A035337 Third column of Wythoff array.

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

0,1

Comments

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)

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

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.

Formula

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

Maple

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

Mathematica

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

Prog

(Python)
from sympy import floor
from mpmath import phi
def a(n): return 3*floor((n + 1)*phi) + 2*n # Indranil Ghosh, Jun 10 2017
(Python)
from math import isqrt
def A035337(n): return 3*(n+isqrt(5*n**2)>>1)+(n-1<<1) # Chai Wah Wu, Aug 11 2022
(PARI) a(n) = 2*n + 3*floor((1+sqrt(5))*(n+1)/2); \\ Altug Alkan, Sep 18 2017

Crossrefs

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.

Sequence in context: A158507 A030765 A198515 * A029500 A243770 A298701

Adjacent sequences: A035334 A035335 A035336 * A035338 A035339 A035340

Keyword

nonn

Author

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

Status

approved