2,2

Let B(a,b) be the tree obtained by joining the centers of two stars with a and b leaves.

For n >= 2, define a = ceil((n-2)/2), b = floor((n-2)/2). Then a(n) is the number of graceful labelings of B(a,b).

A graceful labeling assigns labels 0,1,...,n-1 to the vertices so that all edge differences are distinct.

Douglas B. West, Introduction to Graph Theory, Pearson Education Pte. Ltd, 2002, page 81.

Igor Blokhin, Graceful Prüfer Codes (Python repository).

For n=4 there are 2 such labelings: 1--2--0--3 and 0--3--1--2, so a(4)=2.

Cf. A033472.

Sequence in context: A228315 A120690 A165124 * A288302 A286409 A285610

Adjacent sequences: A394754 A394755 A394758 * A394761 A394762 A394763

nonn,hard,more,new

Igor Blokhin, Mar 31 2026

approved