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

0,5

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).

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 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.

Formula

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

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

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

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   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

Prog

(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)); }

Crossrefs

Cf. A002487, A357743.

Sequence in context: A378261 A305715 A165014 * A357743 A058063 A232094

Adjacent sequences: A358868 A358869 A358870 * A358872 A358873 A358874

Keyword

nonn,tabl

Author

Rémy Sigrist, Dec 04 2022

Status

approved