A356098 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-u, t-v; u-t, t+u+v, v-t; u, u-v, v-u, v].

1, 1, 1, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -3, 3, -3, 0, 0, 3, 3, 3, 3, 0, 0, -3, 3, 3, 3, -3, 0, -1, 0, 3, 3, 3, 3, 0, -1, 1, 1, 0, -3, 3, -3, 0, 1, 1, 1, 1, -1, 0, 0, 0, 0, -1, 1, 1, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 2, 0, 3, 0, 2

Offset

0,8

Comments

We apply the following substitutions to transform T(m) into T(m+1):

t

/ \

/ \

t t-u---t-v

/ \ ___\ / \ / \

/ \ / / \ / \

u-----v u-t--t+u+v--v-t

/ \ / \ / \

/ \ / \ / \

u----u-v---v-u----v

and:

u----u-v---v-u----v

\ / \ / \ /

\ / \ / \ /

u-----v u-t--t+u+v--v-t

\ / ___\ \ / \ /

\ / / \ / \ /

t t-u---t-v

\ /

\ /

t

T(m) has 3^m+1 rows.

As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

Links

Table of n, a(n) for n=0..82.

Rémy Sigrist, Colored representation of T(6) (where the color is function of T(6)(n,k))

Rémy Sigrist, Representation of the multiples of 2 in T7

Rémy Sigrist, Representation of the multiples of 5 in T7

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

Prog

(PARI) See Links section.

Crossrefs

See A355855, A356002, A356096 and A356097 for similar sequences.

Sequence in context: A196306 A107093 A051830 * A106216 A035676 A369283

Adjacent sequences: A356095 A356096 A356097 * A356099 A356100 A356101

Keyword

sign,tabl

Author

Rémy Sigrist, Jul 26 2022

Status

approved