OFFSET
0,3
COMMENTS
We define the family {M_n, n >= 0}, as follows:
- M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
+
/ \
/ \
+ +
O
- for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
+ . . . + . . . +
. B . B .
+ . . . + . . .
. B . .A C.A C.
. . --> + . . . + . . . +
.A C. .C . A.
+ . . . + . B.B .
O .A . C.
+ . . . + . . . +
O
- for any n >= 0, M_n has A087289(n) points,
- the space filling curve M is the limit of M_{2*n} as n tends to infinity.
The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059253, see illustration in Links section).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..8192
F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982.
Larry Riddle, Space Filling Curve
Rémy Sigrist, Illustration of M_6
Rémy Sigrist, Illustration of the connection between the space filling curve M and Hilbert Hamiltonian walk
Rémy Sigrist, PARI program for A341018
FORMULA
EXAMPLE
The curve M starts as follows:
11+ 13+ +19 +21
/ \ / \ / \ / \
10+ 12+ 14+18 +20 +22
\ / \ /
9+ 15+ +17 +23
/ \ / \
8+ 6+ + +26 +24
\ / \ 16 / \ /
7+ 5+ +27 +25
/ \
4+ +28
\ /
1+ 3+ +29 +31
/ \ / \ / \
0+ 2+ +30 +32
- so a(0) = a(8) = a(10) = 0,
a(1) = a(7) = a(9) = a(11) = 1.
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Feb 02 2021
STATUS
approved