

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).


22



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
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OFFSET

0,2


COMMENTS

(n,a(n)) are Wythoff pairs: (0,0), (1,2), (3,5), (4,7), ..., where each difference occurs once.
Selfinverse 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


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: nonattacking 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S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]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, B. Putman, A New Twist on Wythoff's Game, arXiv preprint arXiv:1606.06819 [math.CO], 2016.
Gabriel Nivasch, More on the SpragueGrundy function for Wythoffâ€™s game, pages 377410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009.
R. Silber, Wythoff's Nim and Fibonacci Representations, Fibonacci Quarterly #14 (1977), pp. 8588.
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.
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



