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%I #72 Jan 21 2023 03:17:20
%S 2,6,9,13,17,20,24,27,31,35,38,42,46,49,53,56,60,64,67,71,74,78,82,85,
%T 89,93,96,100,103,107,111,114,118,122,125,129,132,136,140,143,147,150,
%U 154,158,161,165,169,172,176,179,183,187,190,194,197,201,205,208,212
%N Numbers that are not the sum of distinct Lucas numbers 1,3,4,7,11, ... (A000204).
%C Alternatively, Lucas representation of n includes L_0 = 2. - _Fred Lunnon_, Aug 25 2001
%C 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
%C 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
%C Numbers whose minimal Lucas representation (A130310) ends with 1. - _Amiram Eldar_, Jan 21 2023
%H G. C. Greubel, <a href="/A054770/b054770.txt">Table of n, a(n) for n = 1..5000</a>
%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="https://www.fq.math.ca/Scanned/10-1/carlitz2-a.pdf">Lucas representations</a>, Fibonacci Quart. 10 (1972), 29-42, 70, 112.
%H Weiru Chen and Jared Krandel, <a href="https://arxiv.org/abs/1810.11938">Interpolating Classical Partitions of the Set of Positive Integers</a>, arXiv:1810.11938 [math.NT], 2018. See sequence D2 p. 4.
%H Michel Dekking, <a href="https://arxiv.org/abs/1906.08437">Base phi representations and golden mean beta-expansions</a>, arXiv:1906.08437 [math.NT], 2019.
%H Jared Krandel and Weiru Chen, <a href="https://doi.org/10.1007/s11139-019-00196-3">Interpolating classical partitions of the set of positive integers</a>, The Ramanujan Journal (2020).
%F 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)
%F a(n) = A000201(n) + 2*n - 1. - _Michel Dekking_, Sep 07 2017
%F 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
%F 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
%p A054770 := n -> floor(n*(sqrt(5)+5)/2)-1;
%t Complement[Range[220],Total/@Subsets[LucasL[Range[25]],5]] (* _Harvey P. Dale_, Feb 27 2012 *)
%t Table[Floor[n (Sqrt[5] + 5) / 2] - 1, {n, 60}] (* _Vincenzo Librandi_, Oct 30 2018
%o (PARI) a(n)=floor(n*(sqrt(5)+5)/2)-1
%o (Magma) [Floor(n*(Sqrt(5)+5)/2)-1: n in [1..60]]; // _Vincenzo Librandi_, Oct 30 2018
%o (Python)
%o from math import isqrt
%o def A054770(n): return (n+isqrt(5*n**2)>>1)+(n<<1)-1 # _Chai Wah Wu_, Aug 17 2022
%Y Complement of A063732.
%Y Cf. A003263, A003622, A022342, A130310.
%K nonn,easy
%O 1,1
%A Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 28 2000
%E More terms from _James A. Sellers_, May 28 2000
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