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).

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

Author

Rémy Sigrist, Nov 29 2022

Keywords

• nonn
• tabl

Comments

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).

Examples

			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
		

Links

• 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

Crossrefs

See A358871 for a similar sequence.

Cf. A002487, A007306, A355855.

Programs

PARI

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)); }

Formula

A(n, k) = A(k, n).

A(n, 0) = A002487(n).

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