login
A355855
A family of triangles T(m), m > 0, read by triangles and then by rows; triangle T(1) 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+u, t+v; u, u+v, v].
8
1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 2, 4, 2, 3, 4, 4, 3, 1, 3, 2, 3, 1, 1, 4, 4, 3, 6, 3, 5, 7, 7, 5, 2, 6, 4, 6, 2, 5, 6, 8, 8, 6, 5, 3, 7, 4, 8, 4, 7, 3, 4, 6, 7, 6, 6, 7, 6, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 1, 5, 5, 4, 8, 4, 7, 10, 10, 7, 3, 9, 6, 9, 3, 8, 10, 13, 13, 10, 8
OFFSET
1,5
COMMENTS
We apply the following substitutions to transform T(m) into T(m+1):
t
/ \
t / \
/ \ --> t+u---t+v
u---v / \ / \
/ \ / \
u----u+v----v
This sequence can be seen as a two-dimensional variant of A049456.
The base of T(m) corresponds to the m-th row of A049456.
T(m) has 2^(m-1)+1 rows, and largest term 2^(m-1).
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).
EXAMPLE
T(1) is:
1
1 1
T(2) is:
1
2 2
1 2 1
T(3) is:
1
3 3
2 4 2
3 4 4 3
1 3 2 3 1
T(4) is:
1
4 4
3 6 3
5 7 7 5
2 6 4 6 2
5 6 8 8 6 5
3 7 4 8 4 7 3
4 6 7 6 6 7 6 4
1 4 3 5 2 5 3 4 1
PROG
(PARI) See Links section.
(PARI) T(m, n, k) = { if (m==1, 1, my (nn=(n+1)\2, kk=(k+1)\2); if (n%2==1 && k%2==1, T(m-1, nn, kk), n%2==1 && k%2==0, T(m-1, nn, kk) + T(m-1, nn, kk+1), n%2==0 && k%2==1, T(m-1, nn, kk) + T(m-1, nn+1, kk), T(m-1, nn, kk) + T(m-1, nn+1, kk+1))) }
CROSSREFS
Cf. A049456.
Sequence in context: A180312 A178819 A369174 * A046816 A352248 A301475
KEYWORD
nonn,easy,tabf
AUTHOR
Rémy Sigrist, Jul 19 2022
STATUS
approved