A054770 Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, ... (A000204).

2, 6, 9, 13, 17, 20, 24, 27, 31, 35, 38, 42, 46, 49, 53, 56, 60, 64, 67, 71, 74, 78, 82, 85, 89, 93, 96, 100, 103, 107, 111, 114, 118, 122, 125, 129, 132, 136, 140, 143, 147, 150, 154, 158, 161, 165, 169, 172, 176, 179, 183, 187, 190, 194, 197, 201, 205, 208, 212

Offset: 1

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 28 2000

Keywords

• nonn
• easy

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).

Crossrefs

Complement of A063732.

Cf. A003263, A003622, A022342, A130310.

Programs

Magma

[Floor(n*(Sqrt(5)+5)/2)-1: n in [1..60]]; // Vincenzo Librandi, Oct 30 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 *)

PARI

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

Python

from math import isqrt
def A054770(n): return (n+isqrt(5*n**2)>>1)+(n<<1)-1 # Chai Wah Wu, Aug 17 2022

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

Extensions

More terms from James Sellers, May 28 2000