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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.
3
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
1,3
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).
FORMULA
row(n) = union of row(primepi(p)) for each p a prime factor of n, followed by n itself.
EXAMPLE
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
PROG
(PARI) \\ See links.
CROSSREFS
Cf. A317713 (row lengths), A061395 (second last each row).
Cf. A007097 (path).
Sequence in context: A122087 A139642 A264744 * A358532 A143604 A021475
KEYWORD
nonn,tabf
AUTHOR
Kevin Ryde, Jun 08 2022
STATUS
approved