OFFSET
1,1
COMMENTS
Alternatively, Lucas representation of n includes L_0 = 2. - Fred Lunnon, Aug 25 2001
Conjecture: this is the sequence of numbers for which the base phi representation includes phi itself, where phi = (1 + sqrt(5))/2 = the golden ratio. Example: let r = phi; then 6 = r^3 + r + r^(-4). - Clark Kimberling, Oct 17 2012
This conjecture is proved in my paper 'Base phi representations and golden mean beta-expansions', using the formula by Wilson/Agol/Carlitz et al. - Michel Dekking, Jun 25 2019
Numbers whose minimal Lucas representation (A130310) ends with 1. - Amiram Eldar, Jan 21 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Lucas representations, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
Weiru Chen and Jared Krandel, Interpolating Classical Partitions of the Set of Positive Integers, arXiv:1810.11938 [math.NT], 2018. See sequence D2 p. 4.
Michel Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.
Jared Krandel and Weiru Chen, Interpolating classical partitions of the set of positive integers, The Ramanujan Journal (2020).
FORMULA
a(n) = floor(((5+sqrt(5))/2)*n)-1 (conjectured by David W. Wilson; proved by Ian Agol (iagol(AT)math.ucdavis.edu), Jun 08 2000)
a(n) = A000201(n) + 2*n - 1. - Michel Dekking, Sep 07 2017
G.f.: x*(x+1)/(1-x)^2 + Sum_{i>=1} (floor(i*phi)*x^i), where phi = (1 + sqrt(5))/2. - Iain Fox, Dec 19 2017
Ian Agol tells me that David W. Wilson's formula is proved in the Carlitz, Scoville, Hoggatt paper 'Lucas representations'. See Equation (1.12), and use A(A(n))+n = B(n)+n-1 = A(n)+2*n-1, the well known formulas for the lower Wythoff sequence A = A000201, and the upper Wythoff sequence B = A001950. - Michel Dekking, Jan 04 2018
MAPLE
A054770 := n -> floor(n*(sqrt(5)+5)/2)-1;
MATHEMATICA
Complement[Range[220], Total/@Subsets[LucasL[Range[25]], 5]] (* Harvey P. Dale, Feb 27 2012 *)
Table[Floor[n (Sqrt[5] + 5) / 2] - 1, {n, 60}] (* Vincenzo Librandi, Oct 30 2018 *)
PROG
(PARI) a(n)=floor(n*(sqrt(5)+5)/2)-1
(Magma) [Floor(n*(Sqrt(5)+5)/2)-1: n in [1..60]]; // Vincenzo Librandi, Oct 30 2018
(Python)
from math import isqrt
def A054770(n): return (n+isqrt(5*n**2)>>1)+(n<<1)-1 # Chai Wah Wu, Aug 17 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 28 2000
EXTENSIONS
More terms from James A. Sellers, May 28 2000
STATUS
approved