cp's OEIS Frontend

A197957 Odd-index Fibonacci partition triangle read by rows.

%I A197957 #28 Nov 18 2018 09:54:44
%S A197957 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 A197957 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 A197957 1,10,53,187,465,778,581,228,62,11,1,1,11,64,250,704
%N A197957 Odd-index Fibonacci partition triangle read by rows.
%C A197957 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 A197957 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 A197957 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 A197957 The triangle A197956 is obtained by taking differences of suitable pairs in neighboring rows of the triangle.
%H A197957 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 A197957 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 A197957 Triangle starts as follows:
%e A197957   1;
%e A197957   1,  1;
%e A197957   1,  2,  1;
%e A197957   1,  3,  4,  1;
%e A197957   1,  4,  8,  5,  1;
%e A197957   1,  5, 13, 17,  6,  1; ...
%K A197957 nonn,tabl
%O A197957 1,5
%A A197957 _Claus Michael Ringel_, Oct 24 2011