A358871 Square array A(n, k), n, k >= 0, read by antidiagonals: A(0, 0) = 0, A(0, 1) = A(1, 0) = 1, A(1, 1) = 2, 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+1, k+(1+(-1)^(n+k))/2) + A(n, k+(1-(-1)^(n+k))/2).

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

Offset: 0

Author

Rémy Sigrist, Dec 04 2022

Keywords

• nonn
• tabl

Comments

This sequence is a variant of A357743: we can build this sequence:

- by starting from an isosceles right triangle with values 0, 1, 1:

0 <- right angle

/ \

/ \

1-----1

- and repeatedly applying the following substitution to each isosceles right triangle:

t t

/ \ --> /|\

/ \ / | \

u-----v u-u+v-v

^

| right angles

The sequence presents rich patterns (see 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   4   5   4  5   5   7
    2 |  1  3  2   5  3   6   3   7  4   8   4
    3 |  2  3  5   4  5   5   8   6  7   7  10
    4 |  1  4  3   5  2   7   5   8  3   9   6
    5 |  3  4  6   5  7   6   9   7  8   8  11
    6 |  2  5  3   8  5   9   4   9  5  10   5
    7 |  3  4  7   6  8   7   9   6  7   7  12
    8 |  1  5  4   7  3   8   5   7  2   9   7
    9 |  4  5  8   7  9   8  10   7  9   8  13
   10 |  3  7  4  10  6  11   5  12  7  13   6
.
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 4 3 4 3 4 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 2)
• 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)
• 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

Cf. A002487, A357743.

Programs

PARI

A(n,k) = { my (nn = n\2, kk=k\2); if (n<=1 && k<=1, n+k, n%2==0 && k%2==0, A(n/2,k/2), n%2==0, A(n/2,k\2)+A(n/2,k\2+1), k%2==0, A(n\2,k\2)+A(n\2+1,k\2), A(n\2+1,k\2+(1+(-1)^(n\2+k\2))/2) + A(n\2, k\2+(1-(-1)^(n\2+k\2))/2)); }

Formula

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

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

A(n, n) = 2*A002487(n).