A134861 -id:A134861 - cp's OEIS frontend

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

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

A035337 Third column of Wythoff array.

Original entry on oeis.org

3, 11, 16, 24, 32, 37, 45, 50, 58, 66, 71, 79, 87, 92, 100, 105, 113, 121, 126, 134, 139, 147, 155, 160, 168, 176, 181, 189, 194, 202, 210, 215, 223, 231, 236, 244, 249, 257, 265, 270, 278, 283, 291, 299, 304, 312

Offset: 0

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

nonn

Also, positions of 3's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011
The formula a(n) = 3*A003622(n)-n+1 = 3AA(n)-n+1 conjectured by Layman below is correct, since it is well known that AA(n)+1 = B(n) = A(n)+n, where B = A001950, and so 3AA(n)-n+1 = 3B(n)-n-2 = 3A(n)+2n-2. - Michel Dekking, Aug 31 2017
From Amiram Eldar, Mar 21 2022: (Start)
Numbers k for which the Zeckendorf representation A014417(k) ends with 1, 0, 0.
The asymptotic density of this sequence is 1/phi^4 = 2/(7+3*sqrt(5)), where phi is the golden ratio (A001622). (End)

Amiram Eldar, Table of n, a(n) for n = 0..10000
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.
N. J. A. Sloane, Classic Sequences.

Cf. A001622, A014417, A139764.

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.

t := (1+sqrt(5))/2 ; [ seq(3*floor((n+1)*t)+2*n,n=0..80) ];

Table[3 Floor[n GoldenRatio] + 2 n - 2, {n, 46}] (* Michael De Vlieger, Aug 31 2017 *)

a(n) = 2*n + 3*floor((1+sqrt(5))*(n+1)/2); \\ Altug Alkan, Sep 18 2017

from sympy import floor
from mpmath import phi
def a(n): return 3*floor((n + 1)*phi) + 2*n # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A035337(n): return 3*(n+isqrt(5*n**2)>>1)+(n-1<<1) # Chai Wah Wu, Aug 11 2022

a(n) = F(4)A(n)+F(3)(n-1) = 3A(n)+2n-2, where A = A000201 and F = A000045. - Michel Dekking, Aug 31 2017

It appears that a(n) = 3*A003622(n) - n + 1. - John W. Layman, Aug 25 2011

A035338 4th column of Wythoff array.

Original entry on oeis.org

5, 18, 26, 39, 52, 60, 73, 81, 94, 107, 115, 128, 141, 149, 162, 170, 183, 196, 204, 217, 225, 238, 251, 259, 272, 285, 293, 306, 314, 327, 340, 348, 361, 374, 382, 395, 403, 416, 429, 437, 450, 458, 471, 484, 492, 505, 518, 526, 539, 547, 560, 573, 581, 594

Offset: 0

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

nonn

The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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. 5.
John 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.
N. J. A. Sloane, Classic 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.

Cf. A035513, A244593.

t := (1+sqrt(5))/2 ; [ seq(5*floor((n+1)*t)+3*n,n=0..80) ];

f[n_] := 5 Floor[(n + 1) GoldenRatio] + 3n; Array[f, 54, 0] (* Robert G. Wilson v, Dec 11 2017 *)

from math import isqrt
def A035338(n): return 5*(n+1+isqrt(5*(n+1)**2)>>1)+3*n # Chai Wah Wu, Aug 11 2022

A134859 Wythoff AAA numbers.

Original entry on oeis.org

1, 6, 9, 14, 19, 22, 27, 30, 35, 40, 43, 48, 53, 56, 61, 64, 69, 74, 77, 82, 85, 90, 95, 98, 103, 108, 111, 116, 119, 124, 129, 132, 137, 142, 145, 150, 153, 158, 163, 166, 171, 174, 179, 184, 187, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229, 234, 239, 242

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAA = AB - 2 and AAA = A + B - 2.
Also numbers with suffix string 001, when written in Zeckendorf representation (with leading zero for the first term). - A.H.M. Smeets, Mar 20 2024
The asymptotic density of this sequence is 1/phi^3 = phi^3 - 4 = A098317 - 4 = 0.236067... . - Amiram Eldar, Mar 24 2025

			Starting with A=(1,3,4,6,8,9,11,12,14,16,17,19,...), we have A(2)=3, so A(A(2))=4, so A(A(A(2)))=6.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
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.
Aviezri S. Fraenkel, Complementary iterated floor words and the Flora game, SIAM J. Discrete Math. 24 (2010), no. 2, 570-588.
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), Article 15.11.8.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 11 (2008), Article 08.3.3.
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.

Cf. A001622, A000201, A001950, A003622, A003623, A035336, A098317, A101864, A134860, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
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.
Essentially the same as A095098.

# For Maple code for these Wythoff compound sequences see A003622. - N. J. A. Sloane, Mar 30 2016

A[n_] := Floor[n GoldenRatio];
a[n_] := A@ A@ A@ n;
a /@ Range[100] (* Jean-François Alcover, Oct 28 2019 *)

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

from math import isqrt
def A134859(n): return ((n+isqrt(5*n**2)>>1)-1<<1)+n # Chai Wah Wu, Aug 10 2022

a(n) = A(A(A(n))), n >= 1, with A=A000201, the lower Wythoff sequence.
a(n) = 2*floor(n*Phi^2) - n - 2 where Phi = (1+sqrt(5))/2. - Benoit Cloitre, Apr 12 2008; R. J. Mathar, Oct 16 2009
a(n) = A095098(n-1), n > 1. - R. J. Mathar, Oct 16 2009
From A.H.M. Smeets, Mar 23 2024: (Start)
a(n) = A(n) + B(n) - 2 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.
Equals {A003622}\{A134860} (= Wythoff AA \ Wythoff AAB). (End)

Incorrect PARI program removed by R. J. Mathar, Oct 16 2009

A134860 Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.

Original entry on oeis.org

4, 12, 17, 25, 33, 38, 46, 51, 59, 67, 72, 80, 88, 93, 101, 106, 114, 122, 127, 135, 140, 148, 156, 161, 169, 177, 182, 190, 195, 203, 211, 216, 224, 232, 237, 245, 250, 258, 266, 271, 279, 284, 292, 300, 305, 313, 321, 326, 334, 339, 347, 355, 360, 368, 373

Offset: 1

Antti Karttunen, Jun 01 2004 and Clark Kimberling, Nov 14 2007

The lower and upper Wythoff sequences, A and B, satisfy the complementary equations AAB=AA+AB and AAB=A+2B-1.

The asymptotic density of this sequence is 1/phi^4 = 2/(7+3*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Mar 21 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
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
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.3.

Cf. A000201, A001622, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134861, A134862, A134863, A035338, A134864, A035513.
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.
Set-wise difference A003622 \ A095098. Cf. A095089 (fib101 primes).

With[{r = Map[Fibonacci, Range[2, 14]]}, Position[#, {1, 0, 1}][[All, 1]] &@ Table[If[Length@ # < 3, {}, Take[#, -3]] &@ IntegerDigits@ Total@ Map[FromDigits@ PadRight[{1}, Flatten@ #] &@ Reverse@ Position[r, #] &,Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], n + 1, # > 1 &]], {n, 373}]] (* Michael De Vlieger, Jun 09 2017 *)

from sympy import fibonacci
def a(n):
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n): return str(a(n))[-3:]=="101"
print([n for n in range(4, 501) if ok(n)]) # Indranil Ghosh, Jun 08 2017

from math import isqrt
def A134860(n): return 3*(n+isqrt(5*n**2)>>1)+(n<<1)-1 # Chai Wah Wu, Aug 10 2022

a(n) = A(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.

This is the result of merging two sequences which were really the same. - N. J. A. Sloane, Jun 10 2017

A134862 Wythoff ABB numbers.

Original entry on oeis.org

8, 21, 29, 42, 55, 63, 76, 84, 97, 110, 118, 131, 144, 152, 165, 173, 186, 199, 207, 220, 228, 241, 254, 262, 275, 288, 296, 309, 317, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 461, 474, 487, 495, 508, 521, 529, 542, 550, 563, 576, 584, 597

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation ABB = 2A+3B.

The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - Amiram Eldar, Mar 24 2025

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. 5.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134860, A134861, A134863, A035338, A134864, A035513, A244593.

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.

A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := A[B[B[n]]]; Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

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

from math import isqrt
def A134862(n): return 5*(n+isqrt(5*n**2)>>1)+3*n # Chai Wah Wu, Aug 10 2022

a(n) = A(B(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.

A134864 Wythoff BBB numbers.

Original entry on oeis.org

13, 34, 47, 68, 89, 102, 123, 136, 157, 178, 191, 212, 233, 246, 267, 280, 301, 322, 335, 356, 369, 390, 411, 424, 445, 466, 479, 500, 513, 534, 555, 568, 589, 610, 623, 644, 657, 678, 699, 712, 733, 746, 767, 788, 801, 822, 843, 856, 877, 890, 911, 932, 945

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB = 3A+5B.

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134863, A035513.

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.

a:=n->floor(n*((1+sqrt(5))/2)^2): [a(a(a(n)))$n=1..55]; # Muniru A Asiru, Nov 24 2018

Nest[Quotient[#(3+Sqrt@5),2]&,#,3]&/@Range@100 (* Federico Provvedi, Nov 24 2018 *)
b[n_]:=Floor[n GoldenRatio^2]; a[n_]:=b[b[b[n]]]; Array[a, 60] (* Vincenzo Librandi, Nov 24 2018 *)

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

from math import isqrt
def A134864(n): return (m:=5*n)+(((n+isqrt(n*m))&-2)<<2) # Chai Wah Wu, Aug 10 2022

a(n) = B(B(B(n))), n>=1, with B=A001950, the upper Wythoff sequence.

A134863 Wythoff BAB numbers.

Original entry on oeis.org

7, 20, 28, 41, 54, 62, 75, 83, 96, 109, 117, 130, 143, 151, 164, 172, 185, 198, 206, 219, 227, 240, 253, 261, 274, 287, 295, 308, 316, 329, 342, 350, 363, 376, 384, 397, 405, 418, 431, 439, 452, 460, 473, 486, 494, 507, 520, 528, 541, 549, 562, 575, 583, 596

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAB = 2A+3B-1.
Also numbers with suffix string 1010, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 24 2024
The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - Amiram Eldar, Mar 24 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
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. 5.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134864, A035513, A244593.

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.

A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := B[A[B[n]]]; Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

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

from math import isqrt
def A134863(n): return 5*(n+isqrt(5*n**2)>>1)+3*n-1 # Chai Wah Wu, Aug 11 2022

a(n) = B(A(B(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.
From A.H.M. Smeets, Mar 24 2024: (Start)
a(n) = 2*A(n) + 3*B(n) - 1 (see Clark Kimberling 2008), with A=A000201, B=A001950, the lower and upper Wythoff sequences, respectively.
Equals {A035336}\{A134861} (= Wythoff BA \ Wythoff BAA). (End)

A246355 Rectangular array: T(n,k) is the position in the infinite Fibonacci word s = A003849 at which the block s(2)..s(n+1) occurs for the k-th time.

Original entry on oeis.org

2, 5, 2, 7, 5, 2, 10, 7, 7, 2, 13, 10, 10, 7, 2, 15, 13, 15, 10, 7, 2, 18, 15, 20, 15, 10, 10, 2, 20, 18, 23, 20, 15, 15, 10, 2, 23, 20, 28, 23, 20, 23, 15, 10, 2, 26, 23, 31, 28, 23, 31, 23, 15, 10, 2, 28, 26, 36, 31, 28, 36, 31, 23, 15, 10, 2, 31, 28, 41

Offset: 1

Clark Kimberling, Aug 24 2014

Assuming that every row of T is infinite, each row contains the next row as a proper subsequence. Row 1 of A246354 and row 1 of A246355 partition the positive integers.

			The upper Wythoff sequence, A001950 gives the positions of 1 in A003849, which begins thus:  0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1.  For n = 1, the block s(2)..s(2) is simply 1, which occurs at positions 2,5,7,10,13,... as in row 1 of T.  For n = 5, the block s(2)..s(6) is 1,0,0,1,0 which occurs at positions 2,7,10,15,20,23, ...
The first 6 rows follow:
2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
2 .. 5 .. 7 ... 10 .. 13 .. 15 .. 18 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 7 .. 10 .. 15 .. 20 .. 23 .. 28 ...
2 .. 10 . 15 .. 23 .. 31 .. 36 .. 44 ...

Cf. A003849, A246354.

z = 1000; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, 12]]; Flatten[Position[s, 1]]; b[m_, n_] := b[m, n] = Take[s, {m, n}]; z1 = 500; z2 = 12; t[k_] := t[k] = Take[Select[Range[1, z1], b[#, # + k] == b[2, 2 + k] &], z2]; Column[Table[t[k], {k, 0, z2}]](* A246355, array *)
w[n_, k_] := t[n][[k + 1]]; Table[w[n - k, k], {n, 0, z2 - 1}, {k, n, 0, -1}] // Flatten (* A246355, sequence *)

First 2 rows:  A001950 (upper Wythoff numbers);
next 3 rows:  A035336 (Wythoff BA numbers);
next 5 rows:  A134861 (Wythoff BAA numbers);
next 8 rows:  (Wythoff BAAA numbers).
(The patterns continue; in particular the number of identical consecutive rows is always a Fibonacci number, as in A000045.)

A372302 Numbers k for which the Zeckendorf representation A014417(k) ends with "1001".

Original entry on oeis.org

6, 19, 27, 40, 53, 61, 74, 82, 95, 108, 116, 129, 142, 150, 163, 171, 184, 197, 205, 218, 226, 239, 252, 260, 273, 286, 294, 307, 315, 328, 341, 349, 362, 375, 383, 396, 404, 417, 430, 438, 451, 459, 472, 485, 493, 506, 519, 527, 540, 548, 561, 574, 582, 595, 603

Offset: 1

A.H.M. Smeets, Apr 25 2024

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A000045, A005614, A014417.
Tree of Zeckendorf subsequences of positive integers partitioned by their suffix part S (except initial term or offset in some cases). $ is the empty string. length(S) =
0        1        2        3        4        5        6       7
----------------------------------------------------------------------
$:       0:       00:      000:     0000:    00000:   000000:
A000027  A022342  A026274  A101345  A101642  notOEIS  notOEIS
                                                      100000: 0100000:
                                                      A035340  <------
                                             10000:
                                             A035339
                                    1000:    01000:
                                    A035338  <------
                  10:      010:     0010:
                  A035336  <------  A134861
                                    1010:    01010:
                                    A134863  <------
                           100:     0100:
                           A035337  <------
         1:       01:      001:     0001:
         A003622  <------  A134859  A151915
                                    1001:    01001:
                                    A372302  <------
                           101:     0101:
                           A134860  <------
Suffixes 10^n, where ^ means n times repeated concatenation, are the (n+1)-th columns in the Wythoff array A083412 and A035513 (n >= 0).

Equals {A134859}\{A151915}.
a(n) = A134863(n) - 1 = A035338(n) + 1.
a(n) = a(n-1) + 8 + 5*A005614(n-2) = a(n-1) + F(6) + F(5)*A005614(n-2), n > 1, where F(k) is the k-th Fibonacci number (A000045).

cp's OEIS Frontend

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

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A035337 Third column of Wythoff array.

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A035338 4th column of Wythoff array.

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A134859 Wythoff AAA numbers.

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A134860 Wythoff AAB numbers; also, Fib101 numbers: those n for which the Zeckendorf expansion A014417(n) ends with 1,0,1.

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A134862 Wythoff ABB numbers.

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