A063732 -id:A063732 - cp's OEIS frontend

A117407 a(n) = j if n is T(j), else a(n) = k if n is U(k), where T is a Beatty sequence based on (sqrt(5)+5)/2 (A054770) and U is its complement (A063732).

1, 2, 1, 3, 4, 5, 2, 6, 7, 3, 8, 9, 10, 4, 11, 12, 13, 5, 14, 15, 6, 16, 17, 18, 7, 19, 20, 8, 21, 22, 23, 9, 24, 25, 26, 10, 27, 28, 11, 29, 30, 31, 12, 32, 33, 34, 13, 35, 36, 14, 37, 38, 39, 15, 40, 41, 16, 42, 43, 44, 17, 45, 46, 47, 18, 48, 49, 19, 50, 51, 52, 20, 53, 54, 21, 55

Offset: 0

Casey Mongoven, Mar 13 2006

nonn

Every positive integer occurs exactly twice. Taking a Lucas number (A000032) of terms L(n) starting at a(0), the last two terms are a pair of Fibonacci numbers (A000045). If n is even, then the last two terms are F(n+1) followed by F(n-1), if n is odd they are F(n-1) followed by F(n+1), where F is the Fibonacci sequence. For example, the first L(4) = 7 terms of this sequence are (1,2,1,3,4,5,2) and the last members are 5 and 2 which are equal to F(5) and F(3). Note also that L(n) = F(n-1) + F(n+1).

			a(9) = 3 because 9 = T(3).

Cf. A026272, A026242.

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

Original entry on oeis.org

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

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

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

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).

Complement of A063732.

Cf. A003263, A003622, A022342, A130310.

[Floor(n*(Sqrt(5)+5)/2)-1: n in [1..60]]; // Vincenzo Librandi, Oct 30 2018

A054770 := n -> floor(n*(sqrt(5)+5)/2)-1;

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
```

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

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

More terms from James Sellers, May 28 2000

cp's OEIS Frontend

A117407 a(n) = j if n is T(j), else a(n) = k if n is U(k), where T is a Beatty sequence based on (sqrt(5)+5)/2 (A054770) and U is its complement (A063732).

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

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