This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000201 M2322 N0917 #309 Jul 28 2025 20:18:10
%S A000201 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,
%T A000201 42,43,45,46,48,50,51,53,55,56,58,59,61,63,64,66,67,69,71,72,74,76,77,
%U A000201 79,80,82,84,85,87,88,90,92,93,95,97,98,100,101,103,105,106,108,110
%N A000201 Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = A001622.
%C A000201 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
%C A000201 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
%C A000201 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.
%C A000201 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
%C A000201 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
%C A000201 Cumulative sum of A001468 terms. - _Eric Angelini_, Aug 19 2008
%C A000201 The lower Wythoff sequence also can be constructed by playing the so-called Mancala-game: 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
%C A000201 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 Mancala-type game interpretation given by R. Schroeder above. - _John W. Layman_, Feb 03 2011
%C A000201 Also row numbers of A213676 starting with an even number of zeros. - _Reinhard Zumkeller_, Mar 10 2013
%C A000201 From _Jianing Song_, Aug 18 2022: (Start)
%C A000201 Numbers k such that {k*phi} > phi^(-2), where {} denotes the fractional part.
%C A000201 Proof: Write m = floor(k*phi).
%C A000201 If {k*phi} > phi^(-2), take s = m-k+1. From m < k*phi < m+1 we have k < (m-k+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.
%C A000201 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)
%C A000201 The formula from Jianing Song above is a direct consequence of an old result by Carlitz et al. (1972). Their Theorem 11 states that (a(n)) consists of the numbers k such that {k*phi^(-2)} < phi^(-1). One has {k*phi^(-2)} = {k*(2-phi)} = {-k*phi}. Using that 1-phi^(-1) = phi^(-2), the Jianing Song formula follows. - _Michel Dekking_, Oct 14 2023
%C A000201 In the Fokkink-Joshi paper, this sequence is the Cloitre (1,1,2,1)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n-1, else a(n) = a(n-1)+1. - _Michael De Vlieger_, Jul 28 2025
%D A000201 Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, 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?
%D A000201 M. Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, 1989; see p. 107.
%D A000201 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000201 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000201 I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.
%H A000201 N. J. A. Sloane, <a href="/A000201/b000201.txt">Table of n, a(n) for n = 1..10000</a>
%H A000201 J.-P. Allouche and F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%H A000201 J.-P. Allouche, J. Shallit, and G. Skordev, <a href="http://dx.doi.org/10.1016/j.disc.2004.12.004">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A000201 Peter G. Anderson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/58-5/anderson.pdf">The Fibonacci word as a 2-adic number and its continued fraction</a>, Fibonacci Quarterly (2020) Vol. 58, No. 5, 21-24.
%H A000201 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp.756-757.
%H A000201 Shiri Artstein-Avidan, Aviezri S. Fraenkel, and Vera T. Sos, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.070">A two-parameter family of an extension of Beatty</a>, Discr. Math. 308 (2008), 4578-4588.
%H A000201 Shiri Artstein-avidan, Aviezri S. Fraenkel, and Vera T. Sos, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/coatp8.pdf">A two-parameter family of an extension of Beatty sequences</a>, Discrete Math., 308 (2008), 4578-4588.
%H A000201 E. J. Barbeau, J. Chew, and S. Tanny, <a href="http://www.combinatorics.org/Volume_4/Abstracts/v4i1r16.html">A matrix dynamics approach to Golomb's recursion</a>, Electronic J. Combinatorics, #4.1 16 1997.
%H A000201 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 3.
%H A000201 M. Bunder and K. Tognetti, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00147-9">On the self matching properties of [j tau]</a>, Discrete Math., 241 (2001), 139-151.
%H A000201 Lucas Bustos, Hung Viet Chu, Minchae Kim, Uihyeon Lee, Shreya Shankar, and Garrett Tresch, <a href="https://arxiv.org/abs/2504.20286">Integers Having F_{2k} in Both Zeckendorf and Chung-Graham Decompositions</a>, arXiv:2504.20286 [math.NT], 2025. See p. 8.
%H A000201 L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz1.pdf">Fibonacci representations</a>, Fib. Quart., Vol. 10, No. 1 (1972), pp. 1-28.
%H A000201 L. Carlitz, R. Scoville, and T. Vaughan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/carlitz.pdf">Some arithmetic functions related to Fibonacci numbers</a>, Fib. Quart., 11 (1973), 337-386.
%H A000201 Benoit Cloitre, N. J. A. Sloane, and M. J. Vandermast, <a href="http://www.emis.de/journals/JIS/VOL6/Cloitre/cloitre2.html">Numerical analogues of Aronson's sequence</a>, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
%H A000201 Benoit Cloitre, N. J. A. Sloane, and M. J. Vandermast, <a href="http://arXiv.org/abs/math.NT/0305308">Numerical analogues of Aronson's sequence</a>, arXiv:math/0305308 [math.NT], 2003.
%H A000201 Benoit Cloitre, <a href="https://arxiv.org/abs/2506.18103">A study of a family of self-referential sequences</a>, arXiv:2506.18103 [math.GM], 2025. See p. 7.
%H A000201 I. G. Connell, <a href="http://dx.doi.org/10.4153/CMB-1959-025-0">Some properties of Beatty sequences I</a>, Canad. Math. Bull., 2 (1959), 190-197.
%H A000201 J. H. Conway and N. J. A. Sloane, <a href="/A019586/a019586.pdf">Notes on the Para-Fibonacci and related sequences</a>.
%H A000201 H. S. M. Coxeter, <a href="/A000201/a000201.pdf">The Golden Section, Phyllotaxis and Wythoff's Game</a>, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
%H A000201 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H A000201 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A000201 P. J. Downey and R. E. Griswold, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/22-4/downey.pdf">On a family of nested recurrences</a>, Fib. Quart., 22 (1984), 310-317.
%H A000201 Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/WythoffWisdomJune62016.pdf">Wythoff Wisdom</a>, 43 pages, no date, unpublished.
%H A000201 Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="/A001950/a001950.pdf">Wythoff Wisdom</a>, unpublished, no date [Cached copy, with permission]
%H A000201 Larry Ericksen and Peter G. Anderson, <a href="http://www.cs.rit.edu/~pga/k-zeck.pdf">Patterns in differences between rows in k-Zeckendorf arrays</a>, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
%H A000201 Robbert Fokkink and Gandhar Joshi, <a href="https://arxiv.org/abs/2507.16956">On Cloitre's hiccup sequences</a>, arXiv:2507.16956 [math.CO], 2025. See pp. 8, 16-17.
%H A000201 Nathan Fox, <a href="http://arxiv.org/abs/1407.2823">On Aperiodic Subtraction Games with Bounded Nim Sequence</a>, arXiv preprint arXiv:1407.2823 [math.CO], 2014.
%H A000201 A. S. Fraenkel, <a href="http://dx.doi.org/10.4153/CJM-1969-002-7">The bracket function and complementary sets of integers</a>, Canadian J. of Math. 21 (1969) 6-27. [History, references, generalization]
%H A000201 A. S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=1).
%H A000201 A. S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/RationalGames3.pdf">Ratwyt</a>, December 28 2011.
%H A000201 David Garth and Adam Gouge, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Garth/garth14.html">Affinely Self-Generating Sets and Morphisms</a>, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.
%H A000201 M. Griffiths, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.06.497">The Golden String, Zeckendorf Representations, and the Sum of a Series</a>, Amer. Math. Monthly, 118 (2011), 497-507.
%H A000201 Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Griffiths/gr48.html">On a Matrix Arising from a Family of Iterated Self-Compositions</a>, Journal of Integer Sequences, 18 (2015), #15.11.8.
%H A000201 Martin Griffiths, <a href="https://doi.org/10.1017/mag.2018.78">A difference property amongst certain pairs of Beatty sequences</a>, The Mathematical Gazette (2018) Vol. 102, Issue 554, Article 102.36, 348-350.
%H A000201 H. Grossman, <a href="https://www.jstor.org/stable/2311195">A set containing all integers</a>, Amer. Math. Monthly, 69 (1962), 532-533.
%H A000201 A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.
%H A000201 T. Karki, A. Lacroix, and M. Rigo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Rigo/rigo6.html">On the recognizability of self-generating sets</a>, JIS 13 (2010) #10.2.2.
%H A000201 Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/goldentext.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%H A000201 Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Kimberling/kimberling24.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000201 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, Journal of Integer Sequences, 11 (2008) 08.3.3.
%H A000201 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%H A000201 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-5/Problems.pdf">Problem Proposals</a>, The Fibonacci Quarterly, vol. 52 #5, 2015, p5-14.
%H A000201 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%H A000201 Clark Kimberling, <a href="https://doi.org/10.4171/EM/468">Intriguing infinite words composed of zeros and ones</a>, Elemente der Mathematik (2021).
%H A000201 C. Kimberling and K. B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
%H A000201 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. 319-337.[See A317208 for a link.]
%H A000201 U. Larsson and N. Fox, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Larsson/larsson8.html">An Aperiodic Subtraction Game of Nim-Dimension Two</a>, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.
%H A000201 A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoffarray</a>, arXiv:2405.18128 [math.CO], 2024. See page 2.
%H A000201 R. J. Mathar, <a href="/A000201/a000201_1.pdf">Graphical representation among sequences closely related to this one</a> (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
%H A000201 D. J. Newman, <a href="https://www.jstor.org/stable/2298716">Problem 3117</a>, Amer. Math. Monthly, 34 (1927), 158-159.
%H A000201 D. J. Newman, <a href="https://www.jstor.org/stable/2315984">Problem 5252</a>, Amer. Math. Monthly, 72 (1965), 1144-1145.
%H A000201 Gabriel Nivasch, <a href="https://library.slmath.org/books/Book56/files/43nivasch.pdf">More on the Sprague-Grundy function for Wythoff's game</a>, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009.
%H A000201 R. J. Nowakowski, <a href="/A104429/a104429.pdf">Generalizations of the Langford-Skolem problem</a>, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
%H A000201 Michel Rigo, <a href="http://orbi.ulg.be/bitstream/2268/177711/1/Rigo.pdf">Invariant games and non-homogeneous Beatty sequences</a>, Slides of a talk, Journée de Mathématiques Discrètes, 2015.
%H A000201 Vincent Russo and Loren Schwiebert, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/49-2.html">Beatty Sequences, Fibonacci Numbers, and the Golden Ratio</a>, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.
%H A000201 Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024.
%H A000201 Jeffrey Shallit, <a href="https://arxiv.org/abs/2006.04177">Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond</a>, arXiv:2006.04177 [math.CO], 2020.
%H A000201 Jeffrey Shallit, <a href="https://arxiv.org/abs/2103.10904">Frobenius Numbers and Automatic Sequences</a>, arXiv:2103.10904 [math.NT], 2021.
%H A000201 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000201 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%H A000201 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%H A000201 K. B. Stolarsky, <a href="http://dx.doi.org/10.4153/CMB-1976-071-6">Beatty sequences, continued fractions, and certain shift operators</a>, Canadian Math. Bull. 19 (1976) pp. 473-482.
%H A000201 Richard Southwell and Jianwei Huang, <a href="http://arxiv.org/abs/1205.0596">Complex Networks from Simple Rewrite Systems</a>, arXiv preprint arXiv:1205.0596 [cs.SI], 2012. - _N. J. A. Sloane_, Oct 13 2012
%H A000201 X. Sun, <a href="http://dx.doi.org/10.1016/j.disc.2005.07.001">Wythoff's sequence and N-Heap Wythoff's conjectures</a>, Discr. Math., 300 (2005), 180-195.
%H A000201 J. C. Turner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-1/turner.pdf">The alpha and the omega of the Wythoff pairs</a>, Fib. Q., 27 (1989), 76-86.
%H A000201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence</a>
%H A000201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A000201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RabbitConstant.html">Rabbit Constant</a>
%H A000201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffsGame.html">Wythoff's Game</a>
%H A000201 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>
%H A000201 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%H A000201 <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%F A000201 Zeckendorf expansion of n (cf. A035517) ends with an even number of 0's.
%F A000201 Other properties: a(1)=1; for n>1, a(n) is taken to be the smallest integer greater than a(n-1) which is consistent with the condition "n is in the sequence if and only if a(n)+1 is not in the sequence".
%F A000201 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.
%F A000201 a(a(n)) = floor(n*phi^2) - 1 = A003622(n).
%F A000201 {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
%F A000201 {a(n)} is a solution to the recurrence a(a(n)+n) = 2*a(n)+n, a(1)=1 (see Barbeau et al.).
%F A000201 a(n) = A001950(n) - n. - _Philippe Deléham_, May 02 2004
%F A000201 a(0) = 0; a(n) = n + Max_{k : a(k) < n}. - _Vladeta Jovovic_, Jun 11 2004
%F A000201 a(Fibonacci(r-1)+j) = Fibonacci(r)+a(j) for 0 < j <= Fibonacci(r-2); 2 < r. - _Paul Weisenhorn_, Aug 18 2012
%F A000201 With 1 < k and A001950(k-1) < n <= A001950(k): a(n) = 2*n-k; A001950(n) = 3*n-k. - _Paul Weisenhorn_, Aug 21 2012
%e A000201 From Roland Schroeder (florola(AT)gmx.de), Jul 13 2010: (Start)
%e A000201 Example for n = 5; a(5) = 8;
%e A000201 (Start: [1,2,3,4,5]; 8 steps until [5,4,3,2,1]):
%e A000201 [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)
%p A000201 Digits := 100; t := evalf((1+sqrt(5))/2); A000201 := n->floor(t*n);
%t A000201 Table[Floor[N[n*(1+Sqrt[5])/2]], {n, 1, 75}]
%t A000201 Array[ Floor[ #*GoldenRatio] &, 68] (* _Robert G. Wilson v_, Apr 17 2010 *)
%o A000201 (PARI) a(n)=floor(n*(sqrt(5)+1)/2)
%o A000201 (PARI) a(n)=(n+sqrtint(5*n^2))\2 \\ _Charles R Greathouse IV_, Feb 07 2013
%o A000201 (Maxima) makelist(floor(n*(1+sqrt(5))/2),n,1,60); /* _Martin Ettl_, Oct 17 2012 */
%o A000201 (Haskell)
%o A000201 a000201 n = a000201_list !! (n-1)
%o A000201 a000201_list = f [1..] [1..] where
%o A000201 f (x:xs) (y:ys) = y : f xs (delete (x + y) ys)
%o A000201 -- _Reinhard Zumkeller_, Jul 02 2015, Mar 10 2013
%o A000201 (Python)
%o A000201 def aupton(terms):
%o A000201 alst, aset = [None, 1], {1}
%o A000201 for n in range(1, terms):
%o A000201 an = alst[n] + (1 if n not in aset else 2)
%o A000201 alst.append(an); aset.add(an)
%o A000201 return alst[1:]
%o A000201 print(aupton(68)) # _Michael S. Branicky_, May 14 2021
%o A000201 (Python)
%o A000201 from math import isqrt
%o A000201 def A000201(n): return (n+isqrt(5*n**2))//2 # _Chai Wah Wu_, Jan 11 2022
%Y A000201 a(n) = least k such that s(k) = n, where s = A026242. Complement of A001950. See also A058066.
%Y A000201 The permutation A002251 maps between this sequence and A001950, in that A002251(a(n)) = A001950(n), A002251(A001950(n)) = a(n).
%Y A000201 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(n-1). - _Benoit Cloitre_, Mar 08 2003
%Y A000201 A185615 gives values n such that n divides A000201(n)^m for some integer m>0.
%Y A000201 Cf. A183110, A329825.
%Y A000201 Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
%Y A000201 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
%Y A000201 Bisections: A276854, A342279.
%K A000201 nonn,easy,nice
%O A000201 1,2
%A A000201 _N. J. A. Sloane_