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

0,2

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

Table of n, a(n) for n=0..10.

C. Colijn and G. Plazzotta, A metric on phylogenetic tree shapes, Syst. Biol., 67 (2018), 113-126.

N. A. Rosenberg, On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees, Discr. Appl. Math., 291 (2021), 88-98.

Formula

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

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 *)

Crossrefs

First differences give A103410.

Cf. A006894.

Sequence in context: A266541 A301929 A107475 * A259503 A193064 A133832

Adjacent sequences: A108222 A108223 A108224 * A108226 A108227 A108228

Keyword

nonn

Author

N. J. A. Sloane, Jun 16 2005

Status

approved