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A264597 Triangle T(n,m) (n >= 2, 0 <= m <= n-1) read by rows, defined in Comments.

%I A264597 #22 Nov 16 2024 18:49:00
%S A264597 1,1,1,3,2,1,4,2,1,1,7,3,2,1,1,7,3,1,1,1,1,12,4,3,2,1,1,1,8,2,1,1,1,1,
%T A264597 1,1,20,8,4,2,2,1,1,1,1,13,3,2,2,1,1,1,1,1,1,18,4,3,2,2,2,1,1,1,1,1,
%U A264597 18,6,2,2,1,1,1,1,1,1,1,1,31,8,6,4,3,2,2,1,1,1,1,1,1,20,6,3,1,1,1,1,1,1,1,1,1,1,1,31,8,4,3,3,2,2,2,1,1,1,1,1,1,1,24,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,39,10,6,4,3,2,2,2,2,1,1,1,1,1,1,1,1
%N A264597 Triangle T(n,m) (n >= 2, 0 <= m <= n-1) read by rows, defined in Comments.
%C A264597 We work with 2 X 2 matrices X = [a,b; c,d] with integer entries. Let M be the free monoid generated by L = [1,0; 1,1] and R = [1,1; 0,1]. Let t be the map [a,b; c,d] -> (a+b)/(c+d). Then T(n,m) is the number of X in M with trace(X)=n and m <= t(X) < m+1.
%H A264597 M. P. Technau, <a href="http://www.mathematik.uni-wuerzburg.de/~steuding/technau.pdf">The Calkin-Wilf Tree and a Trace Condition</a>, Master's Thesis, Julius-Maximilian’s University of Würzburg, Faculty for Mathematics and Computer Science, 2015.
%e A264597 Triangle begins:
%e A264597    1,
%e A264597    1, 1,
%e A264597    3, 2, 1,
%e A264597    4, 2, 1, 1,
%e A264597    7, 3, 2, 1, 1,
%e A264597    7, 3, 1, 1, 1, 1,
%e A264597   12, 4, 3, 2, 1, 1, 1,
%e A264597    8, 2, 1, 1, 1, 1, 1, 1,
%e A264597   20, 8, 4, 2, 2, 1, 1, 1, 1,
%e A264597   13, 3, 2, 2, 1, 1, 1, 1, 1, 1,
%e A264597   18, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1,
%e A264597   18, 6, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1,
%e A264597   31, 8, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1,
%e A264597   20, 6, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e A264597   ...
%Y A264597 Row sums are A264598. It appears that the row sums are also twice A257007.
%K A264597 nonn,tabl
%O A264597 2,4
%A A264597 _N. J. A. Sloane_, Nov 19 2015