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

0,5

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).

Links

Table of n, a(n) for n=0..90.

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

Formula

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

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

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

Example

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

Prog

(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)); }

Crossrefs

See A358871 for a similar sequence.

Cf. A002487, A007306, A355855.

Sequence in context: A305715 A165014 A358871 * A058063 A232094 A143902

Adjacent sequences: A357740 A357741 A357742 * A357744 A357745 A357746

Keyword

nonn,tabl

Author

Rémy Sigrist, Nov 29 2022

Status

approved