

A000201


Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = A001622.
(Formerly M2322 N0917)


300



1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106, 108, 110
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OFFSET

1,2


COMMENTS

This is the unique sequence a satisfying a'(n)=a(a(n))+1 for all n in the set N of natural numbers, where a' denotes the ordered complement (in N) of a.  Clark Kimberling, Feb 17 2003
This sequence and A001950 may be defined as follows. Consider the maps a > ab, b > a, starting from a(1) = a; then A000201 gives the indices of a, A001950 gives the indices of b. The sequence of letters in the infinite word begins a, b, a, a, b, a, b, a, a, b, a, ... Setting a = 0, b = 1 gives A003849 (offset 0); setting a = 1, b = 0 gives A005614 (offset 0).  Philippe Deléham, Feb 20 2004
These are the numbers whose lazy Fibonacci representation (see A095791) includes 1; the complementary sequence (the upper Wythoff sequence, A001950) are the numbers whose lazy Fibonacci representation includes 2 but not 1.
a(n) is the unique monotonic sequence satisfying a(1)=1 and the condition "if n is in the sequence then n+(rank of n) is not in the sequence" (e.g. a(4)=6 so 6+4=10 and 10 is not in the sequence)  Benoit Cloitre, Mar 31 2006
Write A for A000201 and B for A001950 (the upper Wythoff sequence, complement of A). Then the composite sequences AA, AB, BA, BB, AAA, AAB,...,BBB,... appear in many complementary equations having solution A000201 (or equivalently, A001950). Typical complementary equations: AB=A+B (=A003623), BB=A+2B (=A101864), BBB=3A+5B (=A134864).  Clark Kimberling, Nov 14 2007
Cumulative sum of A001468 terms.  Eric Angelini, Aug 19 2008
The lower Wythoff sequence also can be constructed by playing the socalled Mancalagame: n piles of total d(n) chips are standing in a row. The piles are numbered from left to right by 1, 2, 3, ... . The number of chips in a pile at the beginning of the game is equal to the number of the pile. One step of the game is described as follows: Distribute the pile on the very left one by one to the piles right of it. If chips are remaining, build piles out of one chip subsequently to the right. After f(n) steps the game ends in a constant row of piles. The lower Wythoff sequence is also given by n > f(n).  Roland Schroeder (florola(AT)gmx.de), Jun 19 2010
With the exception of the first term, a(n) gives the number of iterations required to reverse the list {1,2,3,...,n} when using the mapping defined as follows: remove the first term of the list, z(1), and add 1 to each of the next z(1) terms (appending 1's if necessary) to get a new list. See A183110 where this mapping is used and other references given. This appears to be essentially the Mancalatype game interpretation given by R. Schroeder above.  John W. Layman, Feb 03 2011
Also row numbers of A213676 starting with an even number of zeros.  Reinhard Zumkeller, Mar 10 2013
From Jianing Song, Aug 18 2022: (Start)
Numbers k such that {k*phi} > phi^(2), where {} denotes the fractional part.
Proof: Write m = floor(k*phi).
If {k*phi} > phi^(2), take s = mk+1. From m < k*phi < m+1 we have k < (mk+1)*phi < k + phi, so floor(s*phi) = k or k+1. If floor(s*phi) = k+1, then (see A003622) floor((k+1)*phi) = floor(floor(s*phi)*phi) = floor(s*phi^2)1 = s+floor(s*phi)1 = m+1, but actually we have (k+1)*phi > m+phi+phi^(2) = m+2, a contradiction. Hence floor(s*phi) = k.
If floor(s*phi) = k, suppose otherwise that k*phi  m <= phi^(2), then m < (k+1)*phi <= m+2, so floor((k+1)*phi) = m+1. Suppose that A035513(p,q) = k for p,q >= 1, then A035513(p,q+1) = floor((k+1)*phi)  1 = m = A035513(s,1). But it is impossible for one number (m) to occur twice in A035513. (End)


REFERENCES

Eric Friedman, Scott M. Garrabrant, Ilona K. PhippsMorgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?
M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, 1989; see p. 107.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. M. Yaglom, Two games with matchsticks, pp. 17 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
J.P. Allouche, J. Shallit and G. Skordev, Selfgenerating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 115.
Peter G. Anderson, The Fibonacci word as a 2adic number and its continued fraction, Fibonacci Quarterly (2020) Vol. 58, No. 5, 2124.
Joerg Arndt, Matters Computational (The Fxtbook), pp.756757.
Shiri ArtsteinAvidan, Aviezri S. Fraenkel and Vera T. Sos, A twoparameter family of an extension of Beatty, Discr. Math. 308 (2008), 45784588.
Shiri Artsteinavidan, Aviezri S. Fraenkel and Vera T. Sos, A twoparameter family of an extension of Beatty sequences, Discrete Math., 308 (2008), 45784588.
E. J. Barbeau, J. Chew and S. Tanny, A matrix dynamics approach to Golomb's recursion, Electronic J. Combinatorics, #4.1 16 1997.
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139151.
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337386.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190197.
J. H. Conway and N. J. A. Sloane, Notes on the ParaFibonacci and related sequences.
H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135143. [Annotated scanned copy]
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
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.
P. J. Downey and R. E. Griswold, On a family of nested recurrences, Fib. Quart., 22 (1984), 310317.
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. Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 627. [History, references, generalization]
A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353361 (the case a=1).
A. S. Fraenkel, Ratwyt, December 28 2011.
David Garth and Adam Gouge, Affinely SelfGenerating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 113.
M. Griffiths, The Golden String, Zeckendorf Representations, and the Sum of a Series, Amer. Math. Monthly, 118 (2011), 497507.
Martin Griffiths, On a Matrix Arising from a Family of Iterated SelfCompositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
Martin Griffiths, A difference property amongst certain pairs of Beatty sequences, The Mathematical Gazette (2018) Vol. 102, Issue 554, Article 102.36, 348350.
H. Grossman, A set containing all integers, Amer. Math. Monthly, 69 (1962), 532533.
A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
T. Karki, A. Lacroix, and M. Rigo, On the recognizability of selfgenerating sets, JIS 13 (2010) #10.2.2.
Clark Kimberling, A SelfGenerating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 11 (2008) 08.3.3.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.
Clark Kimberling, Problem Proposals, The Fibonacci Quarterly, vol. 52 #5, 2015, p514.
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).
C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267273.
Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319337.[See A317208 for a link.]
U. Larsson and N. Fox, An Aperiodic Subtraction Game of NimDimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.
R. J. Mathar, Graphical representation among sequences closely related to this one (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
D. J. Newman, Problem 3117, Amer. Math. Monthly, 34 (1927), 158159.
D. J. Newman, Problem 5252, Amer. Math. Monthly, 72 (1965), 11441145.
Gabriel Nivasch, More on the SpragueGrundy function for Wythoff’s game, pages 377410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009.
R. J. Nowakowski, Generalizations of the LangfordSkolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
Michel Rigo, Invariant games and nonhomogeneous Beatty sequences, Slides of a talk, Journée de Mathématiques Discrètes, 2015.
Vincent Russo and Loren Schwiebert, Beatty Sequences, Fibonacci Numbers, and the Golden Ratio, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.
Jeffrey Shallit, Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond, arXiv:2006.04177 [math.CO], 2020.
Jeffrey Shallit, Frobenius Numbers and Automatic Sequences, arXiv:2103.10904 [math.NT], 2021.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Canadian Math. Bull. 19 (1976) pp. 473482.
Richard Southwell and Jianwei Huang, Complex Networks from Simple Rewrite Systems, arXiv preprint arXiv:1205.0596 [cs.SI], 2012.  N. J. A. Sloane, Oct 13 2012
X. Sun, Wythoff's sequence and NHeap Wythoff's conjectures, Discr. Math., 300 (2005), 180195.
J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., 27 (1989), 7686.
Eric Weisstein's World of Mathematics, Beatty Sequence
Eric Weisstein's World of Mathematics, Golden Ratio
Eric Weisstein's World of Mathematics, Rabbit Constant
Eric Weisstein's World of Mathematics, Wythoff's Game
Eric Weisstein's World of Mathematics, Wythoff Array
Index entries for sequences related to Beatty sequences
Index entries for sequences of the a(a(n)) = 2n family


FORMULA

Zeckendorf expansion of n (cf. A035517) ends with an even number of 0's.
Other properties: a(1)=1; for n>1, a(n) is taken to be the smallest integer greater than a(n1) which is consistent with the condition "n is in the sequence if and only if a(n)+1 is not in the sequence".
a(1) = 1; for n>0, a(n+1) = a(n)+1 if n is not in the sequence, a(n+1) = a(n)+2 if n is in the sequence.
a(a(n)) = floor(n*phi^2)  1 = A003622(n).
{a(k)} union {a(k)+1} = {1, 2, 3, 4, ...}. Hence a(1) = 1; for n>1, a(a(n)) = a(a(n)1)+2, a(a(n)+1) = a(a(n))+1.  Benoit Cloitre, Mar 08 2003
{a(n)} is a solution to the recurrence a(a(n)+n) = 2*a(n)+n, a(1)=1 (see Barbeau et al.).
a(n) = A001950(n)  n.  Philippe Deléham, May 02 2004
a(0) = 0; a(n) = n + Max_{k : a(k) < n}.  Vladeta Jovovic, Jun 11 2004
a(Fibonacci(r1)+j) = Fibonacci(r)+a(j) for 0 < j <= Fibonacci(r2); 2 < r.  Paul Weisenhorn, Aug 18 2012
With 1 < k and A001950(k1) < n <= A001950(k): a(n) = 2*nk; A001950(n) = 3*nk.  Paul Weisenhorn, Aug 21 2012


EXAMPLE

From Roland Schroeder (florola(AT)gmx.de), Jul 13 2010: (Start)
Example for n = 5; a(5) = 8;
(Start: [1,2,3,4,5]; 8 steps until [5,4,3,2,1]):
[1,2,3,4,5]; [3,3,4,5]; [4,5,6]; [6,7,1,1]; [8,2,2,1,1,1]: [3,3,2,2,2,1,1,1]; [4,3,3,2,1,1,1]; [4,4,3,2,1,1]; [5,4,3,2,1]. (End)


MAPLE

Digits := 100; t := evalf((1+sqrt(5))/2); A000201 := n>floor(t*n);


MATHEMATICA

Table[Floor[N[n*(1+Sqrt[5])/2]], {n, 1, 75}]
Array[ Floor[ #*GoldenRatio] &, 68] (* Robert G. Wilson v, Apr 17 2010 *)


PROG

(PARI) a(n)=floor(n*(sqrt(5)+1)/2)
(PARI) a(n)=(n+sqrtint(5*n^2))\2 \\ Charles R Greathouse IV, Feb 07 2013
(Maxima) makelist(floor(n*(1+sqrt(5))/2), n, 1, 60); /* Martin Ettl, Oct 17 2012 */
(Haskell)
a000201 n = a000201_list !! (n1)
a000201_list = f [1..] [1..] where
f (x:xs) (y:ys) = y : f xs (delete (x + y) ys)
 Reinhard Zumkeller, Jul 02 2015, Mar 10 2013
(Python)
def aupton(terms):
alst, aset = [None, 1], {1}
for n in range(1, terms):
an = alst[n] + (1 if n not in aset else 2)
alst.append(an); aset.add(an)
return alst[1:]
print(aupton(68)) # Michael S. Branicky, May 14 2021
(Python)
from math import isqrt
def A000201(n): return (n+isqrt(5*n**2))//2 # Chai Wah Wu, Jan 11 2022


CROSSREFS

a(n) = least k such that s(k) = n, where s = A026242. Complement of A001950. See also A058066.
The permutation A002251 maps between this sequence and A001950, in that A002251(a(n)) = A001950(n), A002251(A001950(n)) = a(n).
First differences give A014675. a(n) = A022342(n) + 1 = A005206(n) + n + 1. a(2n)a(n)=A007067(n). a(a(a(n)))a(n) = A026274(n1).  Benoit Cloitre, Mar 08 2003
A185615 gives values n such that n divides A000201(n)^m for some integer m>0.
Cf. A183110, A329825.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841.  N. J. A. Sloane, Mar 11 2021
Bisections: A276854, A342279.
Sequence in context: A085270 A330063 A066096 * A090908 A292644 A000202
Adjacent sequences: A000198 A000199 A000200 * A000202 A000203 A000204


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



