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A001950 Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2.
(Formerly M1332 N0509)
2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, 36, 39, 41, 44, 47, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 75, 78, 81, 83, 86, 89, 91, 94, 96, 99, 102, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 143, 146, 149, 151, 154, 157 (list; graph; refs; listen; history; text; internal format)



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

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

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

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

Apart from the initial 0 in A090909, is this the same as that sequence? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007

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

Numbers with an odd number of trailing zeros in their Zeckendorf representation (A014417). - Amiram Eldar, Feb 26 2021


Claude Berge, Graphs and Hypergraphs, North-Holland, 1973; p. 324, Problem 2.

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.

Martin 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. 1-7 of Qvant Selecta: Combinatorics I, Amer Math. Soc., 2001.


N. J. A. Sloane, Table of n, a(n) for n = 1..10000, Mar 30 2016 [First 1000 terms from T. D. Noe]

Jean-Paul Allouche and F. Michel Dekking, Generalized Beatty sequences and complementary triples, Moscow Journal of Combinatorics and Number Theory, Vol. 8, No. 4 (2019), pp. 325-341; arXiv preprint, arXiv:1809.03424 [math.NT], 2018-2019.

L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.

I. G. Connell, Some properties of Beatty sequences I, Canad. Math. Bull., 2 (1959), 190-197.

H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

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]

Nathan Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823, 2014

Aviezri S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, Vol. 89 (1982), pp. 353-361 (the case a=1).

Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1.

Aviezri S. Fraenkel, Ratwyt, December 28 2011.

Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math., Vol. 24, No. 2 (2010), pp. 570-588. - N. J. A. Sloane, May 06 2011

Martin Griffiths, The Golden String, Zeckendorf Representations, and the Sum of a Series, Amer. Math. Monthly, Vol. 118 (2011), pp. 497-507.

Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, Vol. 18 (2015), Article #15.11.8.

Martin Griffiths, A difference property amongst certain pairs of Beatty sequences, The Mathematical Gazette, Vol. 102, Issue 554 (2018), Article 102.36, pp. 348-350.

Tomi Kärki, Anne Lacroix and Michel Rigo, On the recognizability of self-generating sets, JIS, Vol. 13 (2010), Article #10.2.2.

Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, Vol. 3 (2000), Article #00.2.8.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Clark Kimberling, Complementary equations and Wythoff Sequences, JIS, Vol. 11 (2008), Article 08.3.3.

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).

Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), pp. 267-273.

Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (eds.), Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337. [See A317208 for a link.]

Urban Larsson and Nathan Fox, An Aperiodic Subtraction Game of Nim-Dimension Two, Journal of Integer Sequences, 2015, Vol. 18, #15.7.4.

D. J. Newman, Problem 5252, Amer. Math. Monthly, Vol. 72, No. 10 (1965), pp. 1144-1145.

Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff’s game, pages 377-410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009.

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Michel Rigo, Invariant games and non-homogeneous 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, No. 2 (May 2011), pp. 151-154.

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, 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., Vol. 19 (1976), pp. 473-482.

X. Sun, Wythoff's sequence and N-Heap Wythoff's conjectures, Discr. Math., Vol. 300 (2005), pp. 180-195.

J. C. Turner, The alpha and the omega of the Wythoff pairs, Fib. Q., Vol. 27 (1989), pp. 76-86.

Eric Weisstein's World of Mathematics, Beatty Sequence, MathWorld.

Eric Weisstein's World of Mathematics, Golden ratio, MathWorld.

Eric Weisstein's World of Mathematics, Wythoff's Game, MathWorld.

Eric Weisstein's World of Mathematics, Wythoff Array.

Index entries for sequences related to Beatty sequences


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

a(n) = n + floor(n*phi) = n + A000201(n). - Paul Weisenhorn and Philippe Deléham

Append a 0 to the Zeckendorf expansion (cf. A035517) of n-th term of A000201.

a(n) = A003622(n) + 1. - Philippe Deléham, Apr 30 2004

a(n) = Min(m: A134409(m) = A006336(n)). - Reinhard Zumkeller, Oct 24 2007

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

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


From Paul Weisenhorn, Aug 18 2012 and Aug 21 2012: (Start)

a(14)=floor(14*phi^2)=36; a'(14)=floor(14*phi)=22;

with r=9 and j=1: a(13+1)=34+2=36;

with r=8 and j=1: a'(13+1)=21+1=22.

k=6 and a(5)=13 < n <= a(6)=15

a(14)=3*14-6=36; a'(14)=2*14-6=22;

a(15)=3*15-6=39; a'(15)=2*15-6=24. (End)


Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]

Array[ Floor[ #*GoldenRatio^2] &, 60] (* Robert G. Wilson v, Apr 17 2010 *)


(PARI) a(n)=floor(n*(sqrt(5)+3)/2)

(PARI) A001950(n)=(sqrtint(n^2*5)+n*3)\2 \\ M. F. Hasler, Sep 17 2014


a001950 n = a000201 n + n  -- Reinhard Zumkeller, Mar 10 2013

(Magma) [Floor(n*((1+Sqrt(5))/2)^2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2016


from math import isqrt

def A001950(n): return (n+isqrt(5*n**2)>>1)+n # Chai Wah Wu, Aug 10 2022


a(n) = greatest k such that s(k) = n, where s = A026242. Complement of A000201.

A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).

Cf. A001622, A026352, A004976, A004919.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.

First differences give (essentially) A076662.

Bisections: A001962, A001966.

Cf. A014417, A329825.

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

Sequence in context: A018717 A188036 A292645 * A090909 A330064 A022841

Adjacent sequences:  A001947 A001948 A001949 * A001951 A001952 A001953




N. J. A. Sloane


Corrected by Michael Somos, Jun 07 2000



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