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

1,1

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.

Links

Table of n, a(n) for n=1..55.

Craig Knecht, Sum of right and left triangles.

Craig Knecht, Difference between right and left triangles A049581.

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

Example

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;
  ...

Maple

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

Crossrefs

Column k=1 gives A002522.

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

Cf. A071237.

Sequence in context: A173567 A288726 A344572 * A212624 A351475 A034387

Adjacent sequences: A265126 A265127 A265128 * A265130 A265131 A265132

Keyword

nonn,tabl

Author

Craig Knecht, Dec 02 2015

Extensions

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

Status

approved