A005653 - OEIS

%I M0965 #54 Aug 06 2024 05:26:50

%S 2,4,5,7,10,12,13,15,18,20,23,25,26,28,31,33,34,36,38,39,41,44,46,47,

%T 49,52,54,57,59,60,62,65,67,68,70,72,73,75,78,80,81,83,86,88,89,91,93,

%U 94,96,99,101,102,104,107,109,112,114,115,117,120,122,123,125,127,128

%N Lexicographically least increasing sequence, starting with 2, such that the sum of two distinct terms of the sequence is never a Fibonacci number.

%C The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.

%C Positions of 0's in {A078588(n) : n > 0}. - _Clark Kimberling_ and _Jianing Song_, Sep 10 2019

%C 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

%C Jon E. Schoenfield conjectured, and Jeffrey Shallit proved (using the Walnut theorem prover) the characterization in the title. - _Jeffrey Shallit_, Nov 19 2023

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005653/b005653.txt">Table of n, a(n) for n = 1..1000</a>

%H K. Alladi et al., <a href="https://doi.org/10.1016/0012-365X(78)90053-5">On additive partitions of integers</a>, Discrete Math., 22 (1978), 201-211.

%H T. Y. Chow and C. D. Long, <a href="https://web.archive.org/web/20170706084609/http://www-math.mit.edu/~tchow/add.pdf">Additive partitions and continued fractions</a>, Ramanujan J., 3 (1999), 55-72 [set alpha=(1+sqrt(5))/2 in Theorem 2 to get A005652 and A005653]. See also <a href="https://citeseerx.ist.psu.edu/pdf/5d7edc98fb80e240f9fd90c199d5c4ada36856e7">on ResearchGate</a>.

%H Primoz Pirnat, <a href="/A005653/a005653.txt">Mathematica program</a>

%F The set of all n such that the integer multiple of (1+sqrt(5))/2 nearest n is less than n (Chow-Long).

%F Numbers n such that 2{n*phi}={2n*phi}, where { } denotes fractional part. - _Clark Kimberling_, Jan 01 2007

%F Positive integers such that A078588(n) = 0. - _Clark Kimberling_ and _Jianing Song_, Sep 10 2019

%t f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Select[ Range[130], f[ # ] == 0 &]

%t r = (1 + Sqrt[5])/2; z = 300;

%t t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 1, z}] (* {A078588(n) : n > 0} *)

%t Flatten[Position[t, 0]] (* this sequence *)

%t Flatten[Position[t, 1]] (* A005652 *)

%t (* _Clark Kimberling_ and _Jianing Song_, Sep 10 2019 *)

%t r = GoldenRatio;

%t t = Table[If[FractionalPart[n*r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n) : n > 0} *)

%t Flatten[Position[t, 0]] (* this sequence *)

%t Flatten[Position[t, 1]] (* A005652 *)

%t (* _Clark Kimberling_ and _Jianing Song_, Sep 12 2019 *)

%Y Complement of A005652. See A078588 for further comments.

%K nonn,easy

%O 1,1

%A _Simon Plouffe_ and _N. J. A. Sloane_

%E Extended by _Robert G. Wilson v_, Dec 02 2002

%E Definition clarified by _Jeffrey Shallit_, Nov 19 2023