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A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.

%I A001950 M1332 N0509 #264 Jul 29 2025 15:33:47
%S A001950 2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,62,
%T A001950 65,68,70,73,75,78,81,83,86,89,91,94,96,99,102,104,107,109,112,115,
%U A001950 117,120,123,125,128,130,133,136,138,141,143,146,149,151,154,157
%N A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.
%C A001950 Indices at which blocks (1;0) occur in infinite Fibonacci word; i.e., n such that A005614(n-2) = 0 and A005614(n-1) = 1. - _Benoit Cloitre_, Nov 15 2003
%C A001950 A000201 and this sequence 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 A001950 a(n) = n-th integer which is not equal to the floor of any multiple of phi, where phi = (1+sqrt(5))/2 = golden number. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), May 09 2007
%C A001950 Write A for A000201 and B for the present sequence (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, the present sequence). Typical complementary equations: AB=A+B (=A003623), BB=A+2B (=A101864), BBB=3A+5B (=A134864). - _Clark Kimberling_, Nov 14 2007
%C A001950 Apart from the initial 0 in A090909, is this the same as that sequence? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
%C A001950 If we define a base-phi integer as a positive number whose representation in the golden ratio base consists only of nonnegative powers of phi, and if these base-phi integers are ordered in increasing order (beginning 1, phi, ...), then it appears that the difference between the n-th and (n-1)-th base-phi integer is phi-1 if and only if n belongs to this sequence, and the difference is 1 otherwise. Further, if each base-phi integer is written in linear form as a + b*phi (for example, phi^2 is written as 1 + phi), then it appears that there are exactly two base-phi integers with b=n if and only if n belongs to this sequence, and exactly three base-phi integers with b=n otherwise. - _Geoffrey Caveney_, Apr 17 2014
%C A001950 Numbers with an odd number of trailing zeros in their Zeckendorf representation (A014417). - _Amiram Eldar_, Feb 26 2021
%C A001950 Numbers missing from A066096. - _Philippe Deléham_, Jan 19 2023
%D A001950 Claude Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 2.
%D A001950 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, 2019.
%D A001950 Martin Gardner, Penrose Tiles to Trapdoor Ciphers, W. H. Freeman, 1989; see p. 107.
%D A001950 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001950 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001950 I. M. Yaglom, Two games with matchsticks, pp. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.
%H A001950 N. J. A. Sloane, <a href="/A001950/b001950.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A001950 Jean-Paul Allouche and F. Michel Dekking, <a href="https://doi.org/10.2140/moscow.2019.8.325">Generalized Beatty sequences and complementary triples</a>, Moscow Journal of Combinatorics and Number Theory, Vol. 8, No. 4 (2019), pp. 325-341; <a href="https://arxiv.org/abs/1809.03424">arXiv preprint</a>, arXiv:1809.03424 [math.NT], 2018-2019.
%H A001950 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 A001950 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 A001950 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 A001950 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 A001950 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://doi.org/10.37236/8905">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A001950 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 A001950 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 A001950 Robbert Fokkink, <a href="https://arxiv.org/abs/2309.01644">The Pell Tower and Ostronometry</a>, arXiv:2309.01644 [math.CO], 2023.
%H A001950 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 A001950 Aviezri S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, Vol. 89 (1982), pp. 353-361 (the case a=1).
%H A001950 Aviezri S. Fraenkel, <a href="https://www.emis.de/journals/INTEGERS/papers/a13int2005/a13int2005.Abstract.html">The Raleigh game</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1.
%H A001950 Aviezri S. Fraenkel, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/RationalGames3.pdf">Ratwyt</a>, December 28 2011.
%H A001950 Aviezri S. Fraenkel, <a href="http://dx.doi.org/10.1137/090758994">Complementary iterated floor words and the Flora game</a>, SIAM J. Discrete Math., Vol. 24, No. 2 (2010), pp. 570-588. - _N. J. A. Sloane_, May 06 2011
%H A001950 Martin 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, Vol. 118 (2011), pp. 497-507.
%H A001950 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, Vol. 18 (2015), Article #15.11.8.
%H A001950 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, Vol. 102, Issue 554 (2018), Article 102.36, pp. 348-350.
%H A001950 Tomi Kärki, Anne Lacroix, and Michel Rigo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Rigo/rigo6.html">On the recognizability of self-generating sets</a>, JIS, Vol. 13 (2010), Article #10.2.2.
%H A001950 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, Vol. 3 (2000), Article #00.2.8.
%H A001950 Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H A001950 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Kimberling/kimberling719a.html">Complementary equations and Wythoff Sequences</a>, JIS, Vol. 11 (2008), Article 08.3.3.
%H A001950 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 A001950 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 A001950 Clark Kimberling and Kenneth 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, Vol. 123, No. 2 (2016), pp. 267-273.
%H A001950 Johan Kok, <a href="https://arxiv.org/abs/2507.16500">Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs</a>, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.
%H A001950 Wolfdieter Lang, <a href="https://doi.org/10.1007/978-94-009-0223-7_27">The Wythoff and the Zeckendorf representations of numbers are equivalent</a>, in G. E. Bergum et al. (eds.), Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337. [See A317208 for a link.]
%H A001950 Urban Larsson and Nathan 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 A001950 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 A001950 D. J. Newman, <a href="http://www.jstor.org/stable/2315984">Problem 5252</a>, Amer. Math. Monthly, Vol. 72, No. 10 (1965), pp. 1144-1145.
%H A001950 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 A001950 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 A001950 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 A001950 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, No. 2 (May 2011), pp. 151-154.
%H A001950 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 A001950 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 A001950 Jeffrey Shallit, <a href="https://arxiv.org/abs/2103.10904">Frobenius Numbers and Automatic Sequences</a>, arXiv:2103.10904 [math.NT], 2021.
%H A001950 Jeffrey Shallit, <a href="https://arxiv.org/abs/2501.08823">The Hurt-Sada Array and Zeckendorf Representations</a>, arXiv:2501.08823 [math.NT], 2025. See p. 6.
%H A001950 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%H A001950 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., Vol. 19 (1976), pp. 473-482.
%H A001950 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., Vol. 300 (2005), pp. 180-195.
%H A001950 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., Vol. 27 (1989), pp. 76-86.
%H A001950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence</a>.
%H A001950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden ratio</a>.
%H A001950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffsGame.html">Wythoff's Game</a>.
%H A001950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WythoffArray.html">Wythoff Array</a>.
%H A001950 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A001950 a(n) = n + floor(n*phi). In general, floor(n*phi^m) = Fibonacci(m-1)*n + floor(Fibonacci(m)*n*phi). - _Benoit Cloitre_, Mar 18 2003
%F A001950 a(n) = n + floor(n*phi) = n + A000201(n). - _Paul Weisenhorn_ and _Philippe Deléham_
%F A001950 Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of A000201.
%F A001950 a(n) = A003622(n) + 1. - _Philippe Deléham_, Apr 30 2004
%F A001950 a(n) = Min(m: A134409(m) = A006336(n)). - _Reinhard Zumkeller_, Oct 24 2007
%F A001950 If a'=A000201 is the ordered complement (in N) of {a(n)}, then a(Fib(r-2) + j) = Fib(r) + a(j) for 0 < j <= Fib(r-2), 3 < r; and a'(Fib(r-1) + j) = Fib(r) + a'(j) for 0 < j <= Fib(r-2), 2 < r. - _Paul Weisenhorn_, Aug 18 2012
%F A001950 With a(1)=2, a(2)=5, a'(1)=1, a'(2)=3 and 1 < k and a(k-1) < n <= a(k) one gets a(n)=3*n-k, a'(n)=2*n-k. - _Paul Weisenhorn_, Aug 21 2012
%e A001950 From _Paul Weisenhorn_, Aug 18 2012 and Aug 21 2012: (Start)
%e A001950 a(14) = floor(14*phi^2) = 36; a'(14) = floor(14*phi)=22;
%e A001950 with r=9 and j=1: a(13+1) = 34 + 2 = 36;
%e A001950 with r=8 and j=1: a'(13+1) = 21 + 1 = 22.
%e A001950 k=6 and a(5)=13 < n <= a(6)=15
%e A001950 a(14) = 3*14 - 6 = 36; a'(14) = 2*14 - 6 = 22;
%e A001950 a(15) = 3*15 - 6 = 39; a'(15) = 2*15 - 6 = 24. (End)
%p A001950 A001950 := proc(n)
%p A001950     floor(n*(3+sqrt(5))/2) ;
%p A001950 end proc:
%p A001950 seq(A001950(n),n=0..40) ; # _R. J. Mathar_, Jul 16 2024
%t A001950 Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
%t A001950 Array[ Floor[ #*GoldenRatio^2] &, 60] (* _Robert G. Wilson v_, Apr 17 2010 *)
%o A001950 (PARI) a(n)=floor(n*(sqrt(5)+3)/2)
%o A001950 (PARI) A001950(n)=(sqrtint(n^2*5)+n*3)\2 \\ _M. F. Hasler_, Sep 17 2014
%o A001950 (Haskell)
%o A001950 a001950 n = a000201 n + n  -- _Reinhard Zumkeller_, Mar 10 2013
%o A001950 (Magma) [Floor(n*((1+Sqrt(5))/2)^2): n in [1..80]]; // _Vincenzo Librandi_, Nov 19 2016
%o A001950 (Python)
%o A001950 from math import isqrt
%o A001950 def A001950(n): return (n+isqrt(5*n**2)>>1)+n # _Chai Wah Wu_, Aug 10 2022
%Y A001950 a(n) = greatest k such that s(k) = n, where s = A026242.
%Y A001950 Complement of A000201 or A066096.
%Y A001950 A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).
%Y A001950 Cf. A001622, A026352, A004976, A004919.
%Y A001950 Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
%Y A001950 First differences give (essentially) A076662.
%Y A001950 Bisections: A001962, A001966.
%Y A001950 Cf. A014417, A329825.
%Y A001950 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
%K A001950 nonn,easy,nice
%O A001950 1,1
%A A001950 _N. J. A. Sloane_
%E A001950 Corrected by _Michael Somos_, Jun 07 2000