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

1,4

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.

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

Mathematica

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

Prog

(PARI) f=(1+sqrt(5))/2; a(n)=floor(f*floor(n/f))
(Magma) [Floor((1+Sqrt(5))*Floor(2*n/(1+Sqrt(5)))/2): n in [1..100]]; // G. C. Greubel, Oct 23 2018
(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

Crossrefs

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

Sequence in context: A213222 A166737 A088847 * A277193 A240728 A164326

Adjacent sequences: A120610 A120611 A120612 * A120614 A120615 A120616

Keyword

nonn

Author

Benoit Cloitre, Jun 17 2006

Extensions

Offset changed by Michel Dekking, Oct 23 2018

Status

approved