A358650 - OEIS

%I #14 Dec 04 2022 01:43:54

%S 1,2,4,6,12,18,42,78,156,234,546,1014,2886,4758,14118,30966,61932,

%T 92898,216762,402558,1145742,1888926,5604846,12293502,28210026,

%U 45860646,121727346,249864654,813198126,1423166394,4740553974,11234495766,22468991532,33703487298

%N Matula-Goebel tree number of the binomial tree of n vertices.

%C 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.

%C 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)).

%H Kevin Ryde, <a href="/A358650/b358650.txt">Table of n, a(n) for n = 1..117</a>

%H Kevin Ryde, <a href="/A358650/a358650.gp.txt">PARI/GP Code</a>

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

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

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

%e The tree of n=13 vertices numbered 0..12 is

%e   0

%e   | \  \    \

%e   1  2  4     8

%e      |  | \   | \  \

%e      3  5  6  9 10  12

%e            |     |

%e            7    11

%e Vertices 0..7 are the binomial tree of 2^k = 8 vertices, and vertices 8..12 are the binomial tree of 5 vertices.

%e Using the recurrence, a(13) = a(8 + 5) = a(8) * prime(a(5)) = 78*37 = 2886.

%o (PARI) See links.

%Y Cf. A076146, A348067.

%K nonn,easy

%O 1,2

%A _Kevin Ryde_, Nov 25 2022