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A197957 Odd-index Fibonacci partition triangle read by rows. 1

%I #28 Nov 18 2018 09:54:44

%S 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,

%T 60,77,32,8,1,1,8,34,93,162,117,41,9,1,1,9,43,135,288,364,167,51,10,1,

%U 1,10,53,187,465,778,581,228,62,11,1,1,11,64,250,704

%N Odd-index Fibonacci partition triangle read by rows.

%C 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.

%C 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.

%C 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).

%C The triangle A197956 is obtained by taking differences of suitable pairs in neighboring rows of the triangle.

%H Philipp Fahr and Claus Michael Ringel, <a href="http://arxiv.org/abs/1109.2849">The Fibonacci partition triangles</a>, arXiv:1109.2849 [math.CO], 2011.

%H P. Fahr, C. M. Ringel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Ringel/ringel10.html">Categorification of the Fibonacci Numbers Using Representations of Quivers</a>, J. Int. Seq. 15 (2012) # 12.2.1

%e Triangle starts as follows:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 4, 1;

%e 1, 4, 8, 5, 1;

%e 1, 5, 13, 17, 6, 1; ...

%K nonn,tabl

%O 1,5

%A _Claus Michael Ringel_, Oct 24 2011

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)