A004481 -id:A004481 - cp's OEIS frontend

A004482 Tersum n + 1 (answer recorded in base 10).

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

Offset: 0

N. J. A. Sloane

Tersum m + n: write m and n in base 3 and add mod 3 with no carries; e.g., 5 + 8 = "21" + "22" = "10" = 1.

Sprague-Grundy values for game of Wyt Queens.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), Article P1.52.
Andreas Dress, Achim Flammenkamp, and Norbert Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff's game, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

This sequence is row 1 of table A004481.
a(n) = A061347(n+1) + n.
Third column of triangle A296339.

LinearRecurrence[{1,0,1,-1},{1,2,0,4},70] (* or *) Table[3*Floor[n/3]+ Mod[ n+1,3],{n,0,70}] (* Harvey P. Dale, Nov 29 2014 *)

Periodic with period 3 and saltus 3: a(n) = 3*floor(n/3) + ((n+1) mod 3).
a(n) = n - 2*cos(2*(n+1)*Pi/3). - Wesley Ivan Hurt, Sep 29 2017
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/2 - log(2)/3. - Amiram Eldar, Aug 21 2023

More terms from Erich Friedman

A004489 Table of tersums m + n (answers written in base 10).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			Table begins:
  0 1 2 3 4 5 6 ...
  1 2 0 4 5 3 7 ...
  2 0 1 5 3 4 8 ...
  3 4 5 6 7 8 0 ...
  4 5 3 7 8 6 1 ...
  5 3 4 8 6 7 2 ...
  6 7 8 0 1 2 3 ...
  ...

Alois P. Heinz, Antidiagonals d = 0..140, flattened

Similar to but different from A004481.
Main diagonal gives A004488.
Cf. A003987 (analogous sequence for base 2).

T:= proc(n, m) local t, h, r, i;
      t, h, r:= n, m, 0;
      for i from 0 while t>0 or h>0 do
        r:= r +3^i *irem(irem(t, 3, 't') +irem(h, 3, 'h'), 3)
      od; r
    end:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 07 2011

T[n_, m_] := Module[{t, h, r, i, remt, remh}, {t, h, r} = {n, m, 0}; For[i = 0, t>0 || h>0, i++, r = r + 3^i*Mod[({t, remt} = QuotientRemainder[t, 3 ]; remt) + ({h, remh} = QuotientRemainder[h, 3]; remh), 3]]; r]; Table[Table[T[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Jan 07 2014, translated from Maple *)

T(n,m) = fromdigits(Vec(Pol(digits(n,3)) + Pol(digits(m,3)))%3, 3); \\ Kevin Ryde, Apr 06 2021

def T(n, m):
  k, pow3 = 0, 1
  while n + m > 0:
    n, rn = divmod(n, 3)
    m, rm = divmod(m, 3)
    k, pow3 = k + pow3*((rn+rm)%3), pow3*3
  return k
print([T(n, d-n) for d in range(14) for n in range(d+1)]) # Michael S. Branicky, May 04 2021

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g. 5 + 8 = "21" + "22" = "10" = 1.

More terms from Larry Reeves (larryr(AT)acm.org), Jan 23 2001

A018219 Table T(a,b) by antidiagonals of winning positions in 3-pile Wythoff game (a square array).

Original entry on oeis.org

0, 2, 2, 1, 4, 1, 5, 0, 0, 5, 7, 3, 6, 3, 7, 3, 1, 8, 8, 1, 3, 10, 6, 10, 1, 10, 6, 10, 4, 5, 12, 4, 4, 12, 5, 4, 13, 12, 2, 0, 3, 0, 2, 12, 13, 15, 15, 7, 9, 11, 11, 9, 7, 15, 15, 6, 17, 3, 11, 15, 7, 15, 11, 3, 17, 6, 18, 14, 11, 2, 0, 1, 1, 0, 2, 11, 14, 18, 20, 20, 4, 6, 19, 5, 11, 5, 19, 6

Offset: 0

Michael Kleber

(a,b,T(a,b)) are the winning positions in 3-pile Wythoff game. A move in k-pile Wythoff is: pick a subset of the k piles and remove the same number of stones from each. Goal: take the last stone.

T(a,b) = T(b,a). If T(a,b)=c then T(a,c)=b and T(b,c)=a.

			0 2 1 5 7 ...
2 4 0 3 1 ...
1 0 6 8 10 ...
5 3 8 1 4 ...
7 1 10 4 3 ...
T(1,1)=4, since from (114) your opponent can move to (113),(112),(111),(110),(014),(013),(004),(003). You can either win or move to (012) and win a move later.

Rows 0-3: A002251, A018220-A018222. Main diagonal: A051261.

T(a, b)=0 iff A004481(a, b)=0 iff A002251(a)=b.

mex[ s_ ] := Min[ Complement[ Range[ 0, Max[ {s, -1} ]+1 ], Flatten[ s ] ] ]; f[ s_ ] := Join[ s, s+Table[ i, {i, Length[ s ]} ] ]; T[ a_, b_ ] := T[ a, b ] = mex[ { f[ Table[ T[ a-i, b ], {i, a} ] ], f[ Table[ T[ a, b-i ], {i, b} ] ], f[ Table[ T[ a-i, b-i ], {i, Min[ a, b ]} ] ] } ]

Edited and extended by Christian G. Bower, Oct 29 2002

A307296 Array read by antidiagonals: Sprague-Grundy values for the game NimHof with 4 rules [1,0], [3,2], [1,1], [0,1].

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 12 2019

The game NimHof with a list of rules R means that for each rule [a,b] you can move from cell [x,y] to any cell [x-i*a,y-i*b] as long as neither coordinate is negative. See the Friedman et al. article for further details.

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

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?

Rémy Sigrist, Colored representation of T(x,y) for x = 0..999 and y = 0..999 (where the hue is function of T(x,y) and black pixels correspond to zeros)
Rémy Sigrist, PARI program for A307296
N. J. A. Sloane, Maple program for NimHof sequences

List of NimHof sequences:
A-number    Rules R
A003987  [1,0], [0,1]
A004481  [1,0], [1,1], [0,1]
A003987  [1,0], [2,1], [0,1]
A307300  [1,0], [2,2], [0,1]
A307301  [1,0], [3,1], [0,1]
A003987  [1,0], [3,2], [0,1]
A307302  [1,0], [3,3], [0,1]
A307299  [1,0], [1,1], [1,2], [0,1]
A307296  [1,0], [1,1], [3,2], [0,1]
A307297  [1,0], [2,1], [3,3], [0,1]
A307298  [1,0], [1,1], [1,2], [2,3], [0,1]

PARI
```
\\ See Links section.
```

A004483 Tersum n + 2.

Original entry on oeis.org

2, 0, 1, 5, 3, 4, 8, 6, 7, 11, 9, 10, 14, 12, 13, 17, 15, 16, 20, 18, 19, 23, 21, 22, 26, 24, 25, 29, 27, 28, 32, 30, 31, 35, 33, 34, 38, 36, 37, 41, 39, 40, 44, 42, 43, 47, 45, 46, 50, 48, 49, 53, 51, 52, 56, 54, 55, 59, 57, 58, 62, 60, 61, 65, 63, 64, 68, 66, 67

Offset: 0

N. J. A. Sloane

Tersum m + n: write m and n in base 3 and add mod 3 with no carries; e.g., 5 + 8 = "21" + "22" = "10" = 1.

Also Sprague-Grundy values for game of Wyt Queens.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

Andreas Dress, Achim Flammenkamp, and Norbert Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

This sequence is row 2 of table A004481.

Second column of triangle in A296339.

a[n_] := If[Divisible[n, 3], n+2, n-1]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 25 2013 *)
LinearRecurrence[{1,0,1,-1},{2,0,1,5},70] (* Harvey P. Dale, Feb 07 2018 *)

Periodic with period and saltus 3: a(n) = 3*floor(n/3) + ((n+2) mod 3).
a(n) = n + 2*cos(2*n*Pi/3). - Wesley Ivan Hurt, Sep 27 2017
From R. J. Mathar, Dec 14 2017: (Start)
G.f.: ( 2+x^2+2*x^3-2*x ) / ( (1+x+x^2)*(x-1)^2 ).
a(n) = n + A099837(n) if n > 0. (End)
Sum_{n>=2} (-1)^n/a(n) = 2*Pi/(3*sqrt(3)) + log(2)/3 - 1/2. - Amiram Eldar, Aug 21 2023

Edited by N. J. A. Sloane at the suggestion of Philippe Deléham, Nov 20 2007

A047708 Diagonal of Sprague-Grundy function for Wyt Queens (Wythoff's game).

Original entry on oeis.org

0, 2, 1, 6, 7, 8, 3, 5, 4, 16, 14, 15, 10, 9, 11, 20, 13, 21, 12, 25, 17, 18, 19, 30, 31, 38, 35, 36, 22, 23, 43, 45, 48, 49, 24, 26, 27, 28, 29, 33, 60, 32, 61, 57, 66, 37, 63, 34, 64, 67, 40, 39, 41, 42, 82, 44, 74, 79, 47, 46, 87, 86, 50, 95, 96, 52, 101, 51, 102, 53, 54

Offset: 0

N. J. A. Sloane

Since Wythoff(m,n) <= m+n, Wythoff(n,n) <= 2n. It is not known whether there is an efficient (linear in log(m)+log(n)) strategy to compute Wythoff(m,n). Each single row is "easy" in the sense that a+n-Wythoff(a,n) is eventually periodic. - Howard A. Landman.
Inverse of sequence A048850 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001
Comments from Howard A. Landman, Nov 24 2007: (Start)
It is impossible for any integer to appear twice in this sequence because of the way it is constructed. Thus to prove that it is a permutation of the integers, we need only show that every value g appears at least once.
Suppose this were not true; then there must be some g such that for any value of n, G(n,n) is not = g. Since G(n,n) is defined to be the smallest number not found as a G(k,n), G(n,k), or G(k,k) for k < n, this can only happen in one of 2 ways: either there is a number g' smaller than g which is chosen (this can occur at most g times) or g already appears as both G(n,k) and G(k,n) for some k < n (because G(n,k) = G(k,n)) (this can happen at most n/2 times).
Thus we have n <= n/2 + g, or n <= 2g; if g has not appeared within the first 2g terms we have a contradiction. Therefore not only must every integer g appear in the sequence, but it must appear within the first 2g terms (and no sooner than term g/2, since G(n,n) <= 2n). Conversely, this also proves that n/2 <= A(n) = G(n,n) <= 2n. (End)

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
Howard A. Landman, "A Simple FSM-Based Proof of the Additive Periodicity of the Sprague-Grundy Function of Wythoff's Game", in R. Nowakowski (ed.), More Games of No Chance.
Howard A. Landman and Tom Ferguson showed that this is a permutation of the integers at the Jul 24-28 2000 MSRI workshop on combinatorial games.
W. A. Wythoff, "A Modification of the Game of Nim". Nieuw Arch. Wiskunde 8, 199-202, 1907/1909.

Rémy Sigrist, Table of n, a(n) for n = 0..4999
H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, pp. 249-270 (1999).
Rémy Sigrist, PARI program for A047708
Index entries for sequences that are permutations of the natural numbers

Main diagonal of square array in A004481. Sequences A000201 and A001950 give the m and n coordinates of the zeros of Wythoff (i.e., the P-positions of the game, where the previous player has won).

Cf. A048850.

mex[list_] :=  mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[Wnim, {a_, b_}] :=  move[Wnim, {a, b}] =
   Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
    Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
SpragueGrundy[game_, list_] :=  SpragueGrundy[game, list] =
   mex[SpragueGrundy[game, #] & /@ move[game, list]];
Table[SpragueGrundy[Wnim, {i, i}], {i, 0, 64}] (* Birkas Gyorgy, Apr 19 2011 *)

PARI
```
See Links section.
```

More terms from Howard A. Landman

A004486 Sprague-Grundy values for game of Wyt Queens.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Inverse of sequence A064208 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, pp. 249-270 (1999).
Index entries for sequences that are permutations of the natural numbers

This sequence is row 5 of table A004481.

More terms from Howard A. Landman

A004487 Sprague-Grundy values for game of Wyt Queens.

Original entry on oeis.org

6, 7, 8, 1, 9, 10, 3, 4, 5, 13, 0, 2, 16, 17, 18, 12, 20, 14, 15, 11, 24, 25, 26, 27, 21, 29, 30, 19, 32, 31, 22, 23, 36, 37, 38, 28, 33, 41, 42, 43, 44, 45, 34, 35, 48, 49, 50, 39, 40, 53, 54, 55, 56, 57, 46, 47, 60, 61, 62, 51, 52, 65

Offset: 0

N. J. A. Sloane

Inverse of sequence A064211 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, pp. 249-270 (1999).
Index entries for sequences that are permutations of the natural numbers

This sequence is row 6 of table A004481.

More terms from Howard A. Landman

A046876 Length of runs in the sequence of row/column periods of Sprague-Grundy values of Wythoff's Game (A046875).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 4, 2, 3, 2, 3, 23, 24, 3, 20, 18, 84, 25, 43

Offset: 1

Achim Flammenkamp

a(20) >= 12. - Sean A. Irvine, May 02 2021

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).

Cf. A046874, A046875, A004481.

Entry revised by Sean A. Irvine, May 02 2021

A317205 Sprague-Grundy values for Wythoff's game.

Original entry on oeis.org

0, 1, 2, 2, 0, 1, 3, 4, 5, 6, 4, 5, 3, 2, 7, 5, 3, 4, 0, 6, 8, 6, 7, 8, 1, 9, 10, 3, 7, 8, 6, 9, 0, 1, 4, 5, 8, 6, 7, 10, 1, 2, 5, 3, 4, 9, 10, 11, 12, 8, 7, 13, 14, 15, 16, 10, 11, 9, 8, 13, 12, 0, 15, 16, 17, 14, 11, 9, 10, 7, 12, 14, 2, 13, 17, 6, 18, 15

Offset: 0

N. J. A. Sloane, Aug 07 2018

			Triangle begins as:
  0;
  1,  2;
  2,  0,  1;
  3,  4,  5,  6;
  4,  5,  3,  2,  7;
  5,  3,  4,  0,  6,  8;
  6,  7,  8,  1,  9, 10,  3;
  7,  8,  6,  9,  0,  1,  4,  5;

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

Georg Fischer, Table of n, a(n) for n = 0..1274 (First 50 rows)
Uri Blass and Aviezri S. Fraenkel, The Sprague-Grundy function for Wythoff's game, Theoretical Computer Science 75.3 (1990): 311-333. See Table 2.
A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).
R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.
Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff's game, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009. See Table 1.

See A004481 for the full table.

mex[list_] := mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[Wnim, {a_, b_}] := move[Wnim, {a, b}] =
   Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
    Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
SpragueGrundy[game_, list_] := SpragueGrundy[game, list] =
   mex[SpragueGrundy[game, #] & /@ move[game, list]];
t[n_, m_] := SpragueGrundy[Wnim, {n - 1, m - 1}]; (* so far copied from A004481 *)
Flatten[Table[t[n, m], {n, 12}, {m,1, n}]] (* Georg Fischer, Feb 22 2020 *)

More terms from Georg Fischer, Feb 22 2020

cp's OEIS Frontend

A004482 Tersum n + 1 (answer recorded in base 10).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A004489 Table of tersums m + n (answers written in base 10).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A018219 Table T(a,b) by antidiagonals of winning positions in 3-pile Wythoff game (a square array).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A307296 Array read by antidiagonals: Sprague-Grundy values for the game NimHof with 4 rules [1,0], [3,2], [1,1], [0,1].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

A004483 Tersum n + 2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A047708 Diagonal of Sprague-Grundy function for Wyt Queens (Wythoff's game).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A004486 Sprague-Grundy values for game of Wyt Queens.

Views

Author

Keywords

Comments

References

Links

Crossrefs