A173997 Irregular triangle by columns derived from (1, 2, 3, ...) * (1, 2, 3, ...).

1, 2, 3, 2, 4, 4, 5, 6, 3, 6, 8, 6, 7, 10, 9, 4, 8, 12, 12, 8, 9, 14, 15, 12, 5, 10, 16, 18, 16, 10, 11, 18, 21, 20, 15, 6, 12, 20, 24, 24, 20, 12, 13, 22, 27, 28, 25, 18, 7, 14, 24, 30, 32, 30, 24, 14, 15, 26, 33, 36, 35, 30, 21, 8, 16, 28, 36, 40, 40, 36, 28, 16

Offset: 1

Author

Gary W. Adamson, Mar 05 2010

Keywords

• nonn
• easy
• tabf

Comments

Given a (1, 2, 3, ...) * (1, 2, 3, ...) multiplication table; leftmost column of the triangle = (1, 2, 3, ...). Then shift down each successive column of the array twice to get this irregular triangle.

Examples

			Given:
  1,  2,  3,  4,  5, ...
  2,  4,  6,  8, 10, ...
  3,  6,  9, 12, 15, ...
  4,  8, 12, 16, 20, ...
  ...
After the shift twice operation, we obtain:
   1;
   2;
   3,  2;
   4,  4;
   5,  6,  3;
   6,  8,  6;
   7, 10,  9,  4;
   8, 12, 12,  8;
   9, 14, 15, 12,  5;
  10, 16, 18, 16, 10;
  11, 18, 21, 20, 15,  6;
  12, 20, 24, 24, 20, 12;
  ...
		

Links

• Stefano Spezia, First 200 rows of the triangle, flattened

Crossrefs

Cf. A003991, A006918 (row sums).

Programs

Mathematica

Flatten[Table[k(2-2k+n),{n,16},{k,Floor[(n+1)/2]}]] (* Stefano Spezia, Apr 19 2022 *)

Formula

T(n, k) = k*(2 - 2*k + n), with 1 <= k <= floor((n + 1)/2). - Stefano Spezia, Apr 19 2022