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 A334474 #28 Dec 21 2024 10:54:00
%S A334474 0,2,1,8,3,4,9,7,6,5,31,10,11,15,16,32,30,12,14,18,17,33,29,27,13,24,
%T A334474 19,20,35,34,28,26,25,23,22,21,121,36,37,41,42,61,62,63,64,122,120,38,
%U A334474 40,43,44,60,59,66,65,123,119,117,39,47,46,45,58,72,67,68
%N A334474 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve T defined in Comments section; square array T(n, k), n, k >= 0 read by antidiagonals downwards.
%C A334474 We consider a hexagonal lattice with X-axis and Y-axis as follows:
%C A334474 Y
%C A334474 /
%C A334474 /
%C A334474 0 ---- X
%C A334474 We define the family {T_n, n > 0} as follows:
%C A334474 - T_1 contains the origin (0, 0):
%C A334474 +
%C A334474 O
%C A334474 - T_2 contains the points (0, 0), (0, 1) and (1, 0), in that order:
%C A334474 +
%C A334474 / \
%C A334474 / \
%C A334474 + . . +
%C A334474 O
%C A334474 - for n > 1, T_{2*n} is built from 3 copies of T_n and one copy T_{n-1} arranged as follows:
%C A334474 +
%C A334474 / \
%C A334474 / \
%C A334474 / n \
%C A334474 / \
%C A334474 + . . . . +
%C A334474 \ \
%C A334474 + +-----+ +
%C A334474 / \ .n-1/ / .
%C A334474 / \ . / / .
%C A334474 / n \ + / n .
%C A334474 / \ / / .
%C A334474 + . . . . + +---------+
%C A334474 O
%C A334474 - for n > 0, T_{2*n+1} is built from 3 copies of T_n and one copy of T_{n+1} arranged as follows:
%C A334474 +
%C A334474 / \
%C A334474 / \
%C A334474 / \
%C A334474 / n+1 \
%C A334474 / \
%C A334474 + . . . . . +
%C A334474 \ \
%C A334474 + +---------+ +
%C A334474 / \ . / / .
%C A334474 / \ . n / / .
%C A334474 / n \ . / / n .
%C A334474 / \ . / / .
%C A334474 + . . . . +--+ +---------+
%C A334474 O
%C A334474 - for any n > 0, T_n starts at (0, 0) and ends at (n-1, n-1), and contains every point (x, y) such that x, y >= 0 and x + y < n,
%C A334474 - T is the limit of T_{2^k} as k tends to infinity (note that for any k >= 0, T_{2^k} is a prefix of T_{2^(k+1)}),
%C A334474 - T visits exactly once every point (x, y) such that x, y >= 0.
%H A334474 Ideophilus, <a href="https://ideophilus.wordpress.com/2012/09/06/a-triangular-space-filling-curve/">A triangular space-filling curve</a>
%H A334474 Rémy Sigrist, <a href="/A334474/a334474_1.png">Colored representation of T_{2^10}</a> (where the hue is function of the number of steps from the origin)
%H A334474 Rémy Sigrist, <a href="/A334474/a334474.png">Representation of T_{2^k} for k = 1..5</a>
%H A334474 Rémy Sigrist, <a href="/A334474/a334474.gp.txt">PARI program for A334474</a>
%F A334474 T(A334476(n), A334475(n)) = n.
%e A334474 Square array starts:
%e A334474 n\k| 0 1 2 3 4 5 6 7
%e A334474 ---+-------------------------
%e A334474 0| 0 2 8--9 31-32-33 35
%e A334474 | | /| | | | | /|
%e A334474 | |/ | | | | |/ |
%e A334474 1| 1 3 7 10 30-29 34 36
%e A334474 | / / / | /
%e A334474 | / / / | /
%e A334474 2| 4 6 11-12 27-28 37-38
%e A334474 | | / | | |
%e A334474 | |/ | | |
%e A334474 3| 5 15-14-13 26 41-40-39
%e A334474 | / / /
%e A334474 | / / /
%e A334474 4| 16 18 24-25 42-43 47-48
%e A334474 | | /| | / / |
%e A334474 | |/ | | / / |
%e A334474 5| 17 19 23 61 44 46 50-49
%e A334474 | / / /| | / |
%e A334474 | / / / | |/ |
%e A334474 6| 20 22 62 60 45 56 51-52-
%e A334474 | | / / / /|
%o A334474 (PARI) \\ See Links section.
%Y A334474 See A334475 and A334476 for the coordinates of the curve.
%K A334474 nonn,tabl
%O A334474 0,2
%A A334474 _Rémy Sigrist_, May 02 2020