1,3

Also number of partitions of F(n+2) whose highest term is F(n+1) ( or, which is the same, whose number of terms is F(n+1)). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 14 2007

Divide the set of partitions P(i,j) in two subsets : 1) Partitions containing at least one term 1; Deleting a term 1, we prove that their number is P(i-1,j-1) 2). Subtracting 1 from each term of the other partitions we prove that their number is P(i-j,j) Hence P(i,j) - P(i-1,j-1) = P(i-j,j) Replacing successively in this formula i by i-1 and j by j-1 and summing all these equalities we get, if j>= floor((i+1)/2) P(i,j)=sum ({k,1,j}P(i-j;k))= A000041(i-j) As for i=F(n+2) and j=F(n+1) the condition is satisfied : P(F(n+2),F(n+1)) = P (F(n+2),F(n+1)= A000041(n) = 1072214(n) - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 14 2007

Let P(i,j) denote the number of partitions of i whose highest term is j A072214(n) = A000041(F(n)) = P(F(n+2),F(n+1)) - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 14 2007

F(5) = 5, F(4) =3: 5 = 3+2 = 3+1+1 (or 5=3+1+1=2+2+1), then P(5,3) = 2 = A000041(2)= A000041(F(3))=A072214(3)

Table[ PartitionsP[ Fibonacci[n]], {n, 1, 17}]

Cf. A000041, A000045.

Sequence in context: A094062 A038159 A077210 this_sequence A007660 A134412 A005115

Adjacent sequences: A072211 A072212 A072213 this_sequence A072215 A072216 A072217

nonn

Jeff Burch (gburch(AT)erols.com), Jul 03 2002

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 06 2002