A120613 a(n) = floor(phi*floor(n/phi)) where phi=(1+sqrt(5))/2.

0, 1, 1, 3, 4, 4, 6, 6, 8, 9, 9, 11, 12, 12, 14, 14, 16, 17, 17, 19, 19, 21, 22, 22, 24, 25, 25, 27, 27, 29, 30, 30, 32, 33, 33, 35, 35, 37, 38, 38, 40, 40, 42, 43, 43, 45, 46, 46, 48, 48, 50, 51, 51, 53, 53, 55, 56, 56, 58, 59, 59, 61, 61, 63, 64, 64, 66, 67, 67, 69, 69, 71, 72

Offset: 1

Author

Benoit Cloitre, Jun 17 2006

Keywords

• nonn

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (corrected by Michel Dekking, uploaded again by _Georg Fischer_, Jan 31 2019)
• F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
• Martin Griffiths, The Golden String, Zeckendorf Representations, and the Sum of a Series, Amer. Math. Monthly, 118 (2011), 497-507. See p. 502.

Crossrefs

Cf. A001622, A120614 (first differences), A120615 (partial sums), A003842.

Programs

Magma

[Floor((1+Sqrt(5))*Floor(2*n/(1+Sqrt(5)))/2): n in [1..100]]; // G. C. Greubel, Oct 23 2018

Mathematica

Table[Floor[GoldenRatio*Floor[n/GoldenRatio]], {n,1,100}] (* G. C. Greubel, Oct 23 2018 *)

PARI

f=(1+sqrt(5))/2;a(n)=floor(f*floor(n/f))

Python

from math import isqrt
def A120613(n): return (m:=(n+isqrt(5*n**2)>>1)-n)+isqrt(5*m**2)>>1 # Chai Wah Wu, Aug 26 2022

Formula

a(n) = n - A003842(n-2) for n >= 2. [Corrected by Georg Fischer, Jan 31 2019]

In particular, a(n) = n-1 or a(n) = n-2. - Charles R Greathouse IV, Aug 26 2022

Extensions

Offset changed by Michel Dekking, Oct 23 2018