A383333 -id:A383333 - cp's OEIS frontend

A383331 Number of pairs of nonnegative integers, not both equal to 0, with a shortest vectorial addition chain of length n.

Original entry on oeis.org

2, 3, 7, 16, 37, 91, 229, 585, 1528, 4034, 10862

Offset: 0

Pontus von Brömssen, Apr 26 2025

See A383330 for details.

Row 2 of A383333.

Cf. A003065, A383330, A383332.

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.

Original entry on oeis.org

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

Pontus von Brömssen, Apr 26 2025

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.

			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).

Cf. A383333.

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

cp's OEIS Frontend

A383331 Number of pairs of nonnegative integers, not both equal to 0, with a shortest vectorial addition chain of length n.

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