A375388 A family of squares S(m), m > 0, read by squares and then by rows; square S(1) is [1, 1; 1, 1]; for m > 0, square S(m+1) is obtained by replacing each subsquare [t, u; v, w] in S(m) by [t, t+u, u; t+v, t+u+v+w, u+w; v, v+w, w].

1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 3, 2, 3, 1, 3, 9, 6, 9, 3, 2, 6, 4, 6, 2, 3, 9, 6, 9, 3, 1, 3, 2, 3, 1, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 16, 12, 20, 8, 20, 12, 16, 4, 3, 12, 9, 15, 6, 15, 9, 12, 3, 5, 20, 15, 25, 10, 25, 15, 20, 5, 2, 8, 6, 10, 4, 10, 6, 8, 2

Offset: 1

Author

Rémy Sigrist, Aug 13 2024

Keywords

• nonn
• tabf

Comments

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

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

| | |

t--u | t+u |

| | --> t+v----+----u+w

v--w | v+w |

| | |

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

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.

As A355855, this sequence is related to nonperiodic tilings based on tiles decorated with elements of F_p for some odd prime number p; here we use square tiles, there triangular tiles.

Examples

			S(1) is:
             1 1
             1 1
S(2) is:
            1 2 1
            2 4 2
            1 2 1
S(3) is:
          1 3 2 3 1
          3 9 6 9 3
          2 6 4 6 2
          3 9 6 9 3
          1 3 2 3 1
S(4) is:
  1  4  3  5  2  5  3  4  1
  4 16 12 20  8 20 12 16  4
  3 12  9 15  6 15  9 12  3
  5 20 15 25 10 25 15 20  5
  2  8  6 10  4 10  6  8  2
  5 20 15 25 10 25 15 20  5
  3 12  9 15  6 15  9 12  3
  4 16 12 20  8 20 12 16  4
  1  4  3  5  2  5  3  4  1
		

Links

• Rémy Sigrist, Colored representation of S(9) mod 3
• Rémy Sigrist, Colored representation of S(9) mod 7

Crossrefs

Cf. A002487, A049456, A355855.

Programs

PARI

S(n) = { matrix(2^(n-1)+1, 2^(n-1)+1, i,j, A002487(2^(n-1)-1+i) * A002487(2^(n-1)-1+j)); }

Formula

S(m)(n, k) = A049456(m, n) * A049456(m, k).