OFFSET
0,5
COMMENTS
We apply the following substitutions to transform T(m) into T(m+1):
t
/ \
/ \
t 2*t+u 2*t+v
/ \ ___\ / \ / \
/ \ / / \ / \
u-----v t+2*u t+u+v t+2*v
/ \ / \ / \
/ \ / \ / \
u---2*u+v--u+2*v--v
and:
u---2*u+v--u+2*v--v
\ / \ / \ /
\ / \ / \ /
u-----v t+2*u t+u+v t+2*v
\ / ___\ \ / \ /
\ / / \ / \ /
t 2*t+u 2*t+v
\ /
\ /
t
T(m) has 3^m+1 rows, and largest term 3^m.
All terms are odd.
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).
LINKS
Rémy Sigrist, Colored representation of T(6) (the color is function of T(6)(n,k))
Rémy Sigrist, Colored representation of T(6) (the color is function of the 3-adic valuation of T(6)(n,k))
Rémy Sigrist, Representation of the terms congruent to 3 mod 4 in T(6)
Rémy Sigrist, PARI program
Rémy Sigrist, Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp, arXiv:2301.06039 [math.CO], 2023.
EXAMPLE
Triangle T(0) is:
1
1 1
Triangle T(1) is:
1
3 3
3 3 3
1 3 3 1
Triangle T(2) is:
1
5 5
7 7 7
3 9 9 3
9 9 9 9 9
9 9 9 9 9 9
3 9 9 3 9 9 3
7 9 9 9 9 9 9 7
5 7 9 9 9 9 9 7 5
1 5 7 3 9 9 3 7 5 1
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, Jul 22 2022
STATUS
approved