|
|
A005653
|
|
Lexicographically least increasing sequence, starting with 2, such that the sum of two distinct terms of the sequence is never a Fibonacci number.
(Formerly M0965)
|
|
19
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
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
|
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).
|
|
LINKS
|
|
|
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
|
|
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 *)
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|