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A026351

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

1, 2, 4, 5, 7, 9, 10, 12, 13, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 89, 91, 93, 94, 96, 98, 99

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n)=least k such that s(k)=n, where s=A026350.

a(n)=position of n-th 1 in A096270.

From Wolfdieter Lang, Jun 27 2011: (Start)

a(n) = A(n)+1, with Wythoff sequence A(n)=A000201(n), n>=1, and A(0)=0.

a(n) = -floor(-n*phi). Recall that floor(-x) = -(floor(x)+1) if x is not integer and -floor(x) otherwise.

An exhaustive and disjoint decomposition of the integers is given by the following two Wythoff sequences A' and B: A'(0):=-1 (not 0), A'(-n):=-a(n)=-(A(n)+1), n>=1, A'(n) = A(n), n>=1, and B(-n):=-(B(n)+1)= -A026352(n), n>=1, with B(n)=A001950(n), n>=1, and B(0)=0.

(End)

Where odd terms in A060142 occur: A060142(a(n)) = A219608(n). - Reinhard Zumkeller, Nov 26 2012

Positive integers k such that {k*phi} < {(k-1)*phi}, where phi = (1+sqrt(5))/2 and {...} denotes fractional part. - Jeffrey Shallit, Sep 23 2025

LINKS

Carmine Suriano, Table of n, a(n) for n = 0..10000

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. 3.

Benoit Cloitre, N. J. A. Sloane, and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.

Benoit Cloitre, N. J. A. Sloane, and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.

J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences

Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg, and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.

N. J. A. Sloane, Classic Sequences

MATHEMATICA

Table[Floor[n*GoldenRatio] + 1, {n, 0, 100}] (* T. D. Noe, Apr 15 2011 *)

PROG

(Haskell)

import Data.List (findIndices)

a026351 n = a026351_list !! n

a026351_list = findIndices odd a060142_list

-- Reinhard Zumkeller, Nov 26 2012

(Python)

from math import isqrt

def A026351(n): return (n+isqrt(5*n**2)>>1)+1 # Chai Wah Wu, Aug 17 2022