A334576 Joerg Arndt, <a href="/A334576/a334576.pdf">L-system corresponding to P</a>
%H A334576 Robert Ferréol (MathCurve), <a href="https://mathcurve.com/fractals/polya/polya.shtml">Courbe de Polya</a> [in French]
%H A334576 Kevin Ryde, <a href="https://user42.tuxfamily.org/alternate/index.html">Iterations of the Alternate Paperfolding Curve</a>
%H A334576 Rémy Sigrist, <a href="/A334576/a334576.png">Colored line plot of the first 2^14 points of the space filling curve P</a> (where the hue is function of the number of steps from the origin)
%H A334576 Rémy Sigrist, <a href="/A334576/a334576_1.png">Colored scatterplot of the first 2^20 points of the space filling curve P</a> (where the hue is function of the number of steps from the origin)
%H A334576 Rémy Sigrist, <a href="/A334576/a334576.gp.txt">PARI program for A334576</a>
%H A334576 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A334576 a(n+1) = (A020986(n) + A020986(n+1) - 1)/2 for any n >= 0.
%e A334576 The first points of the space filling curve P are as follows:
%e A334576 6| 20...21
%e A334576 | | |
%e A334576 5| 19 22
%e A334576 | | |
%e A334576 4| 16...17...18 23
%e A334576 | | |
%e A334576 3| 15 26...25...24
%e A334576 | | |
%e A334576 2| 4....5 14 27...28...29
%e A334576 | | | | |
%e A334576 1| 3 6 13...12...11 30
%e A334576 | | | | |
%e A334576 0| 0....1....2 7....8....9....10 31..
%e A334576 |
%e A334576 ---+----------------------------------------
%e A334576 y/x| 0 1 2 3 4 5 6 7
%e A334576 - hence a(9) = a(12) = a(17) = a(26) = a(27) = 5.
%o A334576 (PARI) See Links section.
%Y A334576 Cf. A020986, A334577.
%K A334576 nonn
%O A334576 0,3
%A A334576 _Rémy Sigrist_, May 06 2020