A335535 -id:A335535

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A335162

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

+10
19

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

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

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

FORMULA

From Rémy Sigrist, Jun 12 2020: (Start)

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

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

(End)

EXAMPLE

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]

...

PROG

(PARI) See Links section.

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.

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 08 2020

STATUS

approved

A006042

The nim-square of n.
(Formerly M2251)

+10
6

0, 1, 3, 2, 6, 7, 5, 4, 13, 12, 14, 15, 11, 10, 8, 9, 24, 25, 27, 26, 30, 31, 29, 28, 21, 20, 22, 23, 19, 18, 16, 17, 52, 53, 55, 54, 50, 51, 49, 48, 57, 56, 58, 59, 63, 62, 60, 61, 44, 45, 47, 46, 42, 43, 41, 40, 33, 32, 34, 35, 39, 38, 36, 37, 103, 102, 100, 101, 97, 96, 98, 99

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

This is a permutation of the natural numbers; A160679 is the inverse permutation. - Jianing Song, Aug 10 2022

REFERENCES

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..1000

G. P. Michon, Discussion of Conway's On2 [From John W. Layman, Nov 05 2010]

Index entries for sequences related to Nim-multiplication

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = A051775(n,n).

From Jianing Song, Aug 10 2022: (Start)

If n = Sum_j 2^e(j), then a(n) is the XOR of A006017(e(j))'s. Proof: let N+ = XOR and N* denote the nim addition and the nim multiplication, then n N* n = (Sum_j 2^e(j)) N* (Sum_j 2^e(j)) = (Nim-sum_j 2^e(j)) N* (Nim-sum_j 2^e(j)) = (Nim-sum_j (2^e(j) N* 2^e(j))) N+ (Nim-sum_{i<j} ((2^e(i) N* 2^e(j)) N+ (2^e(j) N* 2^e(i)))) = (Nim-sum_j (2^e(j) N* 2^e(j))) N+ (Nim-sum_{i<j} 0) = Nim-sum_j (2^e(j) N* 2^e(j)).

For example, for n = 11 = 2^0 + 2^1 + 2^3, a(11) = A006017(0) XOR A006017(1) XOR A006017(3) = 1 XOR 3 XOR 13 = 15.

More generally, if n = Sum_j 2^e(j), k is a power of 2, then the nim k-th power of n is the XOR of (nim k-th power of 2^e(j))'s. (End)

MAPLE

read("transforms") ;

# insert source of nimprodP2() and A051775() from the b-file at A051776 here...

A006042 := proc(n)

A051775(n, n) ;

end proc:

L := [seq( A006042(n), n=1..1000) ]; # R. J. Mathar, May 28 2011

CROSSREFS

Diagonal of A051775. Without 0, diagonal of A051776.

Column 2 of array in A335162.

Other nim k-th powers: A051917 (k=-1), A160679 (k=1/2), A335170 (k=3), A335535 (k=4), A335171 (k=5), A335172 (k=6), A335173 (k=7), A335536 (k=8).

Cf. A212200, A006017.

KEYWORD

nonn,nice,easy,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(1)-a(49) confirmed, a(50)-a(71) added by John W. Layman, Nov 05 2010

a(0) prepended by Jianing Song, Aug 10 2022

STATUS

approved

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