A354322 Irregular triangle read by rows where row n lists the distinct Matula-Goebel numbers of terminal subtrees occurring in the tree with Matula-Goebel number n.

1, 1, 2, 1, 2, 3, 1, 4, 1, 2, 3, 5, 1, 2, 6, 1, 4, 7, 1, 8, 1, 2, 9, 1, 2, 3, 10, 1, 2, 3, 5, 11, 1, 2, 12, 1, 2, 6, 13, 1, 4, 14, 1, 2, 3, 15, 1, 16, 1, 4, 7, 17, 1, 2, 18, 1, 8, 19, 1, 2, 3, 20, 1, 2, 4, 21, 1, 2, 3, 5, 22, 1, 2, 9, 23, 1, 2, 24, 1, 2, 3, 25

Offset: 1

Author

Kevin Ryde, Jun 08 2022

Keywords

• nonn
• tabf

Comments

A terminal subtree is a vertex and all its descendents.

Row n has length A317713(n).

Row n begins with 1 which is a singleton (single childless vertex), and ends with n itself which is the whole tree.

The second-last term in row n >= 1 is the largest (by tree number) child subtree of the root, which is A061395(n).

A factor of 2 in a tree number is a singleton child, and tree number 2^c is a vertex with c singleton children and no other children.

The second term in each row is T(n,2) = 2^c for the subtree with the fewest singleton children and no other children.

A rooted star is n = 2^c and these are the only rows of length 2.

A path of k vertices down is the prime-th recurrence n = A007097(k-1) and its subtrees are row(n) = A007097(0 .. k-1).

Examples

			Triangle begins:
      k=1  2  3  4
  n=1:  1,
  n=2:  1, 2,
  n=3:  1, 2, 3,
  n=4:  1, 4,
  n=5:  1, 2, 3, 5,
  n=6:  1, 2, 6,
  n=7:  1, 4, 7,
For n=78, tree 78 and its subtree numbers are
      78
    / | \
   1  2  6      distinct tree numbers
      |  | \    row(78) = {1, 2, 6, 78}
      1  1  2
            |
            1
		

Links

• Kevin Ryde, PARI/GP Code
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A317713 (row lengths), A061395 (second last each row).

Cf. A007097 (path).

Programs

PARI

\\ See links.

Formula

row(n) = union of row(primepi(p)) for each p a prime factor of n, followed by n itself.