A108225 a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.

0, 2, 3, 5, 12, 68, 2280, 2598062, 3374961778893, 5695183504492614029263280, 16217557574922386301420536972254869595782763547562

Offset: 0

Author

N. J. A. Sloane, Jun 16 2005

Keywords

• nonn

Comments

From a posting by Antreas P. Hatzipolakis to the Yahoo news group "Hyacinthos", circa Jun 10 2005.

The next term has 99 digits. - Harvey P. Dale, Jun 09 2011

a(n) for n>0 gives the rank of the unlabeled binary rooted tree, among those with n+1 leaves, that has the largest rank according to the bijection of Colijn and Plazzotta (2018) between unlabeled binary rooted trees and positive integers. - Noah A Rosenberg, Jun 03 2022

Links

• Robert Israel, Table of n, a(n) for n = 0..14
• C. Colijn and G. Plazzotta, A metric on phylogenetic tree shapes, Syst. Biol., 67 (2018), 113-126.
• Luc Devroye, Michael R. Doboli, Noah A. Rosenberg, and Stephan Wagner, Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees, arXiv:2409.18956 [math.CO], 2024. See p. 3.
• N. A. Rosenberg, On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees, Discr. Appl. Math., 291 (2021), 88-98.
• J.S. Seneschal, Iteration of Semi-Complete Graphs

Crossrefs

First differences give A103410.

Cf. A006894.

Cf. A000217, A007501, A002658, A072638.

Programs

Maple

F:=proc(n) option remember; if n <= 1 then RETURN(2*n) fi; (F(n-1)+F(n-2))*(F(n-1)-F(n-2)+1)/2; end;
a[ -2]:=-2:a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n]+2, n=-2..8); # Zerinvary Lajos, Jun 08 2007

Mathematica

RecurrenceTable[{a[0]==0,a[1]==2,a[n]==(a[n-1]+a[n-2])(a[n-1]- a[n-2]+1)/2},a[n],{n,15}] (* Harvey P. Dale, Jun 09 2011 *)

Formula

Conjecture: a(n) = A006894(n) + 1. - R. J. Mathar, Apr 23 2007

From J.S. Seneschal, Jul 17 2025 (Start)

a(n) = A000217(a(n)) - A072638(n) = A072638(n-1) + 2.

a(n) = A002658(n-1) + a(n-1) for n > 1. (End)