A165162 Triangle T(n,m) with 2n-1 entries per row, read by rows: the first n entries count down from n to 1, the remaining n-1 entries down from n-1 to 1.

1, 2, 1, 1, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Author

Paul Curtz, Sep 06 2009

Keywords

• nonn
• easy
• tabf

Comments

Arose in a study of saddle-point quantities (see A057058 and references therein).

In conjunction with denominators defined in A165200 this constitutes a triangle of fractions:

1;

2,1/2,1/4;

3,2/2,1/3,2/6,1/9;

4,3/2,2/3,1/4,3/8,2/12,1/16;

Examples

			1;
2,1,1;
3,2,1,2,1;
4,3,2,1,3,2,1;
5,4,3,2,1,4,3,2,1;
		

References

• P. Curtz, Stabilite locale des systemes quadratiques. Ann. sc. Ecole Normale Sup., 1980, 293-302.

Programs

Mathematica

Flatten[ Table[ Range[k, 1, -1], {n, 1, 10}, {k, {n, n-1}}]] (* Jean-François Alcover, Aug 02 2012 *)

Formula

T(n,m) = n-m+1 for 1 <= m <= n. T(n,m) = 2n-m for n< m <= 2n-1. [R. J. Mathar, Nov 24 2010]

sum_{m=1..2n-1} T(n,m) = n^2.