

A005653


Sum of 2 terms is never a Fibonacci number.
(Formerly M0965)


18



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

1,1


COMMENTS

The ChowLong 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


REFERENCES

K. Alladi et al., On additive partitions of integers, Discrete Math., 22 (1978), 201211.
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
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 less than n (ChowLong).
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


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]] (* 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 A005652. See A078588 for further comments.
Sequence in context: A188029 A187951 A047495 * A188468 A285251 A325124
Adjacent sequences: A005650 A005651 A005652 * A005654 A005655 A005656


KEYWORD

nonn,easy


AUTHOR

Simon Plouffe, N. J. A. Sloane.


EXTENSIONS

Extended by Robert G. Wilson v, Dec 02 2002


STATUS

approved



