A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.

1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 40, 42, 43, 45, 48, 50, 51, 53, 55, 56, 58, 61, 63, 64, 66, 69, 71, 74, 76, 77, 79, 82, 84, 85, 87, 90, 92, 95, 97, 98, 100, 103, 105, 106, 108, 110, 111, 113, 116, 118, 119, 121, 124, 126, 129, 131

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

Also, k such that k = 2*ceiling(k*phi) - ceiling(k*sqrt(5)) where phi = (1+sqrt(5))/2. - Benoit Cloitre, Dec 05 2002

The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.

Positions of 1's in {A078588(n) : n > 0}. - Clark Kimberling and Jianing Song, Sep 10 2019

Also positive integers k such that {k*r} > 1/2, where r = golden ratio = (1 + sqrt(5))/2 and { } = fractional part. - Clark Kimberling and Jianing Song, Sep 12 2019

The lexicographically least property can be proved with the Walnut theorem prover. - Jeffrey Shallit, Nov 20 2023

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211.
• T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653].
• Primoz Pirnat, Mathematica program

Crossrefs

Complement of A005653.

Equals A279934 - 1.

See A078588 for further comments.

Programs

Mathematica

f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Select[ Range[131], f[ # ] == 1 &]
r = (1 + Sqrt[5])/2; z = 300;
t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 1, z}] (* {A078588(n) : n > 0} *)
Flatten[Position[t, 0]] (* A005653 *)
Flatten[Position[t, 1]] (* this sequence *)
(* Clark Kimberling and Jianing Song, Sep 10 2019 *)
r = GoldenRatio;
t = Table[If[FractionalPart[n*r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n) : n > 0} *)
Flatten[Position[t, 0]] (* A005653 *)
Flatten[Position[t, 1]] (* this sequence *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)

Formula

The set of all k such that the integer multiple of (1+sqrt(5))/2 nearest k is greater than k (Chow-Long).

Numbers k such that 2*{k*phi} - {2k*phi} = 1, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007

Positive integers k such that A078588(k) = 1. - Clark Kimberling and Jianing Song, Sep 10 2019

Extensions

Extended by Robert G. Wilson v, Dec 02 2002

Definition clarified by Jeffrey Shallit, Nov 19 2023