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A334577 a(n) is the Y-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334576 gives X-coordinates.

%I A334577 #16 May 09 2020 02:31:32
%S A334577 0,0,0,1,2,2,1,0,0,0,0,1,1,1,2,3,4,4,4,5,6,6,5,4,3,3,3,2,2,2,1,0,0,0,
%T A334577 0,1,2,2,1,0,0,0,0,1,1,1,2,3,3,3,3,2,1,1,2,3,4,4,4,5,5,5,6,7,8,8,8,9,
%U A334577 10,10,9,8,8,8,8,9,9,9,10,11,12,12,12,13
%N A334577 a(n) is the Y-coordinate of the n-th point of the space filling curve P defined in Comments section; sequence A334576 gives X-coordinates.
%C A334577 The space filling curve P corresponds to the midpoint curve of the alternate paperfolding curve and can be built as follows:
%C A334577 - we define the family {P_k, k > 0}:
%C A334577     - P_1 corresponds to the points (0, 0), (1, 0), (2, 0) and (2, 1), in that order:
%C A334577                     +
%C A334577                     |
%C A334577                     |
%C A334577           +----+----+
%C A334577          O
%C A334577     - for any k > 0, P_{n+1} is built from four copies of P_n as follows:
%C A334577                                             +
%C A334577                                             |A
%C A334577                 +                           |
%C A334577                C|                   +----+  |
%C A334577          A     B|    --->           |C  B|  |B  C
%C A334577         +-------+                   +    |  +----+-+
%C A334577        O                           C|    |        C|
%C A334577                              A     B|   A|  A     B|
%C A334577                             +-------+    +-+-------+
%C A334577                            O
%C A334577 - the space filling curve P is the limit of P_k as k tends to infinity.
%C A334577 We can also describe the space filling curve P by mean of an L-system (see Links section).
%H A334577 Rémy Sigrist, <a href="/A334577/b334577.txt">Table of n, a(n) for n = 0..4095</a>
%H A334577 Joerg Arndt, <a href="/A334576/a334576.pdf">L-system corresponding to P</a>
%H A334577 Kevin Ryde, <a href="https://user42.tuxfamily.org/alternate/index.html">Iterations of the Alternate Paperfolding Curve</a>
%H A334577 Rémy Sigrist, <a href="/A334577/a334577.gp.txt">PARI program for A334577</a>
%H A334577 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A334577 a(n+1) = (A020990(n) + A020990(n+1) - 1)/2 for any n >= 0.
%e A334577 The first points of the space filling curve P are as follows:
%e A334577       6|                                  20...21
%e A334577        |                                  |    |
%e A334577       5|                                  19   22
%e A334577        |                                  |    |
%e A334577       4|                        16...17...18   23
%e A334577        |                        |              |
%e A334577       3|                        15   26...25...24
%e A334577        |                        |    |
%e A334577       2|              4....5    14   27...28...29
%e A334577        |              |    |    |              |
%e A334577       1|              3    6    13...12...11   30
%e A334577        |              |    |              |    |
%e A334577       0|    0....1....2    7....8....9....10   31..
%e A334577        |
%e A334577     ---+----------------------------------------
%e A334577     y/x|    0    1    2    3    4    5    6    7
%e A334577 - hence a(15) = a(24) = a(25) = a(26) = 3.
%o A334577 (PARI) See Links section.
%Y A334577 Cf. A020990, A334576.
%K A334577 nonn
%O A334577 0,5
%A A334577 _Rémy Sigrist_, May 06 2020