A335162 Array read by upward antidiagonals: T(n,k) (n >= 0, k >= 0) = nim k-th power of n.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 2, 1, 1, 0, 1, 5, 6, 1, 2, 1, 0, 1, 6, 7, 14, 3, 3, 1, 0, 1, 7, 5, 13, 5, 2, 1, 1, 0, 1, 8, 4, 8, 4, 2, 1, 2, 1, 0, 1, 9, 13, 10, 7, 2, 8, 3, 3, 1, 0, 1, 10, 12, 14, 6, 3, 10, 11, 2, 1, 1, 0, 1, 11, 14, 10, 10, 3, 13, 9, 7, 1, 2, 1, 0, 1, 12, 15, 13, 11, 1, 14, 15, 6, 10, 3, 3, 1, 0

Offset: 0

Author

N. J. A. Sloane, Jun 08 2020

Keywords

• nonn
• tabl

Comments

Although the nim-addition table (A003987) and nim-multiplication table (A051775) can be found in Conway's "On Numbers and Games", and in the Berlekamp-Conway-Guy "Winning Ways", this exponentiation-table seems to have been omitted.

The n-th row is A212200(n)-periodic. - Rémy Sigrist, Jun 12 2020

Examples

			The array begins:
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, ...,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, ...,
  1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1,  2, 3, ...,
  1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1,  3, 2, ...,
  1, 4, 6,14, 5, 2, 8,11, 7,10, 3,12,13, 9,15, 1,  4, 6, ...,
  1, 5, 7,13, 4, 2,10, 9, 6, 8, 3,15,14,11,12, 1,  5, 7, ...,
  1, 6, 5, 8, 7, 3,13,15, 4,14, 2,11,10,12, 9, 1,  6, 5, ...,
  1, 7, 4,10, 6, 3,14,12, 5,13, 2, 9, 8,15,11, 1,  7, 4, ...,
  1, 8,13,14,10, 1, 8,13,14,10, 1, 8,13,14,10, 1,  8,13, ...,
  1, 9,12,10,11, 2,14, 4,15,13, 3, 7, 8, 5, 6, 1,  9,12, ...,
  1,10,14,13, 8, 1,10,14,13, 8, 1,10,14,13, 8, 1, 10,14, ...,
  1,11,15, 8, 9, 2,13, 5,12,14, 3, 6,10, 4, 7, 1, 11,15, ...,
  1,12,11,14,15, 3, 8, 6, 9,10, 2, 4,13, 7, 5, 1, 12,11, ...,
  1,13,10, 8,14, 1,13,10, 8,14, 1,13,10, 8,14, 1, 13,10, ...,
  1,14, 8,10,13, 1,14, 8,10,13, 1,14, 8,10,13, 1, 14, 8, ...,
  1,15, 9,13,12, 3,10, 7,11, 8, 2, 5,14, 6, 4, 1, 15, 9, ...
  ...
The initial antidiagonals are:
  [1]
  [1,  0]
  [1,  1,  0]
  [1,  2,  1,  0]
  [1,  3,  3,  1,  0]
  [1,  4,  2,  1,  1,  0]
  [1,  5,  6,  1,  2,  1,  0]
  [1,  6,  7, 14,  3,  3,  1,  0]
  [1,  7,  5, 13,  5,  2,  1,  1,  0]
  [1,  8,  4,  8,  4,  2,  1,  2,  1,  0]
  [1,  9, 13, 10,  7,  2,  8,  3,  3,  1,  0]
  [1, 10, 12, 14,  6,  3, 10, 11,  2,  1,  1,  0]
  [1, 11, 14, 10, 10,  3, 13,  9,  7,  1,  2,  1,  0]
  [1, 12, 15, 13, 11,  1, 14, 15,  6, 10,  3,  3,  1,  0]
  ...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..20300
• J. H. Conway, Integral lexicographic codes, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.
• Rémy Sigrist, PARI program for A335162
• Index entries for sequences related to Nim-multiplication

Crossrefs

Cf. A003987, A051775, A059971, A212200, A223541.

Rows: for nim-powers of 4 through 10 see A335163-A335169.

Columns: for nim-squares, cubes, fourth, fifth, sixth, seventh and eighth powers see A006042, A335170, A335535, A335171, A335172, A335173 and A335536.

Programs

PARI

See Links section.

Formula

From Rémy Sigrist, Jun 12 2020: (Start)

T(n, A212200(n)) = 1 for any n > 0.

T(n, n) = A059971(n).

(End)