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

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

1,1

Comments

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

Links

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.

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A045042 A367930 A095099 * A301001 A047958 A300747

Adjacent sequences: A134857 A134858 A134859 * A134861 A134862 A134863

Keyword

nonn,base,changed

Author

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

Extensions

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

Status

approved