

A005652


Sum of 2 terms is never a Fibonacci number.
(Formerly M2517)


24



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
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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 ChowLong 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


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), 201211.
T. Y. Chow and C. D. Long, Additive partitions and continued fractions, Ramanujan J., 3 (1999), 5572 [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 (ChowLong).
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]] (* 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 *)


CROSSREFS

Complement of A005653.
Equals A279934  1.
See A078588 for further comments.
Sequence in context: A231010 A285250 A188469 * A047401 A187952 A188028
Adjacent sequences: A005649 A005650 A005651 * A005653 A005654 A005655


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane, Simon Plouffe


EXTENSIONS

Extended by Robert G. Wilson v, Dec 02 2002


STATUS

approved



