cp's OEIS Frontend

A341163 a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.

%I A341163 #17 Feb 09 2021 22:01:43
%S A341163 0,1,3,2,0,1,3,4,6,5,3,4,6,5,3,2,0,1,3,2,0,1,3,4,6,7,9,8,6,7,9,10,12,
%T A341163 11,9,10,12,11,9,8,6,7,9,8,6,7,9,10,12,11,9,10,12,11,9,8,6,5,3,4,6,5,
%U A341163 3,2,0,1,3,2,0,1,3,4,6,5,3,4,6,5,3,2,0,1,3
%N A341163 a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.
%C A341163 Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
%C A341163            Y
%C A341163           /
%C A341163          /
%C A341163         0 ---- X
%C A341163 We define the family {A_n, n >= 0} as follows:
%C A341163 - A_0 corresponds to the points (0, 0), (1, 1) and (3, 0), in that order:
%C A341163            .   __+__   .
%C A341163          __----     ----__
%C A341163         +     .     .     +
%C A341163        0
%C A341163 - for any n >= 0, A_{n+1} is obtained by arranging 4 copies of A_n as follows:
%C A341163                          +
%C A341163                         /B\
%C A341163         +              /   \
%C A341163        /B\            /A   C\
%C A341163       /   \    -->   +-------+
%C A341163      /A   C\        /B\C   B/A\
%C A341163     +-------+      /   \   /   \
%C A341163    O              /A   C\A/B   C\
%C A341163                  +-------+-------+
%C A341163                 O
%C A341163 - the space filling curve A is the limit of A_n as n tends to infinity.
%C A341163 This sequence has similarities with A341018.
%H A341163 Rémy Sigrist, <a href="/A341163/b341163.txt">Table of n, a(n) for n = 0..8192</a>
%H A341163 Zbigniew Fiedorowicz, <a href="https://people.math.osu.edu/fiedorowicz.1/math655/peano_t.html">The Peano Curve Theorem</a>
%H A341163 Rémy Sigrist, <a href="/A341163/a341163.png">Illustration of A_5</a>
%H A341163 Rémy Sigrist, <a href="/A341163/a341163_1.png">Illustration of the first bisection of A</a>
%H A341163 Rémy Sigrist, <a href="/A341163/a341163_2.png">Illustration of the first quadrisection of A</a>
%H A341163 Rémy Sigrist, <a href="/A341163/a341163_3.png">Illustration of the fourth quadrisection of A</a>
%H A341163 Rémy Sigrist, <a href="/A341163/a341163.gp.txt">PARI program for A341163</a>
%H A341163 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A341163 The curve A starts as follows:
%e A341163                   .
%e A341163                 .   .
%e A341163               .   5   .
%e A341163             4   .   .   6
%e A341163           .   .   3   .   .
%e A341163         .   1   .   .   7   .
%e A341163       0   .   .   2   .   .   8
%e A341163 - so a(0) = a(4) = 0,
%e A341163      a(1) = a(5) = 1,
%e A341163      a(3) = 2,
%e A341163      a(2) = a(6) = 3,
%e A341163      a(8) = 6.
%o A341163 (PARI) See Links section.
%Y A341163 Cf. A341018, A341164.
%K A341163 nonn,look
%O A341163 0,3
%A A341163 _Rémy Sigrist_, Feb 06 2021