A197956 -id:A197956 - cp's OEIS frontend

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

Claus Michael Ringel, Oct 24 2011

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.

			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; ...

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

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A197957 Odd-index Fibonacci partition triangle read by rows.

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