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A005653
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Sum of 2 terms is never a Fibonacci number.
(Formerly M0965)
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18
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201-211.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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].
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FORMULA
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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
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MATHEMATICA
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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]] (* A005653 *)
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]] (* A005653 *)
Flatten[Position[t, 1]] (* A005652 *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)
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CROSSREFS
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Complement of A005652. See A078588 for further comments.
Sequence in context: A188029 A187951 A047495 * A188468 A285251 A325124
Adjacent sequences: A005650 A005651 A005652 * A005654 A005655 A005656
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KEYWORD
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nonn,easy
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AUTHOR
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Simon Plouffe, N. J. A. Sloane.
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EXTENSIONS
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Extended by Robert G. Wilson v, Dec 02 2002
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STATUS
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approved
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