A356096 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; 2*t-u, 2*t-v; 2*u-t, t+u+v, 2*v-t; u, 2*u-v, 2*v-u, v].

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, 5, 5, 5, 1, 1, -1, 5, 3, 5, -1, 1, 1, 1, 5, 5, 5, 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

Offset

0,8

Comments

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

t

/ \

/ \

t 2*t-u 2*t-v

/ \ ___\ / \ / \

/ \ / / \ / \

u-----v 2*u-t t+u+v 2*v-t

/ \ / \ / \

/ \ / \ / \

u---2*u-v--2*v-u--v

and:

u---2*u-v--2*v-u--v

\ / \ / \ /

\ / \ / \ /

u-----v 2*u-t t+u+v 2*v-t

\ / ___\ \ / \ /

\ / / \ / \ /

t 2*t-u 2*t-v

\ /

\ /

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).

Links

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

Rémy Sigrist, Colored representation of T6 (the color is function of T(6)(n, k))

Rémy Sigrist, Representation of the multiples of 3 in T(7)

Rémy Sigrist, Representation of the negative terms in T(7)

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
                       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   5   5   5   1
            1  -1   5   3   5  -1   1
          1   1   5   5   5   5   1   1
        1   3   1  -1   5  -1   1   3   1
      1   1   1   1   1   1   1   1   1   1

Prog

(PARI) See Links section.

Crossrefs

See A355855, A356002, A356097 and A356098 for similar sequences.

Sequence in context: A086997 A074298 A356097 * A326029 A356167 A176028

Adjacent sequences: A356093 A356094 A356095 * A356097 A356098 A356099

Keyword

sign,tabf

Author

Rémy Sigrist, Jul 26 2022

Status

approved