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A356097
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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; t, t; u, t+u+v, v; u, u, v, v].
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4
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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, 3, 3, 5, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 5, 3, 3, 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, 1, 5, 3, 3, 5, 1
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OFFSET
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0,8
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COMMENTS
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We apply the following substitutions to transform T(m) into T(m+1):
t
/ \
/ \
t t-----t
/ \ ___\ / \ / \
/ \ / / \ / \
u-----v u---t+u+v---v
/ \ / \ / \
/ \ / \ / \
u-----u-----v-----v
and:
u-----u-----v-----v
\ / \ / \ /
\ / \ / \ /
u-----v u---t+u+v---v
\ / ___\ \ / \ /
\ / / \ / \ /
t t-----t
\ /
\ /
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).
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LINKS
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EXAMPLE
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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 3 3 5 1
1 1 3 3 3 1 1
1 1 5 3 3 5 1 1
1 3 1 1 5 1 1 3 1
1 1 1 1 1 1 1 1 1 1
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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