A352289 a(1) = 1 and thereafter a(n) = 2*prime(a(n-1)).

1, 4, 14, 86, 886, 13766, 298154, 8455786, 300427382, 12942000398, 659492202274, 38995629272042, 2634767648759954, 200877694833442486, 17101872791349773894, 1611548301646589127698, 166820830882144877428234

Offset: 1

Author

Kevin Ryde, Mar 16 2022

Keywords

• nonn
• more
• hard

Comments

In Matula-Goebel tree codes, a(n) is a rooted caterpillar consisting of a path of n-1 internal vertices down, and n childless vertices under them so each has exactly 2 children.

Mir, Rosselló, and Rotger show that among phylogenic trees (meaning series-reduced, no vertex with just 1 child) with n childless vertices, tree a(n) has the largest total cophenetic index A352288(a(n)) = binomial(n,3).

Examples

			For n=3, a(3) = 14 is the Matula-Goebel code of the following tree
  root  14
       /  \     tree numbers of subtrees shown,
      4    1    with "1" being childless,
     / \        and n=3 of those
    1   1
		

Links

• Lucas A. Brown, Python program.
• Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [q-bio.PE], 2012.

Crossrefs

Cf. A352288 (total cophenetic index).

Cf. A331937.

Programs

Mathematica

NestList[2 Prime[#] &, 1, 10] (* Michael De Vlieger, Apr 18 2022 *)

PARI

a(n) = my(ret=1); for(i=2,n, ret=2*prime(ret)); ret;

Python

from functools import lru_cache
from sympy import prime
@lru_cache(maxsize=None)
def A352289(n): return 1 if n == 1 else 2*prime(A352289(n-1)) # Chai Wah Wu, Apr 18 2022

Extensions

a(10)-a(15) from Daniel Suteu, Mar 19 2022

a(16) and a(17) from Lucas A. Brown, Sep 04 2024