

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.
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
There is a 7state 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

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S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]
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



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



KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



