A197957 Odd-index Fibonacci partition triangle read by rows.

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

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

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