A383334 Square array read by antidiagonals: T(n,k) is the smallest positive weight of an n-tuple of nonnegative integers with a shortest vectorial addition chain of length k; n >= 1, k >= 0.

1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 7, 4, 3, 2, 1, 11, 6, 4, 3, 2, 1, 19, 8, 5, 4, 3, 2, 1, 29, 12, 7, 5, 4, 3, 2, 1, 47, 20, 9, 6, 5, 4, 3, 2, 1, 71, 29, 13, 8, 6, 5, 4, 3, 2, 1, 127, 44, 20, 10, 7, 6, 5, 4, 3, 2, 1, 191, 70, 30, 14, 9, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Author

Pontus von Brömssen, Apr 26 2025

Keywords

• nonn
• tabl

Comments

See A383333 for details.

T(n,k) is the smallest positive degree of a monomial x_1^e_1*...*x_n^e_n that requires k multiplications, given x_1, ..., x_n.

Examples

			Array begins:
  n\k| 0  1  2  3  4  5  6  7  8
  ---+--------------------------
  1  | 1  2  3  5  7 11 19 29 47
  2  | 1  2  3  4  6  8 12 20 29
  3  | 1  2  3  4  5  7  9 13 20
  4  | 1  2  3  4  5  6  8 10 14
  5  | 1  2  3  4  5  6  7  9 11
The smallest positive weight of a triple of nonnegative integers with a shortest addition chain of length 8 is T(3,8) = 20. Up to permutations, (3,4,13) is the only such triple, with a shortest addition chain [(1,0,0), (0,1,0), (0,0,1),] (0,0,2), (0,0,4), (0,1,4), (1,1,4), (2,2,8), (3,3,12), (3,3,13), (3,4,13).
		

Crossrefs

Cf. A383333.

Rows: A003064 (n=1), A383332 (n=2).