|
|
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).
|
|
23
|
|
|
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
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
|
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S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, 43 pages, no date, unpublished.
Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, Wythoff Wisdom, unpublished, no date [Cached copy, with permission]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
|
_Michael Kleber_
|
|
EXTENSIONS
|
Edited by _Christian G. Bower_, Oct 29 2002
|
|
STATUS
|
approved
|
|
|
|