A358650 Matula-Goebel tree number of the binomial tree of n vertices.

1, 2, 4, 6, 12, 18, 42, 78, 156, 234, 546, 1014, 2886, 4758, 14118, 30966, 61932, 92898, 216762, 402558, 1145742, 1888926, 5604846, 12293502, 28210026, 45860646, 121727346, 249864654, 813198126, 1423166394, 4740553974, 11234495766, 22468991532, 33703487298

Offset: 1

Author

Kevin Ryde, Nov 25 2022

Keywords

• nonn
• easy

Comments

The top-down definition of the binomial tree suits Matula-Goebel numbering: The tree of n = 2^k + r vertices, for 1 <= r <= 2^k is the binomial tree of 2^k vertices and a child subtree under the root which is the binomial tree of r vertices.

In the tree of n vertices, adding a new singleton child under each vertex gives the tree of 2*n vertices, so that a(2*n) = A348067(a(n)).

Examples

			The tree of n=13 vertices numbered 0..12 is
  0
  | \  \    \
  1  2  4     8
     |  | \   | \  \
     3  5  6  9 10  12
           |     |
           7    11
Vertices 0..7 are the binomial tree of 2^k = 8 vertices, and vertices 8..12 are the binomial tree of 5 vertices.
Using the recurrence, a(13) = a(8 + 5) = a(8) * prime(a(5)) = 78*37 = 2886.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..117
• Kevin Ryde, PARI/GP Code
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A076146, A348067.

Programs

PARI

\\ See links.

Formula

a(2^k + r) = a(2^k) * prime(a(r)) for 1 <= r <= 2^k.

a(2^k) = A076146(k+1), being a tree of order k.