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

2,4

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.

Links

Table of n, a(n) for n=2..154.

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.

Example

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,
  ...

Crossrefs

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

Sequence in context: A194528 A194520 A082727 * A112603 A097294 A060848

Adjacent sequences: A264594 A264595 A264596 * A264598 A264599 A264600

Keyword

nonn,tabl

Author

N. J. A. Sloane, Nov 19 2015

Status

approved