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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). 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 (list; graph; refs; listen; history; text; internal format)
 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 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S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished. Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission] Alex Meadows, B. Putman, A New Twist on Wythoff's Game, arXiv preprint arXiv:1606.06819 [math.CO], 2016.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 Wythoffâ€™s game, pages 377-410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009. 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] 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

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Last modified November 28 23:44 EST 2020. Contains 338755 sequences. (Running on oeis4.)