A134860 - OEIS (original) (raw)

A134860

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

15

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

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

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.

EXTENSIONS

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