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(n-1,n) for all n, and d(i,n) = 2d(i-1,n-1) + d(i,n-1) - d(i-1,n-2) for 0 < i <= n/2, and d(i,n) = d(i-1,n-1) + 2d(i,n-1) - d(i-1,n-2) for n/2 < i < n.
The numbers d(i,n-1) and d(i,n) form the dimension vector of the Fibonacci modules R(n), these are indecomposable quiver representations of the 3-regular 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
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
KEYWORD
nonn,tabl
AUTHOR
Claus Michael Ringel, Oct 24 2011
STATUS
approved