A355855 -id:A355855 - cp's OEIS frontend

A356002 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; 2t+u, 2t+v; t+2u, t+u+v, t+2v; u, 2u+v, u+2v, v].

1, 1, 1, 1, 3, 3, 3, 3, 3, 1, 3, 3, 1, 1, 5, 5, 7, 7, 7, 3, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 3, 9, 9, 3, 7, 9, 9, 9, 9, 9, 9, 7, 5, 7, 9, 9, 9, 9, 9, 7, 5, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 7, 11, 11, 11, 5, 15, 15, 5, 17, 17, 17, 17, 17

Offset: 0

Rémy Sigrist, Jul 22 2022

We apply the following substitutions to transform T(m) into T(m+1):
                            t
                           / \
                          /   \
       t               2*t+u 2*t+v
      / \    _\       / \   / \
     /   \      /      /   \ /   \
    u-----v         t+2*u t+u+v t+2*v
                     / \   / \   / \
                    /   \ /   \ /   \
                   u---2*u+v--u+2*v--v
and:
                   u---2*u+v--u+2*v--v
                    \   / \   / \   /
                     \ /   \ /   \ /
    u-----v         t+2*u t+u+v t+2*v
     \   /   _\      \   / \   /
      \ /       /       \ /   \ /
       t               2*t+u 2*t+v
                          \   /
                           \ /
                            t
T(m) has 3^m+1 rows, and largest term 3^m.
All terms are odd.
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

			Triangle T(0) is:
              1
             1 1
Triangle T(1) is:
              1
             3 3
            3 3 3
           1 3 3 1
Triangle T(2) is:
              1
             5 5
            7 7 7
           3 9 9 3
          9 9 9 9 9
         9 9 9 9 9 9
        3 9 9 3 9 9 3
       7 9 9 9 9 9 9 7
      5 7 9 9 9 9 9 7 5
     1 5 7 3 9 9 3 7 5 1

Rémy Sigrist, Colored representation of T(6) (the color is function of T(6)(n,k))
Rémy Sigrist, Colored representation of T(6) (the color is function of the 3-adic valuation of T(6)(n,k))
Rémy Sigrist, Representation of the terms congruent to 3 mod 4 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.

See A355855 for a similar sequence.

Cf. A177407.

PARI
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See Links section.
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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; 2t-u, 2t-v; 2u-t, t+u+v, 2v-t; u, 2u-v, 2v-u, v].

Original entry on oeis.org

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

Rémy Sigrist, Jul 26 2022

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).

			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

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.

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

PARI
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See Links section.
```

A356097 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, t; u, t+u+v, v; u, u, v, v].

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jul 26 2022

We apply the following substitutions to transform T(m) into T(m+1):
                            t
                           / \
                          /   \
       t                 t-----t
      / \    _\       / \   / \
     /   \      /      /   \ /   \
    u-----v           u---t+u+v---v
                     / \   / \   / \
                    /   \ /   \ /   \
                   u-----u-----v-----v
and:
                   u-----u-----v-----v
                    \   / \   / \   /
                     \ /   \ /   \ /
    u-----v           u---t+u+v---v
     \   /   _\      \   / \   /
      \ /       /       \ /   \ /
       t                 t-----t
                          \   /
                           \ /
                            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).

			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 3 3 5 1
        1 1 3 3 3 1 1
       1 1 5 3 3 5 1 1
      1 3 1 1 5 1 1 3 1
     1 1 1 1 1 1 1 1 1 1

Rémy Sigrist, Colored representation of T(6) (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 multiples of 5 in T(7)
Rémy Sigrist, Representation of the multiples of 7 in T(7)
Rémy Sigrist, Representation of the 1's in T(7)
Rémy Sigrist, Representation of the terms congruent to 1 mod 4 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.
Wikipedia, Hexaflake

See A355855, A356002, A356096 and A356098 for similar sequences.

Cf. A353174.

PARI
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See Links section.
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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].

Original entry on oeis.org

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

Rémy Sigrist, Jul 26 2022

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).

			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

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.

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

PARI
```
See Links section.
```

A357743 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, for n, k >= 0, A(2n, 2k) = A(n, k), A(2n, 2k+1) = A(n, k) + A(n, k+1), A(2n+1, 2k) = A(n, k) + A(n+1, k), A(2n+1, 2k+1) = A(n, k+1) + A(n+1, k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 1, 3, 2, 3, 1, 3, 4, 5, 5, 4, 3, 2, 5, 3, 6, 3, 5, 2, 3, 5, 6, 5, 5, 6, 5, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 5, 7, 8, 7, 7, 8, 7, 5, 4, 3, 7, 4, 9, 5, 10, 5, 9, 4, 7, 3, 5, 8, 9, 7, 8, 11, 11, 8, 7, 9, 8, 5, 2, 7, 5, 8, 3, 9, 6, 9, 3, 8, 5, 7, 2

Offset: 0

Rémy Sigrist, Nov 29 2022

This sequence is closely related to A002487 and A355855: we can build this sequence:
- by starting from an equilateral triangle with values 0, 1, 1:
        0
       / \
      1---1
- and repeatedly applying the following substitution:
                           t
                          / \
        t                /   \
       / \     -->     t+u---t+v
      u---v            / \   / \
                      /   \ /   \
                     u----u+v----v
The sequence reduced modulo an odd prime number presents rich nonperiodic patterns (see illustrations in Links section).

			Array A(n, k) begins:
  n\k |  0  1  2   3  4   5   6   7  8   9  10
  ----+---------------------------------------
    0 |  0  1  1   2  1   3   2   3  1   4   3
    1 |  1  2  3   3  4   5   5   4  5   7   8
    2 |  1  3  2   5  3   6   3   7  4   9   5
    3 |  2  3  5   6  5   5   8   9  7   8  11
    4 |  1  4  3   5  2   7   5   8  3   9   6
    5 |  3  5  6   5  7  10  11   9  8  11  11
    6 |  2  5  3   8  5  11   6  11  5  10   5
    7 |  3  4  7   9  8   9  11  10  7   7  12
    8 |  1  5  4   7  3   8   5   7  2   9   7
    9 |  4  7  9   8  9  11  10   7  9  14  17
   10 |  3  8  5  11  6  11   5  12  7  17  10
.
The first antidiagonals are:
              0
             1 1
            1 2 1
           2 3 3 2
          1 3 2 3 1
         3 4 5 5 4 3
        2 5 3 6 3 5 2
       3 5 6 5 5 6 5 3
      1 4 3 5 2 5 3 4 1
     4 5 7 8 7 7 8 7 5 4

Rémy Sigrist, Colored representation of the first 512 antidiagonals (where the color is function of A(n, k) mod 3)
Rémy Sigrist, Colored representation of the first 512 antidiagonals (where the color is function of A(n, k) mod 5)
Index entries for sequences related to Stern's sequences

See A358871 for a similar sequence.

Cf. A002487, A007306, A355855.

A(n,k) = { if (n==0 && k==0, 0, n==1 && k==0, 1, n==0 && k==1, 1, n%2==0 && k%2==0, A(n/2,k/2), n%2==0, A(n/2,(k-1)/2) + A(n/2,(k+1)/2), k%2==0, A((n-1)/2,k/2) + A((n+1)/2,k/2), A((n+1)/2,(k-1)/2) + A((n-1)/2,(k+1)/2)); }

A(n, k) = A(k, n).
A(n, 0) = A002487(n).
A(n, 1) = A007306(n+1) for any n > 0.

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).

Original entry on oeis.org

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

Rémy Sigrist, Jul 30 2022

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).

			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

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.

See A355855, A356002, A356096, A356097 and A356098 for similar sequences.

PARI
```
See Links section.
```

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].

Original entry on oeis.org

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

Rémy Sigrist, Aug 13 2024

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.

			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

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

Cf. A002487, A049456, A355855.

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)); }

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

cp's OEIS Frontend

A356002 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; t+2*u, t+u+v, t+2*v; u, 2*u+v, u+2*v, v].

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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].

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A356097 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, t; u, t+u+v, v; u, u, v, v].

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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].

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A357743 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, for n, k >= 0, A(2*n, 2*k) = A(n, k), A(2*n, 2*k+1) = A(n, k) + A(n, k+1), A(2*n+1, 2*k) = A(n, k) + A(n+1, k), A(2*n+1, 2*k+1) = A(n, k+1) + A(n+1, k).

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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).

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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].

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PARI

Formula

A356002 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; 2t+u, 2t+v; t+2u, t+u+v, t+2v; u, 2u+v, u+2v, v].

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

A357743 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, for n, k >= 0, A(2n, 2k) = A(n, k), A(2n, 2k+1) = A(n, k) + A(n, k+1), A(2n+1, 2k) = A(n, k) + A(n+1, k), A(2n+1, 2k+1) = A(n, k+1) + A(n+1, k).