A066096 -id:A066096 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane

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

			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)

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 (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.
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.
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.
Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.
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.
Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See p. 6.
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.
Index entries for sequences related to Beatty sequences

a(n) = greatest k such that s(k) = n, where s = A026242.
Complement of A000201 or A066096.
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

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

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

A001950 := proc(n)
    floor(n*(3+sqrt(5))/2) ;
end proc:
seq(A001950(n),n=0..40) ; # R. J. Mathar, Jul 16 2024

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)
```

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

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

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

Corrected by Michael Somos, Jun 07 2000

A003622 The Wythoff compound sequence AA: a(n) = floor(n*phi^2) - 1, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98, 101, 103, 106, 108, 111, 114, 116, 119, 122, 124, 127, 129, 132, 135, 137, 140, 142, 145, 148, 150, 153, 156, 158, 161, 163, 166

Offset: 1

N. J. A. Sloane, Mira Bernstein, Marc LeBrun

Also, integers with "odd" Zeckendorf expansions (end with ...+F_2 = ...+1) (Fibonacci-odd numbers); first column of Wythoff array A035513; from a 3-way splitting of positive integers. [Edited by Peter Munn, Sep 16 2022]
Also, numbers k such that A005206(k) = A005206(k+1). Also k such that A022342(A005206(k)) = k+1 (for all other k's this is k). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001
Also, positions of 1's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011
From Amiram Eldar, Sep 03 2022: (Start)
Numbers with an odd number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is odd.
The asymptotic density of this sequence is 1 - 1/phi (A132338). (End)
{a(n)} is the unique monotonic sequence of positive integers such that {a(n)} and {b(n)}: b(n) = a(n) - n form a partition of the nonnegative integers. - Yifan Xie, Jan 25 2025

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 62.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.
C. Kimberling, "Stolarsky interspersions", Ars Combinatoria 39 (1995) 129-138.
D. R. Morrison, "A Stolarsky array of Wythoff pairs", in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10.
N. J. A. Sloane and Simon Plouffe, Encyclopedia of Integer Sequences, Academic Press, 1995: this sequence appears twice, as both M3277 and M3278.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (terms 1.1000 from T. D. Noe)
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 62.
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - _N. J. A. Sloane_, Jun 10 2012
Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1.
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), Article 15.11.8.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Clark Kimberling, Interspersions.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS 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 K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.
L. Lindroos, A. Sills, and H. Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., 52 (2014), 61-65.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 3.
Mathematics Stack Exchange, Golden ratio and floor function floor(phi^2*n) - floor(phi*floor(phi*n)) = 1.
M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.5.
N. J. A. Sloane, Classic Sequences
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), Article P2.52.

Positions of 1's in A003849.
Complement of A022342.
Cf. A066096, A139764, A035516, A026273, A104326, A132338, A356749.
The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.
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

a003622 n = a003622_list !! (n-1)
a003622_list = filter ((elem 1) . a035516_row) [1..]
-- Reinhard Zumkeller, Mar 10 2013

A003622 := proc(n)
    n+floor(n*(1+sqrt(5))/2)-1 ;
end proc: # R. J. Mathar, Jan 25 2015
# Maple code for the Wythoff compound sequences, from N. J. A. Sloane, Mar 30 2016
# The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.
# Assume files out1, out2 contain lists of the terms in the base sequences A and B from their b-files
read out1; read out2; b[0]:=b1: b[1]:=b2:
w2:=(i,j,n)->b[i][b[j][n]];
w3:=(i,j,k,n)->b[i][b[j][b[k][n]]];
for i from 0 to 1 do
lprint("name=",i);
lprint([seq(b[i][n],n=1..100)]):
od:
for i from 0 to 1 do for j from 0 to 1 do
lprint("name=",i,j);
lprint([seq(w2(i,j,n),n=1..100)]);
od: od:
for i from 0 to 1 do for j from 0 to 1 do for k from 0 to 1 do
lprint("name=",i,j,k);
lprint([seq(w3(i,j,k,n),n=1..100)]);
od: od: od:

With[{c=GoldenRatio^2},Table[Floor[n c]-1,{n,70}]] (* Harvey P. Dale, Jun 11 2011 *)
Range[70]//Floor[#*GoldenRatio^2]-1& (* Waldemar Puszkarz, Oct 10 2017 *)

PARI
```
a(n)=floor(n*(sqrt(5)+3)/2)-1
```

a(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ Michel Marcus, Sep 17 2022

from sympy import floor
from mpmath import phi
def a(n): return floor(n*phi**2) - 1 # Indranil Ghosh, Jun 09 2017

from math import isqrt
def A003622(n): return (n+isqrt(5*n**2)>>1)+n-1 # Chai Wah Wu, Aug 11 2022

a(n) = floor(n*phi) + n - 1. [Corrected by Jianing Song, Aug 18 2022]
a(n) = floor(floor(n*phi)*phi) = A000201(A000201(n)). [See the Mathematics Stack Exchange link for a proof of the equivalence of the definition. - Jianing Song, Aug 18 2022]
a(n) = 1 + A022342(1 + A022342(n)).
G.f.: 1 - (1-x)*Sum_{n>=1} x^a(n) = 1/1 + x/1 + x^2/1 + x^3/1 + x^5/1 + x^8/1 + ... + x^F(n)/1 + ... (continued fraction where F(n)=n-th Fibonacci number). - Paul D. Hanna, Aug 16 2002
a(n) = A001950(n) - 1. - Philippe Deléham, Apr 30 2004
a(n) = A022342(n) + n. - Philippe Deléham, May 03 2004
a(n) = a(n-1) + 2 + A005614(n-2); also a(n) =  a(n-1) + 1 + A001468(n-1). - A.H.M. Smeets, Apr 26 2024

A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105

Offset: 0

N. J. A. Sloane, Mira Bernstein

First column of inverse Stolarsky array.

A rectangle of size a(n) X n approximates a golden rectangle. So does A295282(n) X n, which targets the golden ratio's underlying objective. These approximations differ first for n = 4 and generally if n = F(6*k)/2, where F(n) = A000045(n) is the n-th Fibonacci number and k >= 1. - Peter Munn, Jan 12 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iain Fox, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vincenzo Librandi)
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences.

Cf. A166946 (characteristic function), A007064 (complement).
Different from A026355.
Sequences with similar terms: A022342, A295282.
Other roundings of n*tau: A000201, A004956, A066096.
Cf. A000045 (Fibonacci numbers), A001622 (value of tau).

A007067:=n->round(n*(1+sqrt(5))/2); seq(A007067(n), n=0..100); # Wesley Ivan Hurt, Nov 27 2013

a[n_] := Round[n*GoldenRatio]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 27 2012 *)

a(n) = round(n*(1+sqrt(5))/2) \\ Michel Marcus, May 20 2013

from math import isqrt
def A007067(n): return (isqrt(5*n**2<<2)>>1)+n+1>>1 # Chai Wah Wu, Aug 26 2022

Satisfies a(a(n)) = a(n) + n. - Franklin T. Adams-Watters, Aug 14 2006

a(n) = floor((A066096(2*n) + 1)/2). - Peter Munn, Jan 12 2018

A022342 Integers with "even" Zeckendorf expansions (do not end with ...+F_2 = ...+1) (the Fibonacci-even numbers); also, apart from first term, a(n) = Fibonacci successor to n-1.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107

Offset: 1

Marc LeBrun

The Zeckendorf expansion of n is obtained by repeatedly subtracting the largest Fibonacci number you can until nothing remains; for example, 100 = 89 + 8 + 3.
The Fibonacci successor to n is found by replacing each F_i in the Zeckendorf expansion by F_{i+1}; for example, the successor to 100 is 144 + 13 + 5 = 162.
If k appears, k + (rank of k) does not (10 is the 7th term in the sequence but 10 + 7 = 17 is not a term of the sequence). - Benoit Cloitre, Jun 18 2002
From Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001: (Start)
a(n) = Sum_{k in A_n} F_{k+1}, where a(n)= Sum_{k in A_n} F_k is the (unique) expression of n as a sum of "noncontiguous" Fibonacci numbers (with index >= 2).
a(10^n) gives the first few digits of g = (sqrt(5)+1)/2.
The sequences given by b(n+1) = a(b(n)) obey the general recursion law of Fibonacci numbers. In particular the (sub)sequence (of a(-)) yielded by a starting value of 2=a(1), is the sequence of Fibonacci numbers >= 2. Starting points of all such subsequences are given by A035336.
a(n) = floor(phi*n+1/phi); phi = (sqrt(5)+1)/2. a(F_n)=F_{n+1} if F_n is the n-th Fibonacci number.
(End)
From Amiram Eldar, Sep 03 2022: (Start)
Numbers with an even number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is even.
The asymptotic density of this sequence is 1/phi (A094214). (End)

			The successors to 1, 2, 3, 4=3+1 are 2, 3, 5, 7=5+2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.
E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook)
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 3, 6.
M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.
Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See p. 2.
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)
Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), Article P2.52.

Positions of 0's in A003849.
Complement of A003622.
Cf. A000201, A005206, A035336, A066096, A001950, A062879, A035516, A026274.
Cf. A035612, A094214, A104326, A356749.
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

a022342 n = a022342_list !! (n-1)
a022342_list = filter ((notElem 1) . a035516_row) [0..]
-- Reinhard Zumkeller, Mar 10 2013

[Floor(n*(Sqrt(5)+1)/2)-1: n in [1..100]]; // Vincenzo Librandi, Feb 16 2015

A022342 := proc(n)
      local g;
      g := (1+sqrt(5))/2 ;
    floor(n*g)-1 ;
end proc: # R. J. Mathar, Aug 04 2013

With[{t=GoldenRatio^2},Table[Floor[n*t]-n-1,{n,70}]] (* Harvey P. Dale, Aug 08 2012 *)

PARI
```
a(n)=floor(n*(sqrt(5)+1)/2)-1
```

a(n)=(sqrtint(5*n^2)+n-2)\2 \\ Charles R Greathouse IV, Feb 27 2014

from math import isqrt
def A022342(n): return (n+isqrt(5*n**2)>>1)-1 # Chai Wah Wu, Aug 17 2022

a(n) = floor(n*phi^2) - n - 1 = floor(n*phi) - 1 = A000201(n) - 1, where phi is the golden ratio.
a(n) = A003622(n) - n. - Philippe Deléham, May 03 2004
a(n+1) = A022290(2*A003714(n)). - R. J. Mathar, Jan 31 2015
For n > 1: A035612(a(n)) > 1. - Reinhard Zumkeller, Feb 03 2015
a(n) = A000201(n) - 1. First differences are given in A014675 (or A001468, ignoring its first term). - M. F. Hasler, Oct 13 2017
a(n) = a(n-1) + 1 + A005614(n-2) for n > 1; also  a(n) = a(n-1) + A014675(n-2) = a(n-1) + A001468(n-1). - A.H.M. Smeets, Apr 26 2024

Name edited by Peter Munn, Dec 07 2021

A035336 a(n) = 2floor(nphi) + n - 1, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 7, 10, 15, 20, 23, 28, 31, 36, 41, 44, 49, 54, 57, 62, 65, 70, 75, 78, 83, 86, 91, 96, 99, 104, 109, 112, 117, 120, 125, 130, 133, 138, 143, 146, 151, 154, 159, 164, 167, 172, 175, 180, 185, 188, 193, 198, 201, 206, 209, 214, 219, 222, 227, 230, 235, 240

Offset: 1

N. J. A. Sloane and J. H. Conway

nonn

Second column of Wythoff array.
These are the numbers in A022342 that are not images of another value of the same sequence if it is given offset 0. - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001
Also, positions of 2's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011
From Amiram Eldar, Mar 21 2022: (Start)
Numbers k for which the Zeckendorf representation A014417(k) ends with 0, 1, 0.
The asymptotic density of this sequence is sqrt(5)-2. (End)

Reinhard Zumkeller, 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.
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS, Vol. 11 (2008), Article 08.3.3.
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.
Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 5.
N. J. A. Sloane, Classic Sequences.

Cf. A001622, A014417, A022342, A066096.
Cf. A139764, A089910, A194584.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.

import Data.List (elemIndices)
a035336 n = a035336_list !! (n-1)
a035336_list = elemIndices 0 a005713_list
-- Reinhard Zumkeller, Dec 30 2011

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

Digits := 100: t := (1+sqrt(5))/2; [ seq(2*floor((n+1)*t)+n,n=0..80) ];

Table[2*Floor[n*(1 + Sqrt[5])/2] + n - 1, {n, 50}] (* Wesley Ivan Hurt, Nov 21 2017 *)
Array[2 Floor[# GoldenRatio] + # - 1 &, 60] (* Robert G. Wilson v, Dec 12 2017 *)

from sympy import floor
from mpmath import phi
def a(n): return 2*floor(n*phi) + n - 1 # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A035336(n): return (n+isqrt(5*n**2)&-2)+n-1 # Chai Wah Wu, Aug 17 2022

a(n) = B(A(n)), with A(k)=A000201(k) and B(k)=A001950(k) (Wythoff BA-numbers).
a(n) = A(n) + A(A(n)), with A(A(n))=A003622(n) (Wythoff AA-numbers).
Equals A022342(A003622(n)+1). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001, sequence reference updated by Peter Munn, Nov 23 2017
a(n) = 2*A003622(n) - (n - 1) = A003623(n) - 1. - Franklin T. Adams-Watters, Jun 30 2009
A005713(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = A089910(n) - 2. - Bob Selcoe, Sep 21 2014

A004922 a(n) = floor(n*phi^7), where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 871, 900, 929, 958, 987, 1016, 1045, 1074, 1103, 1132, 1161, 1190, 1219, 1248, 1277, 1306, 1335, 1364

Offset: 0

N. J. A. Sloane

nonn

Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A004919, A004920, A004921, A004923, A004924, A004925, A004926.
Cf. A004927, A004928, A004929, A004930, A004931, A004932, A004933.
Cf. A004934, A004935, A004976, A066096, A090909.
Cf. A001622, A004962 (ceiling).

[Floor(n*((1 + Sqrt(5))/2)^7): n in [0..50]]; // Vincenzo Librandi, Jul 22 2015

Table[Floor[n ((1 + Sqrt[5])/2)^7], {n, 0, 50}] (* Vincenzo Librandi, Jul 22 2015 *)

from sympy import sqrt
phi = (1 + sqrt(5))/2
for n in range(0,101): print(int(n*phi**7), end=', ') # Karl V. Keller, Jr., Jul 22 2015

A090909 Terms a(k) of A073869 for which a(k-1) = a(k), and a(k) and a(k+1) are distinct.

Original entry on oeis.org

Offset: 1

Amarnath Murthy, Dec 14 2003

nonn

Is this the same as A001950? - Alec Mihailovs (alec(AT)mihailovs.com), Jul 23 2007
Identical to n + A066096(n)? - Ed Russell (times145(AT)hotmail.com), May 09 2009
From Michel Dekking, Dec 18 2024: (Start)
Proof of Mihailovs's conjecture: This follows immediately from the result in my 2023 paper in JIS that A073869 is equal to Hofstadter’s G-sequence A005206, and my recent comment in A005206 on the pairs of duplicate values in A005206.
The answer to Russell’s question is well-known, and also Detlef’s formula is well-known.
Originally, this sequence was given the name “Terms a(k) of A073869 for which a(k-1), a(k) and a(k+1) are distinct.’’ These are the triples (1,2,3),(4,5,6),(6,7,8), (9,10,11), ... occurring at  k = 3, k = 8, k = 11, k = 16,... in A005206. Note that if a duplicate pair (a(m-1), a(m)) is followed directly by another duplicate pair, then a(m-3), a(m-2) and a(m-1) are distinct, and only so. This corresponds to the block 00 occurring in the Fibonacci word obtained by projecting A005206 on the Fibonacci word (see Corollary in my recent comment in A005206). These occurrences are at the Wythoff AB numbers A003623 according to Wolfdieter Lang’s comment in A003623. Conclusion: the sequence of terms a(k) of A073869 for which a(k-1), a(k), and a(k+1) are distinct is given by the Wythoff AB-numbers. (End)

			A073869 = A005206 = 0,1,1,2,3,3,4,4,5,6,6,... The pair (1,1) occurs at k = 2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
F. M. Dekking, On Hofstadter's G-sequence, Journal of Integer Sequences 26 (2023), Article 23.9.2, 1-11.

Cf. A001622, A001950, A002251, A005206, A073869, A090908.
Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004935, A004976, A066096.

[Floor((3+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

(* First program *)
A002251= Fold[Append[#1, #2 Ceiling[#2/GoldenRatio] -Total[#1]] &, {1}, Range[2, 500]] - 1; (* Birkas Gyorgy's code of A019444, modified *)
A090909= Join[{0}, Select[Partition[A002251, 2, 1], #[[2]] > #[[1]] &][[All, 2]]] (* G. C. Greubel, Sep 12 2023 *)
(* Second program *)
Floor[GoldenRatio^2*Range[0,80]] (* G. C. Greubel, Sep 12 2023 *)

[floor(golden_ratio^2*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

a(n) = floor(phi^2*n), where phi = (1+sqrt(5))/2. - Gary Detlefs, Mar 10 2011

More terms from R. J. Mathar, Sep 29 2017

Name corrected by Michel Dekking, Dec 13 2024

cp's OEIS Frontend

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

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A003622 The Wythoff compound sequence AA: a(n) = floor(n*phi^2) - 1, where phi = (1+sqrt(5))/2.

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A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

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A022342 Integers with "even" Zeckendorf expansions (do not end with ...+F_2 = ...+1) (the Fibonacci-even numbers); also, apart from first term, a(n) = Fibonacci successor to n-1.

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A035336 a(n) = 2*floor(n*phi) + n - 1, where phi = (1+sqrt(5))/2.

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A035336 a(n) = 2floor(nphi) + n - 1, where phi = (1+sqrt(5))/2.