A018219 -id:A018219 - cp's OEIS frontend

A051261 Main diagonal of A018219, i.e., (n, n, a_n) is a winning position.

0, 4, 6, 1, 3, 7, 11, 2, 5, 10, 17, 23, 21, 9, 28, 26, 24, 8, 34, 12, 35, 13, 32, 29, 44, 19, 45, 15, 18, 14, 57, 56, 16, 20, 58, 48, 25, 54, 73, 76, 72, 27, 75, 82, 62, 66, 84, 87, 71, 78, 95, 31, 74, 94, 97, 103, 22, 40, 39, 30, 86, 33, 41, 91, 46, 42, 117, 119, 122, 37

Offset: 0

Michael Kleber

Index entries for sequences that are permutations of the natural numbers

Cf. A018219, Inverse is A077226.

mex[s_] := Min[Complement[Range[0, Max[{s, -1}] + 1], Flatten[s]]];
f[s_] := Join[s, s + Table[i, {i, Length[s]}]];
T[n_, k_] := T[n, k] = mex[{f[Table[T[n - i, k], {i, n}]], f[Table[T[n, k - i], {i, k}]], f[Table[T[n - i, k - i], {i, Min[n, k]}]]}];
a[n_] := T[n, n];
Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Aug 19 2019, from A018219 Mma code *)

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

A018220 Row 1 of A018219, i.e., (1, n, a_n) is a winning position.

Original entry on oeis.org

2, 4, 0, 3, 1, 6, 5, 12, 15, 17, 14, 20, 7, 19, 10, 8, 28, 9, 31, 13, 11, 32, 38, 33, 41, 43, 40, 42, 16, 49, 51, 18, 21, 23, 53, 59, 61, 63, 22, 66, 26, 24, 27, 25, 67, 75, 68, 78, 76, 29, 79, 30, 85, 34, 88, 87, 93, 95, 97, 35, 100, 36, 98, 37, 99, 106, 39, 44, 46, 111, 114

Offset: 0

Michael Kleber

nonn

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n.

Index entries for sequences that are permutations of the natural numbers

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

A018222 Row 3 of A018219, i.e., (3,n,a_n) is a winning position.

Original entry on oeis.org

5, 3, 8, 1, 4, 0, 9, 11, 2, 6, 18, 7, 13, 12, 27, 22, 25, 29, 10, 33, 30, 38, 15, 39, 35, 16, 44, 14, 43, 17, 20, 50, 54, 19, 57, 24, 56, 62, 21, 23, 64, 69, 63, 28, 26, 74, 76, 73, 75, 81, 31, 87, 86, 88, 32, 93, 36, 34, 91, 90, 101, 100, 37, 42, 40, 102, 106, 110, 112, 41

Offset: 0

Michael Kleber

nonn

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n.

Index entries for sequences that are permutations of the natural numbers

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

A018221 Row 2 of A018219, i.e., (2,n,a_n) is a winning position.

Original entry on oeis.org

1, 0, 6, 8, 10, 12, 2, 7, 3, 11, 4, 9, 5, 16, 24, 23, 13, 26, 29, 34, 32, 35, 40, 15, 14, 38, 17, 47, 44, 18, 53, 48, 20, 52, 19, 21, 57, 65, 25, 61, 22, 66, 69, 67, 28, 71, 78, 27, 31, 79, 84, 82, 33, 30, 83, 90, 89, 36, 99, 95, 97, 39, 100, 103, 107, 37, 41, 43, 110, 42

Offset: 0

Michael Kleber

nonn

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n.

Index entries for sequences that are permutations of the natural numbers

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

A002251 Start with the nonnegative integers; then swap L(k) and U(k) for all k >= 1, where L = A000201, U = A001950 (lower and upper Wythoff sequences).

Original entry on oeis.org

0, 2, 1, 5, 7, 3, 10, 4, 13, 15, 6, 18, 20, 8, 23, 9, 26, 28, 11, 31, 12, 34, 36, 14, 39, 41, 16, 44, 17, 47, 49, 19, 52, 54, 21, 57, 22, 60, 62, 24, 65, 25, 68, 70, 27, 73, 75, 29, 78, 30, 81, 83, 32, 86, 33, 89, 91, 35, 94, 96, 37, 99, 38, 102, 104, 40, 107, 109

Offset: 0

Michael Kleber

(n,a(n)) are Wythoff pairs: (0,0), (1,2), (3,5), (4,7), ..., where each difference occurs once.
Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001
If the offset is 1, the sequence can also be obtained by rearranging the natural numbers so that sum of n terms is a multiple of n, or equivalently so that the arithmetic mean of the first n terms is an integer. - Amarnath Murthy, Aug 16 2002
For n = 1, 2, 3, ..., let p(n)=least natural number not already an a(k), q(n) = n + p(n); then a(p(n)) = q(n), a(q(n)) = p(n). - Clark Kimberling
Also, indices of powers of 2 in A086482. - Amarnath Murthy, Jul 26 2003
There is a 7-state Fibonacci automaton (see a002251_1.pdf) that accepts, in parallel, the Zeckendorf representations of n and a(n). - Jeffrey Shallit, Jul 14 2023

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

T. D. Noe, Table of n, a(n) for n = 0..10000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 18.
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), #P1.52.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]
Alex Meadows and B. Putman, A New Twist on Wythoff's Game, arXiv preprint arXiv:1606.06819 [math.CO], 2016.
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.
Jeffrey Shallit, Automaton for A002251
Jeffrey Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], 2023.
R. Silber, Wythoff's Nim and Fibonacci Representations, Fibonacci Quarterly #14 (1977), pp. 85-88.
N. J. A. Sloane, Scatterplot of first 100 terms [The points are symmetrically placed about the diagonal, although that is hard to see here because the scales on the axes are different]
Index entries for sequences that are permutations of the natural numbers

The sequence maps between A000201 and A001950, in that a(A000201(n)) = A001950(n), a(A001950(n)) = A000201(n).
Row 0 of A018219.
Cf. A073869, A342297.

With[{n = 42}, {0}~Join~Take[Values@ #, LengthWhile[#, # == 1 &] &@ Differences@ Keys@ #] &@ Sort@ Flatten@ Map[{#1 -> #2, #2 -> #1} & @@ # &, Transpose@ {Array[Floor[# GoldenRatio] &, n], Array[Floor[# GoldenRatio^2] &, n]}]] (* Michael De Vlieger, Nov 14 2017 *)

A002251_upto(N,c=0,A=Vec(0,N))={for(n=1,N, A[n]||(#AA002251[1]=2, a(0)=0 is not included. - M. F. Hasler, Nov 27 2019, replacing earlier code from Sep 17 2014

a(n) = A019444(n+1) - 1.

Edited by Christian G. Bower, Oct 29 2002

A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.

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

Offset: 0

N. J. A. Sloane

T(a,b) = T(b,a).

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..5049
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.
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), #P1.52.
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).
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.
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 A004481

A004482-A004487 are rows 1 to 6. Cf. A047708 (main diagonal).
See A317205 for triangle of values on or below main diagonal.
Similar to but different from A004489.
T(a, b)=0 iff A018219(a, b)=0 iff A002251(a)=b.

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}];
Flatten@Table[t[n - m + 1, m], {n, 11}, {m, n}] (* Birkas Gyorgy, Apr 19 2011 *)

PARI
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See Links section.
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A077226 Inverse of A051261, i.e., (n, a(n), a(n)) is a winning position in 3-pile Wythoff game.

Original entry on oeis.org

0, 3, 7, 4, 1, 8, 2, 5, 17, 13, 9, 6, 19, 21, 29, 27, 32, 10, 28, 25, 33, 12, 56, 11, 16, 36, 15, 41, 14, 23, 59, 51, 22, 61, 18, 20, 73, 69, 84, 58, 57, 62, 65, 81, 24, 26, 64, 89, 35, 100, 75, 78, 76, 109, 37, 116, 31, 30, 34, 111, 126, 92, 44, 98, 86, 124, 45, 121, 94, 99

Offset: 0

Christian G. Bower, Oct 29 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A018219.

cp's OEIS Frontend

A051261 Main diagonal of A018219, i.e., (n, n, a_n) is a winning position.

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A018220 Row 1 of A018219, i.e., (1, n, a_n) is a winning position.

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A018222 Row 3 of A018219, i.e., (3,n,a_n) is a winning position.

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A018221 Row 2 of A018219, i.e., (2,n,a_n) is a winning position.

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A002251 Start with the nonnegative integers; then swap L(k) and U(k) for all k >= 1, where L = A000201, U = A001950 (lower and upper Wythoff sequences).

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A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.

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A077226 Inverse of A051261, i.e., (n, a(n), a(n)) is a winning position in 3-pile Wythoff game.

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