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A197957 Odd-index 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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)