

A356096


A family of triangles T(m), m >= 0, read by triangles and then by rows; triangle T(0) is [1; 1, 1]; for m >= 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; 2*tu, 2*tv; 2*ut, t+u+v, 2*vt; u, 2*uv, 2*vu, v].


4



1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 5, 3, 5, 1, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1
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OFFSET

0,8


COMMENTS

We apply the following substitutions to transform T(m) into T(m+1):
t
/ \
/ \
t 2*tu 2*tv
/ \ ___\ / \ / \
/ \ / / \ / \
uv 2*ut t+u+v 2*vt
/ \ / \ / \
/ \ / \ / \
u2*uv2*vuv
and:
u2*uv2*vuv
\ / \ / \ /
\ / \ / \ /
uv 2*ut t+u+v 2*vt
\ / ___\ \ / \ /
\ / / \ / \ /
t 2*tu 2*tv
\ /
\ /
t
T(m) has 3^m+1 rows.
All terms are odd.
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).


LINKS



EXAMPLE

Triangle T(0) is:
1
1 1
Triangle T(1) is:
1
1 1
1 3 1
1 1 1 1
Triangle T(2) is:
1
1 1
1 3 1
1 1 1 1
1 1 5 1 1
1 5 5 5 5 1
1 1 5 3 5 1 1
1 1 5 5 5 5 1 1
1 3 1 1 5 1 1 3 1
1 1 1 1 1 1 1 1 1 1


PROG

(PARI) See Links section.


CROSSREFS



KEYWORD

sign,tabf


AUTHOR



STATUS

approved



