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).

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

0,2

Comments

(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

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

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S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.

Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]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

Formula

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

Mathematica

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 *)

Prog

(PARI) A002251_upto(N, c=0, A=Vec(0, N))={for(n=1, N, A[n]||(#A<A[n]=n+c++)|| A[n+c]=n); A} \\ The resulting vector starts with A002251[1]=2, a(0)=0 is not included. - M. F. Hasler, Nov 27 2019, replacing earlier code from Sep 17 2014

Crossrefs

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.

Sequence in context: A332022 A109261 A085240 * A093545 A293845 A249265

Adjacent sequences: A002248 A002249 A002250 * A002252 A002253 A002254

Keyword

nonn,easy,nice

Author

Michael Kleber

Extensions

Edited by Christian G. Bower, Oct 29 2002

Status

approved