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).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..11313
Rémy Sigrist, Representation of multiples of 2 in T(12)
Rémy Sigrist, Representation of multiples of 2^2 in T(12)
Rémy Sigrist, Representation of multiples of 3 in T(12)
Rémy Sigrist, Representation of multiples of 5 in T(12)
Rémy Sigrist, Colored representation of T(10) (the color is function of T(10)(n,k))
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
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
KEYWORD
nonn,easy,tabf
AUTHOR
Rémy Sigrist, Jul 19 2022
STATUS
approved