A005653 Lexicographically least increasing sequence, starting with 2, such that the sum of two distinct terms of the sequence is never a Fibonacci number.

2, 4, 5, 7, 10, 12, 13, 15, 18, 20, 23, 25, 26, 28, 31, 33, 34, 36, 38, 39, 41, 44, 46, 47, 49, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 93, 94, 96, 99, 101, 102, 104, 107, 109, 112, 114, 115, 117, 120, 122, 123, 125, 127, 128

Offset: 1

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Positions of 0'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

Jon E. Schoenfield conjectured, and Jeffrey Shallit proved (using the Walnut theorem prover) the characterization in the title. - Jeffrey Shallit, Nov 19 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]. See also on ResearchGate.
• Primoz Pirnat, Mathematica program

Crossrefs

Complement of A005652. 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[130], f[ # ] == 0 &]
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]] (* this sequence *)
Flatten[Position[t, 1]] (* A005652 *)
(* 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]] (* this sequence *)
Flatten[Position[t, 1]] (* A005652 *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)

Formula

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

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

Positive integers such that A078588(n) = 0. - 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