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A356245
A family of squares A(m), m >= 0, read by squares and then by rows; A(0) is [1, 1; 1, 1]; for m >= 0, square A(m+1) is obtained by replacing each subsquare [t, u; v, w] by [t, t+u, t+u, u; t+v, t+u+v, t+u+w, u+w; t+v, t+v+w, u+v+w, u+w; v, v+w, v+w, w] in A(m).
1
1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 2, 2, 1, 1, 3, 3, 2, 4, 4, 2, 3, 3, 1, 3, 5, 6, 5, 7, 7, 5, 6, 5, 3, 3, 6, 7, 5, 8, 8, 5, 7, 6, 3, 2, 5, 5, 3, 6, 6, 3, 5, 5, 2, 4, 7, 8, 6, 9, 9, 6, 8, 7, 4, 4, 7, 8, 6, 9, 9, 6, 8, 7, 4, 2, 5, 5, 3, 6, 6, 3, 5, 5, 2
OFFSET
0,6
COMMENTS
We apply the following substitutions to transform A(m) into A(m+1):
t----t+u---t+u----u
| | | |
| | | |
t-----u t+v--t+u+v-t+u+w--u+w
| | ___\ | | | |
| | / | | | |
v-----w t+v--t+v+w-u+v+w--u+w
| | | |
| | | |
v----v+w---v+w----w
A(m) has 3^m+1 rows.
As m gets larger, A(m) exhibits interesting fractal features (see illustrations in Links section).
EXAMPLE
Square A(0) is:
1 1
1 1
Square A(1) is:
1 2 2 1
2 3 3 2
2 3 3 2
1 2 2 1
Square A(2) is:
1 3 3 2 4 4 2 3 3 1
3 5 6 5 7 7 5 6 5 3
3 6 7 5 8 8 5 7 6 3
2 5 5 3 6 6 3 5 5 2
4 7 8 6 9 9 6 8 7 4
4 7 8 6 9 9 6 8 7 4
2 5 5 3 6 6 3 5 5 2
3 6 7 5 8 8 5 7 6 3
3 5 6 5 7 7 5 6 5 3
1 3 3 2 4 4 2 3 3 1
PROG
(PARI) See Links section.
CROSSREFS
See A355855, A356002, A356096, A356097 and A356098 for similar sequences.
Sequence in context: A256790 A337225 A256478 * A106638 A329400 A131909
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, Jul 30 2022
STATUS
approved