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A197956 Even-index Fibonacci partition triangle read by rows. 1
1, 1, 1, 2, 1, 3, 1, 4, 7, 1, 5, 12, 1, 6, 18, 29, 1, 7, 25, 53, 1, 8, 33, 85, 130, 1, 9, 42, 126, 247, 1, 10, 52, 177, 414, 611, 1, 11, 63, 239, 642, 1192, 1, 12, 75, 313, 943, 2062, 2965, 1, 13, 88, 400, 1330, 3306, 5897, 1, 14, 102, 501, 1817, 5023, 10447, 14726 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

These are the entries of a triangle which starts

           1,

           1,

       1,  2,

       1,  3,

   1,  4,  7,

   1,  5, 12,

1, 6, 18, 29,...

The numbers d(i,n) in the row with index n are recursively defined for 0 <= n and 0 <= i <= n/2, by d(0,n) = 1 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,2i) = 3d(i-1,n-1) - d(i-1,n-2).

The numbers d(i,n-1) and d(i,n) form the dimension vector of the Fibonacci modules P(n), these are indecomposable quiver representations of the 3-regular tree with bipartite orientation.

A linear combination of the row n (with coefficients either 1 or of the form 3*2^t) gives a partition of the Fibonacci number f_{2n+2}, see A000045 and A001906.

The sequence A177011 is obtained by reading the rows with even index from the right.

The sequence A177020 is obtained by reading the rows with odd index from the right.

The sequence of the entries in the last column of the triangle, with even row index is recorded as A132262, the sequence of the entries in the last column of the triangle, with odd row index is A110122 (Number of Delannoy paths of length n with no EE's crossing the line y=x)

The sequence A197957 is obtained by taking differences of pairs of numbers in neighboring rows of the triangle.

LINKS

Table of n, a(n) for n=1..64.

Philipp Fahr and Claus Michael Ringel, The Fibonacci partition triangles, http://arxiv.org/abs/1109.2849

P. Fahr, C. M. Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, J. Int. Seq. 15 (2012) # 12.2.1

CROSSREFS

Sequence in context: A082904 A034868 A050382 * A054072 A325642 A110977

Adjacent sequences:  A197953 A197954 A197955 * A197957 A197958 A197959

KEYWORD

nonn,tabf

AUTHOR

Claus Michael Ringel, Oct 24 2011

STATUS

approved

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Last modified July 9 22:42 EDT 2020. Contains 335555 sequences. (Running on oeis4.)