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

Author

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

Keywords

• nonn

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.
• Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
• 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.
• Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 5.
• N. J. A. Sloane, Classic Sequences.

Crossrefs

Cf. A001622, A014417, A022342, A066096.

Cf. A139764, A089910, A194584.

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

Programs

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

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 *)

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

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