OFFSET
1,1
COMMENTS
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
Numbers missing from A066096. - Philippe Deléham, Jan 19 2023
REFERENCES
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.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (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]
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Nathan Fox, On Aperiodic Subtraction Games with Bounded Nim Sequence, arXiv preprint arXiv:1407.2823 [math.CO], 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.
A. J. Macfarlane, On the fibbinary numbers and the Wythoffarray, arXiv:2405.18128 [math.CO], 2024. See page 2.
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.
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024.
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.
Eric Weisstein's World of Mathematics, Golden ratio.
Eric Weisstein's World of Mathematics, Wythoff's Game.
Eric Weisstein's World of Mathematics, Wythoff Array.
FORMULA
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) = A003622(n) + 1. - Philippe Deléham, Apr 30 2004
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
EXAMPLE
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)
MAPLE
A001950 := proc(n)
floor(n*(3+sqrt(5))/2) ;
end proc:
seq(A001950(n), n=0..40) ; # R. J. Mathar, Jul 16 2024
MATHEMATICA
Table[Floor[N[n*(1+Sqrt[5])^2/4]], {n, 1, 75}]
Array[ Floor[ #*GoldenRatio^2] &, 60] (* Robert G. Wilson v, Apr 17 2010 *)
PROG
(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
(Haskell)
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
(Python)
from math import isqrt
def A001950(n): return (n+isqrt(5*n**2)>>1)+n # Chai Wah Wu, Aug 10 2022
CROSSREFS
a(n) = greatest k such that s(k) = n, where s = A026242.
A002251 maps between A000201 and A001950, in that A002251(A000201(n)) = A001950(n), A002251(A001950(n)) = A000201(n).
First differences give (essentially) A076662.
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
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Corrected by Michael Somos, Jun 07 2000
STATUS
approved