A356245 A family of squares A(m), m >= 0, read by squares and then by rows; A(0) is [1, 1; 1, 1]; for m >= 0, square A(m+1) is obtained by replacing each subsquare [t, u; v, w] by [t, t+u, t+u, u; t+v, t+u+v, t+u+w, u+w; t+v, t+v+w, u+v+w, u+w; v, v+w, v+w, w] in A(m).

1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 2, 2, 1, 1, 3, 3, 2, 4, 4, 2, 3, 3, 1, 3, 5, 6, 5, 7, 7, 5, 6, 5, 3, 3, 6, 7, 5, 8, 8, 5, 7, 6, 3, 2, 5, 5, 3, 6, 6, 3, 5, 5, 2, 4, 7, 8, 6, 9, 9, 6, 8, 7, 4, 4, 7, 8, 6, 9, 9, 6, 8, 7, 4, 2, 5, 5, 3, 6, 6, 3, 5, 5, 2

Offset: 0

Author

Rémy Sigrist, Jul 30 2022

Keywords

• nonn
• tabf

Comments

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

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

| | | |

| | | |

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

| | _\ | | | |

| | / | | | |

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

| | | |

| | | |

v----v+w---v+w----w

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

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

Examples

			Square A(0) is:
     1 1
     1 1
Square A(1) is:
     1 2 2 1
     2 3 3 2
     2 3 3 2
     1 2 2 1
Square A(2) is:
     1 3 3 2 4 4 2 3 3 1
     3 5 6 5 7 7 5 6 5 3
     3 6 7 5 8 8 5 7 6 3
     2 5 5 3 6 6 3 5 5 2
     4 7 8 6 9 9 6 8 7 4
     4 7 8 6 9 9 6 8 7 4
     2 5 5 3 6 6 3 5 5 2
     3 6 7 5 8 8 5 7 6 3
     3 5 6 5 7 7 5 6 5 3
     1 3 3 2 4 4 2 3 3 1
		

Links

• Rémy Sigrist, Representation of the multiples of 2 in T(6)
• Rémy Sigrist, Representation of the multiples of 3 in T(6)
• Rémy Sigrist, Representation of the multiples of 5 in T(6)
• 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, A356096, A356097 and A356098 for similar sequences.

Programs

PARI

See Links section.