A035336 a(n) = 2*floor(n*phi) + n - 1, where phi = (1+sqrt(5))/2.

2, 7, 10, 15, 20, 23, 28, 31, 36, 41, 44, 49, 54, 57, 62, 65, 70, 75, 78, 83, 86, 91, 96, 99, 104, 109, 112, 117, 120, 125, 130, 133, 138, 143, 146, 151, 154, 159, 164, 167, 172, 175, 180, 185, 188, 193, 198, 201, 206, 209, 214, 219, 222, 227, 230, 235, 240

Offset

1,1

Comments

Second column of Wythoff array.

These are the numbers in A022342 that are not images of another value of the same sequence if it is given offset 0. - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001

Also, positions of 2's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011

From Amiram Eldar, Mar 21 2022: (Start)

Numbers k for which the Zeckendorf representation A014417(k) ends with 0, 1, 0.

The asymptotic density of this sequence is sqrt(5)-2. (End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.

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.

Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).

Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), pp. 267-273.

N. J. A. Sloane, Classic Sequences.

Formula

a(n) = B(A(n)), with A(k)=A000201(k) and B(k)=A001950(k) (Wythoff BA-numbers).

a(n) = A(n) + A(A(n)), with A(A(n))=A003622(n) (Wythoff AA-numbers).

Equals A022342(A003622(n)+1). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001, sequence reference updated by Peter Munn, Nov 23 2017

a(n) = 2*A003622(n) - (n - 1) = A003623(n) - 1. - Franklin T. Adams-Watters, Jun 30 2009

A005713(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011

a(n) = A089910(n) - 2. - Bob Selcoe, Sep 21 2014

Maple

Digits := 100: t := (1+sqrt(5))/2; [ seq(2*floor((n+1)*t)+n, n=0..80) ];

Mathematica

Table[2*Floor[n*(1 + Sqrt[5])/2] + n - 1, {n, 50}] (* Wesley Ivan Hurt, Nov 21 2017 *)
Array[2 Floor[# GoldenRatio] + # - 1 &, 60] (* Robert G. Wilson v, Dec 12 2017 *)

Prog

(Haskell)
import Data.List (elemIndices)
a035336 n = a035336_list !! (n-1)
a035336_list = elemIndices 0 a005713_list
-- Reinhard Zumkeller, Dec 30 2011
(Magma) [2*Floor(n*(1+Sqrt(5))/2)+n-1: n in [1..80]]; // Vincenzo Librandi, Nov 19 2016
(Python)
from sympy import floor
from mpmath import phi
def a(n): return 2*floor(n*phi) + n - 1 # Indranil Ghosh, Jun 10 2017
(Python)
from math import isqrt
def A035336(n): return (n+isqrt(5*n**2)&-2)+n-1 # Chai Wah Wu, Aug 17 2022

Crossrefs

Cf. A001622, A014417, A022342, A066096.

Cf. A139764, A089910, A194584.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.

Sequence in context: A190447 A190375 A066097 * A351388 A246128 A343990

Adjacent sequences: A035333 A035334 A035335 * A035337 A035338 A035339

Keyword

nonn

Author

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

Status

approved