cp's OEIS Frontend

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.

%I A354322 #10 Aug 27 2024 18:30:48
%S A354322 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,
%T A354322 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,
%U A354322 2,4,21,1,2,3,5,22,1,2,9,23,1,2,24,1,2,3,25
%N 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.
%C A354322 A terminal subtree is a vertex and all its descendents.
%C A354322 Row n has length A317713(n).
%C A354322 Row n begins with 1 which is a singleton (single childless vertex), and ends with n itself which is the whole tree.
%C A354322 The second-last term in row n >= 1 is the largest (by tree number) child subtree of the root, which is A061395(n).
%C A354322 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.
%C A354322 The second term in each row is T(n,2) = 2^c for the subtree with the fewest singleton children and no other children.
%C A354322 A rooted star is n = 2^c and these are the only rows of length 2.
%C A354322 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).
%H A354322 Kevin Ryde, <a href="/A354322/a354322.gp.txt">PARI/GP Code</a>
%H A354322 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A354322 row(n) = union of row(primepi(p)) for each p a prime factor of n, followed by n itself.
%e A354322 Triangle begins:
%e A354322       k=1  2  3  4
%e A354322   n=1:  1,
%e A354322   n=2:  1, 2,
%e A354322   n=3:  1, 2, 3,
%e A354322   n=4:  1, 4,
%e A354322   n=5:  1, 2, 3, 5,
%e A354322   n=6:  1, 2, 6,
%e A354322   n=7:  1, 4, 7,
%e A354322 For n=78, tree 78 and its subtree numbers are
%e A354322       78
%e A354322     / | \
%e A354322    1  2  6      distinct tree numbers
%e A354322       |  | \    row(78) = {1, 2, 6, 78}
%e A354322       1  1  2
%e A354322             |
%e A354322             1
%o A354322 (PARI) \\ See links.
%Y A354322 Cf. A317713 (row lengths), A061395 (second last each row).
%Y A354322 Cf. A007097 (path).
%K A354322 nonn,tabf
%O A354322 1,3
%A A354322 _Kevin Ryde_, Jun 08 2022