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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. 13
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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..55.

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

Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences, 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):

k=0

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 1 if str(a(n))[-3:]=="101" else 0

print [n for n in xrange(12, 501) if ok(n)] # Indranil Ghosh, Jun 08 2017

CROSSREFS

Cf. A000201, 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: A036047 A045042 A095099 * A047958 A175704 A111371

Adjacent sequences:  A134857 A134858 A134859 * A134861 A134862 A134863

KEYWORD

nonn

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

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Last modified February 23 07:28 EST 2018. Contains 299473 sequences. (Running on oeis4.)