A392670 Triangle read by rows where row n is the numerators of the node values in a preorder traversal of the Calkin-Wilf tree with n levels, columns 0 <= k < 2^n-1.

1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 3, 1, 1, 1, 1, 4, 3, 3, 5, 2, 2, 2, 5, 3, 3, 4, 1, 1, 1, 1, 1, 5, 4, 4, 7, 3, 3, 3, 8, 5, 5, 7, 2, 2, 2, 2, 7, 5, 5, 8, 3, 3, 3, 7, 4, 4, 5, 1, 1, 1, 1, 1, 1, 6, 5, 5, 9, 4, 4, 4, 11, 7, 7, 10, 3, 3, 3, 3, 11, 8, 8, 13, 5, 5, 5, 12, 7, 7, 9, 2, 2, 2, 2, 2, 9, 7, 7, 12, 5, 5, 5, 13, 8, 8, 11, 3, 3, 3, 3, 10, 7, 7, 11, 4, 4, 4, 9, 5, 5, 6

Offset

1,4

Comments

The Calkin-Wilf tree read breadth-first is fractions d(i)/d(i+1) where d(i) = A002487(i) is Stern's diatomic sequence, and here the n levels are read preorder.

Triangle row n comprises repetitions of the numerators in the bottom-most level (level n) of its tree since a left child has the same numerator as its parent.

This bottom-most level is Stern diatomic terms d(i) for 2^(n-1) <= i < 2^n, and the number of repetitions is the number of consecutive left children which is the ruler function A001511(i).

Consequently the whole sequence is the diatomic sequence with terms repeated ruler function many times, and so with flat indexing a(n) = A002487(A046699(n+1)).

Links

Alois P. Heinz, Rows n = 1..14, flattened

Wikipedia, Calkin-Wilf tree.

Example

Triangle begins:
    k = 0 1 2 3 4 5 6 7 8 9 . . .
  n=1:  1;
  n=2:  1,1,2;
  n=3:  1,1,1,3,2,2,3;
  n=4:  1,1,1,1,4,3,3,5,2,2,2,5,3,3,4;
For row n=4, the Calkin-Wilf tree of 4 levels is as follows and row 4 is the numerators traversed in preorder
                           1/1
                 /                    \
             1/2                        2/1
          /       \                  /        \
      1/3          3/2            2/3          3/1
    /    \        /    \        /    \        /    \
  1/4    4/3    3/5    5/2    2/5    5/3    3/4    4/1
Notice each left descent has numerator unchanged and those repetitions are
  row  1,1,1,1, 4, 3,3, 5, 2,2,2, 5, 3,3, 4
       \-----/  ^  \-/  ^  \---/  ^  \-/  ^
  term    1     4   3   5    2    5   3   4   diatomic
  reps    4     1   2   1    3    1   2   1   ruler

Maple

b:= (n, u, d)-> [u/d, `if`(n=1, [], [b(n-1, u, u+d)[], b(n-1, u+d, d)[]])[]]:
T:= n-> map(numer, b(n, 1$2))[]:
seq(T(n), n=1..6);  # Alois P. Heinz, Jan 19 2026

Crossrefs

Cf. A002487, A001511, A046699, A392671 (denominators).

Cf. A003462 (row sums).

Sequence in context: A292869 A106348 A161092 * A029332 A358172 A344058

Adjacent sequences: A392667 A392668 A392669 * A392671 A392672 A392673

Keyword

nonn,tabf,look,frac

Author

V. V. Muromtsev, Jan 19 2026

Status

approved