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A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.
(Formerly M2517)
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also, n such that n = 2*ceiling(n*phi) - ceiling(n*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 theoremm 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
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].
FORMULA
The set of all n such that the integer multiple of (1+sqrt(5))/2 nearest n is greater than n (Chow-Long).
Numbers n such that 2{n*phi}-{2n*phi}=1, where { } denotes fractional part. - Clark Kimberling, Jan 01 2007
Positive integers such that A078588(n) = 1. - Clark Kimberling and Jianing Song, Sep 10 2019
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 *)
CROSSREFS
Complement of A005653.
Equals A279934 - 1.
See A078588 for further comments.
Sequence in context: A231010 A285250 A188469 * A047401 A187952 A188028
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Extended by Robert G. Wilson v, Dec 02 2002
Definition clarified by Jeffrey Shallit, Nov 19 2023
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)