A134864 Wythoff BBB numbers.

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

1,1

Comments

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

Links

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.

Formula

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A000201, A001950, A003622, A003623, A035336, 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.

Sequence in context: A124659 A164539 A245170 * A093100 A292472 A081271

Adjacent sequences: A134861 A134862 A134863 * A134865 A134866 A134867

Keyword

nonn

Author

Clark Kimberling, Nov 14 2007

Status

approved