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
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), 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]. See also on ResearchGate.
Primoz Pirnat, Mathematica program
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 *)
(* 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]] (* this sequence *)
Flatten[Position[t, 1]] (* A005652 *)
(* Clark Kimberling and Jianing Song, Sep 12 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended by Robert G. Wilson v, Dec 02 2002
Definition clarified by Jeffrey Shallit, Nov 19 2023
STATUS
approved