

A197957


Oddindex Fibonacci partition triangle read by rows.


1



1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 5, 1, 1, 5, 13, 17, 6, 1, 1, 6, 19, 35, 24, 7, 1, 1, 7, 26, 60, 77, 32, 8, 1, 1, 8, 34, 93, 162, 117, 41, 9, 1, 1, 9, 43, 135, 288, 364, 167, 51, 10, 1, 1, 10, 53, 187, 465, 778, 581, 228, 62, 11, 1, 1, 11, 64, 250, 704
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OFFSET

1,5


COMMENTS

The numbers d(i,n) in the row with index n are recursively defined for 1 <= n and 0 <= i < n, by d(0,n) = 1 = d(n1,n) for all n, and d(i,n) = 2d(i1,n1) + d(i,n1)  d(i1,n2) for 0 < i <= n/2, and d(i,n) = d(i1,n1) + 2d(i,n1)  d(i1,n2) for n/2 < i < n.
The numbers d(i,n1) and d(i,n) form the dimension vector of the Fibonacci modules R(n), these are indecomposable quiver representations of the 3regular tree with bipartite orientation.
A linear combination of the row n (with all coefficients of the form 2^t) gives a partition of the Fibonacci number f_{2n+1} (A000045, A001519).
The triangle A197956 is obtained by taking differences of suitable pairs in neighboring rows of the triangle.


LINKS

Table of n, a(n) for n=1..71.
Philipp Fahr and Claus Michael Ringel, The Fibonacci partition triangles, arXiv:1109.2849 [math.CO], 2011.
P. Fahr, C. M. Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, J. Int. Seq. 15 (2012) # 12.2.1


EXAMPLE

Triangle starts as follows:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 8, 5, 1;
1, 5, 13, 17, 6, 1; ...


CROSSREFS

Sequence in context: A112564 A244911 A258309 * A089899 A092422 A096465
Adjacent sequences: A197954 A197955 A197956 * A197958 A197959 A197960


KEYWORD

nonn,tabl


AUTHOR

Claus Michael Ringel, Oct 24 2011


STATUS

approved



