A051775 -id:A051775 - cp's OEIS frontend

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

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

Offset: 0

Marc LeBrun

Another way to construct the array: construct an infinite square matrix starting in the top left corner using the rule that each entry is the smallest nonnegative number that is not in the row to your left or in the column above you.
After a few moves the [symmetric] matrix looks like this:
  0 1 2 3 4 5 ...
  1 0 3 2 5 ...
  2 3 0 1 ?
  3 2 1
  4 5 ?
  5
The ? is then replaced with a 6.

			Table begins
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, ...
   2,  3,  0,  1,  6,  7,  4,  5, 10, 11,  8, ...
   3,  2,  1,  0,  7,  6,  5,  4, 11, 10, ...
   4,  5,  6,  7,  0,  1,  2,  3, 12, ...
   5,  4,  7,  6,  1,  0,  3,  2, ...
   6,  7,  4,  5,  2,  3,  0, ...
   7,  6,  5,  4,  3,  2, ...
   8,  9, 10, 11, 12, ...
   9,  8, 11, 10, ...
  10, 11,  8, ...
  11, 10, ...
  12, ...
  ...
The first few antidiagonals are
   0;
   1,  1;
   2,  0,  2;
   3,  3,  3,  3;
   4,  2,  0,  2,  4;
   5,  5,  1,  1,  5,  5;
   6,  4,  6,  0,  6,  4,  6;
   7,  7,  7,  7,  7,  7,  7,  7;
   8,  6,  4,  6,  0,  6,  4,  6,  8;
   9,  9,  5,  5,  1,  1,  5,  5,  9,  9;
  10,  8, 10,  4,  2,  0,  2,  4, 10,  8, 10;
  11, 11, 11, 11,  3,  3,  3,  3, 11, 11, 11, 11;
  12, 10,  8, 10, 12,  2,  0,  2, 12, 10,  8, 10, 12;
  ...
[Symmetric] matrix in base 2:
     0    1   10   11  100  101,  110  111 1000 1001 1010 1011 ...
     1    0   11   10  101  100,  111  110 1001 1000 1011  ...
    10   11    0    1  110  111,  100  101 1010 1011  ...
    11   10    1    0  111  110,  101  100 1011  ...
   100  101  110  111    0    1    10   11  ...
   101  100  111  110    1    0    11  ...
   110  111  100  101   10   11   ...
   111  110  101  100   11  ...
  1000 1001 1010 1011  ...
  1001 1000 1011  ...
  1010 1011  ...
  1011  ...
   ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From N. J. A. Sloane, Jul 14 2009]
R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.

T. D. Noe, Rows n = 0..100 of triangle, flattened
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Rémy Sigrist, Colored representation of T(x,y) for x = 0..1023 and y = 0..1023 (where the hue is function of T(x,y) and black pixels correspond to zeros)
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, The OEIS: A Fingerprint File for Mathematics, arXiv:2105.05111 [math.HO], 2021. Mentions this sequence.
Index entries for sequences related to Nim-sums

Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.
Cf. A003986 (OR), A004198 (AND), A221146 (carries).
Antidiagonal sums are in A006582.

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:
# alternative:
read("transforms") :
A003987 := proc(n,m)
    XORnos(n,m) ;
end proc: # R. J. Mathar, Apr 17 2013
seq(seq(Bits:-Xor(k,m-k),k=0..m),m=0..20); # Robert Israel, Dec 31 2015

Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (* BitXor and Nim Sum are equivalent *)

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitxor(k, n - k),", ");); print(););};
tabl(13) \\ Indranil Ghosh, Mar 31 2017

for n in range(14):
    print([k^(n - k) for k in range(n + 1)]) # Indranil Ghosh, Mar 31 2017

T(2i,2j) = 2T(i,j), T(2i+1,2j) = 2T(i,j) + 1.

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

Original entry on oeis.org

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

N. J. A. Sloane, Jun 08 2020

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

			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]
  ...

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

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.

PARI
```
See Links section.
```

From Rémy Sigrist, Jun 12 2020: (Start)
T(n, A212200(n)) = 1 for any n > 0.
T(n, n) = A059971(n).
(End)

A051776 Table T(n,m) = Nim-product of n and m, read by antidiagonals, for n >= 1, m >= 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 19 1999

			Table begins:
  1  2  3  4  5  6 ...
  2  3  1  8 10 11 ...
  3  1  2 12 15 13 ...
  4  8 12  6  2 14 ...

J. H. Conway, On Numbers and Games, Academic Press, p. 52.

R. J. Mathar, Table of n, a(n) for n = 1..1830
H. W. Lenstra, Nim multiplication, (1978)
Index entries for sequences related to Nim-multiplication

Cf. A051775, A003987, A051910.

We continue from A003987: to compute a Nim-multiplication table using (a) an addition table AT := array(0..NA, 0..NA) and (b) a nimsum procedure for larger values; MT := array(0..N,0..N); for a from 0 to N do MT[a,0] := 0; MT[0,a] := 0; MT[a,1] := a; MT[1,a] := a; od: for a from 2 to N do for b from a to N do t1 := {}; for i from 0 to a-1 do for j from 0 to b-1 do u1 := MT[i,b]; u2 := MT[a,j];
if u1<=NA and u2<=NA then u12 := AT[u1,u2]; else u12 := nimsum(u1,u2); fi; u3 := MT[i,j]; if u12<=NA and u3<=NA then u4 := AT[u12,u3]; else u4 := nimsum(u12,u3); fi; t1 := { op(t1), u4}; #t1 := { op(t1), AT[ AT[ MT[i,b], MT[a,j] ], MT[i,j] ] }; od; od;
t2 := sort(convert(t1,list)); j := nops(t2); for i from 1 to nops(t2) do if t2[i] <> i-1 then j := i-1; break; fi; od; MT[a,b] := j; MT[b,a] := j; od; od;

T(n,m) = A051775(n,m).

A053398 Nim-values from game of Kopper's Nim.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 3, 3, 3, 3, 3, 3, 3, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 0, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 0, 2, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1

Offset: 1

David W. Wilson

Rows/columns 1-10 are A007814, A050603, A053399, A053384-A053890.

Comment from R. K. Guy: David Singmaster (zingmast(AT)sbu.ac.uk) sent me, about 5 years ago, a game he'd received from Bodo Koppers. It is played with two heaps of beans. The move is to remove one heap and split the other into two nonempty heaps. I'm not sure if Koppers invented it, or got it from elsewhere. I do not think that he analyzed it, but Singmaster did.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A007814, A050603, A053399, A053384-A053890.

Cf. A003986, A007814 (both edges & central terms & minima per row), A000523 (max per row), A245836 (row sums), A003987, A051775.

a053398 :: Int -> Int -> Int
a053398 n k = a007814 $ a003986 (n - 1) (k - 1) + 1
a053398_row n = map (a053398 n) [1..n]
a053398_tabl = map a053398_row [1..]
-- Reinhard Zumkeller, Aug 04 2014

a(x, y) = place of last zero bit of (x-1) OR (y-1).

T(n,k) = A007814(A003986(n-1,k-1)+1). - Reinhard Zumkeller, Aug 04 2014

A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 8, 8, 8, 8, 16, 12, 6, 12, 16, 32, 32, 11, 11, 32, 32, 64, 48, 64, 13, 64, 48, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 192, 96, 192, 24, 192, 96, 192, 256, 512, 512, 176, 176, 44, 44, 176, 176, 512, 512, 1024, 768

Offset: 0

Tilman Piesk, Mar 21 2013

Nimber multiplication is commutative, so this array is symmetric, and can be represented in a more compact way by the rows of the lower triangle (A223540).
The diagonal is A006017 (nim-squares of powers of 2).
The elements of this array are listed in A223543. In the key-matrix A223542 the entries of this array (which become very large) are replaced by the corresponding index numbers of A223543. (Surprisingly, the key-matrix seems to be interesting on its own.)
The number of different entries per antidiagonal is probably A002487. That would mean that there are exactly A002487(n+1) numbers that can be expressed as a nim-product(2^a,2^b) with a+b=n. - Tilman Piesk, Mar 27 2013

			T(1,7) = T(3,5) = 192, the result of the nim-multiplications 2^1*2^7 and 2^3*2^5.
The array begins:
1,2,4,8,16,32,64,128,256,...
2,3,8,12,32,48,128,192,512,...
4,8,6,11,64,128,96,176,1024,...
8,12,11,13,128,192,176,208,2048,...
16,32,64,128,24,44,75,141,4096,...
32,48,128,192,44,52,141,198,8192,...
64,128,96,176,75,141,103,185,16384,...
128,192,176,208,141,198,185,222,32768,...
256,512,1024,2048,4096,8192,16384,32768,384,...
...

J. H. Conway, "Integral lexicographic codes." Discrete Mathematics 83.2-3 (1990): 219-235. See Table 4.

Tilman Piesk, First 128 rows of the matrix, flattened
Tilman Piesk, Elements of dual matrix (256 SVGs)
Tilman Piesk, Walsh permutation; nimber multiplication (Wikiversity)
Tilman Piesk, Class bin and function nimprod (MATLAB code)

Cf. A051775, A223540, A006017 (main diagonal), A223543, A223542, A000079, A002487.

For rows 1,2,3,4, see A134683, A335159, A335160, A335161.

T(m,n) = A051775(A000079(m),A000079(n)).

T(m,n) = A223543(A223542(m,n)).

Edited by N. J. A. Sloane, Jun 08 2020

A212200 Multiplicative order of n in nim-multiplication.

Original entry on oeis.org

1, 3, 3, 15, 15, 15, 15, 5, 15, 5, 15, 15, 5, 5, 15, 85, 85, 255, 255, 85, 85, 255, 255, 85, 85, 255, 255, 255, 255, 85, 85, 255, 255, 255, 255, 85, 255, 85, 255, 255, 255, 255, 255, 255, 85, 255, 85, 255, 85, 85, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 85, 85, 255, 255, 51, 255, 255, 255, 51, 255, 255, 17, 255, 85, 255, 17, 255, 85

Offset: 1

N. J. A. Sloane, May 03 2012

nonn

For n <= 255, computed using R. J. Mathar's Maple programs from A051775. a(256) = 21845 from J. H. Conway and Alex Ryba, May 04 2012
Apparently, all terms belong to A001317, and A001317(k) appears 2^k times. - Rémy Sigrist, Jun 14 2020
From Jianing Song, Aug 10 2022: (Start)
The observation above is incorrect. Note that {0,1,...,2^2^k-1} together with the nim operations makes a field isomorphic to GF(2^2^k). This means that:
- Every number is a divisor of a number of the form 2^2^k-1, and every divisor of 2^2^k-1 for some k appears;
- If d is a divisor of 2^2^k-1 for some k, then d appears phi(d) times among {a(1),a(2),...,a(2^2^m-1)} for all m >= k, phi = A000010. This means that if d > 1, and k is the smallest number such that d | 2^2^k-1, then d can only appear among {a(2^2^(k-1)),...a(2^2^k-1)}.
So the correct result should be: all terms are divisors of numbers of the form 2^2^k-1, and each divisor d appears phi(d) times.
For example, 641 would appear 640 times in this sequence, among {a(2^32),...,a(2^64-1)}, although to determine their positions is hard. (End)

			The nim-products 4*4*...*4 are (cf. A051775): 4, 4^2=6, 4^3=4*6=14, 4^4=4*14=5, 4^5=2, 4^6=8, ..., 4^14=15, 4^15=1, so 4 has order a(4) = 15.

J. H. Conway, On Numbers and Games, Academic Press, Chapter 6.

Alex Ryba, Table of n, a(n) for n = 1..65999

Cf. A001317, A051775, A051776, A212201, A212202, A212203, A212204.

A006042 The nim-square of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

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).

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

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.

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

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_{iFor 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)

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

A004468 a(n) = Nim product 3 * n.

Original entry on oeis.org

0, 3, 1, 2, 12, 15, 13, 14, 4, 7, 5, 6, 8, 11, 9, 10, 48, 51, 49, 50, 60, 63, 61, 62, 52, 55, 53, 54, 56, 59, 57, 58, 16, 19, 17, 18, 28, 31, 29, 30, 20, 23, 21, 22, 24, 27, 25, 26, 32, 35, 33, 34, 44, 47, 45, 46, 36, 39, 37, 38, 40, 43, 41, 42, 192, 195, 193, 194, 204, 207, 205

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 3,1,2.
This is a permutation of the natural numbers; A006015 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

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

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 3 of array in A051775.

Cf. A006015, A004469-A004480, A000695, A062880, A001196.

read("transforms") ;
# insert Maple procedures nimprodP2() and A051775() of the b-file in A051775 here.
A004468 := proc(n)
        A051775(3,n) ;
end proc:
L := [seq(A004468(n),n=0..1000)] ; # R. J. Mathar, May 28 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 3, 1, 2][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 3, 1, 2}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,3,1,2]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 3, 1, 2]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

a(n) = A051775(3,n).
From Jianing Song, Aug 10 2022: (Start)
a(n) = 3*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 3*n.
a(n) = n/2 if n has only digits 0 or 2 in quaternary (n is in A062880). Otherwise, a(n) > n/2.
a(n) = 2*n/3 if and only if n has only digits 0 or 3 in quaternary (n is in A001196). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=2} 4^i, then a(n) = 2*n/3 if and only if 3*A + B = 2/3*(A + 2*B), or B = 7*A. If A != 0, then A is of the form (4*s+1)*4^t, but 7*A is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 3. (End)

More terms from Erich Friedman

A006015 Nim product 2*n.

Original entry on oeis.org

0, 2, 3, 1, 8, 10, 11, 9, 12, 14, 15, 13, 4, 6, 7, 5, 32, 34, 35, 33, 40, 42, 43, 41, 44, 46, 47, 45, 36, 38, 39, 37, 48, 50, 51, 49, 56, 58, 59, 57, 60, 62, 63, 61, 52, 54, 55, 53, 16, 18, 19, 17, 24, 26, 27, 25, 28, 30, 31, 29, 20, 22, 23, 21, 128, 130, 131, 129, 136, 138, 139

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 2,3,1.
This is a permutation of the natural numbers; A004468 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

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).

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 2 of array in A051775.

Cf. A004468-A004480, A000695, A062880, A001196.

a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 2, 3, 1][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,2,3,1]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 2, 3, 1]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

From Jianing Song, Aug 10 2022: (Start)
a(n) = A051775(2,n).
a(n) = 2*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 2*n.
a(n) = n/3 if n has only digits 0 or 3 in quaternary (n is in A001196). Otherwise, a(n) > n/3.
a(n) = 3*n/2 if and only if n has only digits 0 or 2 in quaternary (n is in A062880). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=3} 4^i, then a(n) = 3*n/2 if and only if 2*A + B = 3/2*(A + 3*B), or A = 7*B. If B != 0, then B is of the form (4*s+1)*4^t, but 7*B is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 2. (End)

More terms from Erich Friedman.

A051910 Triangle T(n,m) = Nim-product of n and m, read by rows, 0<=m<=n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 20 1999

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

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games, Academic Press, p. 52.

R. J. Mathar, Table of n, a(n) for n = 0..3320
Index entries for sequences related to Nim-multiplication

Cf. A051776, A003987, A051775, A051911.

We continue from A003987: to compute a Nim-multiplication table using (a) an addition table AT := array(0..NA, 0..NA) and (b) a nimsum procedure for larger values; MT := array(0..N,0..N); for a from 0 to N do MT[a,0] := 0; MT[0,a] := 0; MT[a,1] := a; MT[1,a] := a; od: for a from 2 to N do for b from a to N do t1 := {}; for i from 0 to a-1 do for j from 0 to b-1 do u1 := MT[i,b]; u2 := MT[a,j];
if u1<=NA and u2<=NA then u12 := AT[u1,u2]; else u12 := nimsum(u1,u2); fi; u3 := MT[i,j]; if u12<=NA and u3<=NA then u4 := AT[u12,u3]; else u4 := nimsum(u12,u3); fi; t1 := { op(t1), u4}; #t1 := { op(t1), AT[ AT[ MT[i,b], MT[a,j] ], MT[i,j] ] }; od; od;
t2 := sort(convert(t1,list)); j := nops(t2); for i from 1 to nops(t2) do if t2[i] <> i-1 then j := i-1; break; fi; od; MT[a,b] := j; MT[b,a] := j; od; od;

T(n,m) = A051775(n,m) = A051776(n,m).

Showing 1-10 of 29 results. Next

cp's OEIS Frontend

A003987 Table of n XOR m (or Nim-sum of n and m) read by antidiagonals with m>=0, n>=0.

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

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A051776 Table T(n,m) = Nim-product of n and m, read by antidiagonals, for n >= 1, m >= 1.

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A053398 Nim-values from game of Kopper's Nim.

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A223541 Array T(m,n) = nim-product(2^m,2^n) (m>=0, n>=0) read by antidiagonals.

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A212200 Multiplicative order of n in nim-multiplication.

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A006042 The nim-square of n.

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