|
| |
|
|
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.
|
|
5
|
|
|
|
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, 19, 20, 35, 34, 28, 26, 25, 23, 22, 21, 121, 36, 37, 41, 42, 61, 62, 63, 64, 122, 120, 38, 40, 43, 44, 60, 59, 66, 65, 123, 119, 117, 39, 47, 46, 45, 58, 72, 67, 68
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
We consider a hexagonal lattice with X-axis and Y-axis as follows:
Y
/
/
0 ---- X
We define the family {T_n, n > 0} as follows:
- T_1 contains the origin (0, 0):
+
O
- T_2 contains the points (0, 0), (0, 1) and (1, 0), in that order:
+
/ \
/ \
+ . . +
O
- for n > 1, T_{2*n} is built from 3 copies of T_n and one copy T_{n-1} arranged as follows:
+
/ \
/ \
/ n \
/ \
+ . . . . +
\ \
+ +-----+ +
/ \ .n-1/ / .
/ \ . / / .
/ n \ + / n .
/ \ / / .
+ . . . . + +---------+
O
- 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:
+
/ \
/ \
/ \
/ n+1 \
/ \
+ . . . . . +
\ \
+ +---------+ +
/ \ . / / .
/ \ . n / / .
/ n \ . / / n .
/ \ . / / .
+ . . . . +--+ +---------+
O
- 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,
- 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)},
- T visits exactly once every point (x, y) such that x, y >= 0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
Square array starts:
n\k| 0 1 2 3 4 5 6 7
---+-------------------------
0| 0 2 8--9 31-32-33 35
| | /| | | | | /|
| |/ | | | | |/ |
1| 1 3 7 10 30-29 34 36
| / / / | /
| / / / | /
2| 4 6 11-12 27-28 37-38
| | / | | |
| |/ | | |
3| 5 15-14-13 26 41-40-39
| / / /
| / / /
4| 16 18 24-25 42-43 47-48
| | /| | / / |
| |/ | | / / |
5| 17 19 23 61 44 46 50-49
| / / /| | / |
| / / / | |/ |
6| 20 22 62 60 45 56 51-52-
| | / / / /|
|
|
|
PROG
|
(PARI) See Links section.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
