|
|
A054770
|
|
Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, ... (A000204).
|
|
16
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
LINKS
|
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Lucas representations, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
|
|
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)
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 *)
|
|
PROG
|
(PARI) a(n)=floor(n*(sqrt(5)+5)/2)-1
(Python)
from math import isqrt
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 28 2000
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|