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

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, 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, 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

Offset: 2

Author

N. J. A. Sloane, Nov 19 2015

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
   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, 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,
  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,
  ...
		

Links

• M. P. Technau, The Calkin-Wilf Tree and a Trace Condition, Master's Thesis, Julius-Maximilian’s University of Würzburg, Faculty for Mathematics and Computer Science, 2015.

Crossrefs

Row sums are A264598. It appears that the row sums are also twice A257007.