A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

2, 5, 5, 10, 10, 10, 17, 17, 17, 17, 26, 26, 26, 26, 26, 37, 37, 37, 37, 37, 37, 50, 50, 50, 50, 50, 50, 50, 65, 65, 65, 65, 65, 65, 65, 65, 82, 82, 82, 82, 82, 82, 82, 82, 82, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101

Offset: 1

Author

Craig Knecht, Dec 02 2015

Keywords

• nonn
• tabl

Comments

The natural numbers can sequentially fill a right- or left-handed equilateral triangle. Componentwise addition of the values of these two triangles produces the present triangle.

The row sums for this triangle give A034262.

The difference between the right- and left-handed triangles produces A049581.

Examples

			Displayed as a triangle:
   2;
   5  5;
  10 10 10;
  17 17 17 17;
  26 26 26 26 26;
  37 37 37 37 37 37;
  ...
		

Links

• Craig Knecht, Sum of right and left triangles.
• Craig Knecht, Difference between right and left triangles A049581.

Crossrefs

Column k=1 gives A002522.

Cf. A049581 (difference of triangles), A034262 (row sum of triangle), A069894 (center column).

Cf. A071237.

Programs

Maple

seq(seq(n^2+1,k=1..n),n=1..10); # Georg Fischer, Oct 01 2021

Formula

T(n,k) = n^2 + 1 for k = 1..n and n >= 1. - Georg Fischer, Oct 01 2021

Sum_{k=1..n} k * T(n,k) = A071237(n). - Alois P. Heinz, Oct 01 2021

Extensions

Row 6 with T(6,k)=37 inserted by Georg Fischer, Oct 01 2021