A335162 - OEIS

A335162

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

%I #40 Aug 25 2022 03:39:07

%S 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,

%T 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,

%U 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

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

%C 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.

%C The n-th row is A212200(n)-periodic. - _Rémy Sigrist_, Jun 12 2020

%H Rémy Sigrist, <a href="/A335162/b335162.txt">Table of n, a(n) for n = 0..20300</a>

%H J. H. Conway, <a href="https://doi.org/10.1016/0012-365X(90)90008-6">Integral lexicographic codes</a>, Discrete Mathematics 83.2-3 (1990): 219-235. See Table 3.

%H Rémy Sigrist, <a href="/A335162/a335162.gp.txt">PARI program for A335162</a>

%H <a href="/index/Ni#Nimmult">Index entries for sequences related to Nim-multiplication</a>

%F From _Rémy Sigrist_, Jun 12 2020: (Start)

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

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

%F (End)

%e The array begins:

%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...,

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...,

%e 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, ...,

%e 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, ...,

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

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

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

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

%e 1, 8,13,14,10, 1, 8,13,14,10, 1, 8,13,14,10, 1, 8,13, ...,

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

%e 1,10,14,13, 8, 1,10,14,13, 8, 1,10,14,13, 8, 1, 10,14, ...,

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

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

%e 1,13,10, 8,14, 1,13,10, 8,14, 1,13,10, 8,14, 1, 13,10, ...,

%e 1,14, 8,10,13, 1,14, 8,10,13, 1,14, 8,10,13, 1, 14, 8, ...,

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

%e ...

%e The initial antidiagonals are:

%e [1]

%e [1, 0]

%e [1, 1, 0]

%e [1, 2, 1, 0]

%e [1, 3, 3, 1, 0]

%e [1, 4, 2, 1, 1, 0]

%e [1, 5, 6, 1, 2, 1, 0]

%e [1, 6, 7, 14, 3, 3, 1, 0]

%e [1, 7, 5, 13, 5, 2, 1, 1, 0]

%e [1, 8, 4, 8, 4, 2, 1, 2, 1, 0]

%e [1, 9, 13, 10, 7, 2, 8, 3, 3, 1, 0]

%e [1, 10, 12, 14, 6, 3, 10, 11, 2, 1, 1, 0]

%e [1, 11, 14, 10, 10, 3, 13, 9, 7, 1, 2, 1, 0]

%e [1, 12, 15, 13, 11, 1, 14, 15, 6, 10, 3, 3, 1, 0]

%e ...

%o (PARI) See Links section.

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

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

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

%K nonn,tabl

%O 0,8

%A _N. J. A. Sloane_, Jun 08 2020