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

Author

Rémy Sigrist, Jul 26 2022

Keywords

• sign
• tabf

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).

Examples

			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
		

Links

• 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.

Crossrefs

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

Programs

PARI

See Links section.