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A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.

%I A005652 M2517 #77 Aug 06 2024 05:25:54
%S A005652 1,3,6,8,9,11,14,16,17,19,21,22,24,27,29,30,32,35,37,40,42,43,45,48,
%T A005652 50,51,53,55,56,58,61,63,64,66,69,71,74,76,77,79,82,84,85,87,90,92,95,
%U A005652 97,98,100,103,105,106,108,110,111,113,116,118,119,121,124,126,129,131
%N A005652 Lexicographically least increasing sequence, starting with 1, such that the sum of 2 distinct terms is never a Fibonacci number.
%C A005652 Also, k such that k = 2*ceiling(k*phi) - ceiling(k*sqrt(5)) where phi = (1+sqrt(5))/2. - _Benoit Cloitre_, Dec 05 2002
%C A005652 The Chow-Long paper gives a connection with continued fractions, as well as generalizations and other references for this and related sequences.
%C A005652 Positions of 1's in {A078588(n) : n > 0}. - _Clark Kimberling_ and _Jianing Song_, Sep 10 2019
%C A005652 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 A005652 The lexicographically least property can be proved with the Walnut theorem prover. - _Jeffrey Shallit_, Nov 20 2023
%D A005652 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005652 T. D. Noe, <a href="/A005652/b005652.txt">Table of n, a(n) for n = 1..1000</a>
%H A005652 K. Alladi et al., <a href="http://dx.doi.org/10.1016/0012-365X(78)90053-5">On additive partitions of integers</a>, Discrete Math., 22 (1978), 201-211.
%H A005652 T. Y. Chow and C. D. Long, <a href="https://citeseerx.ist.psu.edu/pdf/8bd27d7422b6e0154cc6ab53a917590031228540">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].
%H A005652 Primoz Pirnat, <a href="/A005652/a005652_1.txt">Mathematica program</a>
%F A005652 The set of all k such that the integer multiple of (1+sqrt(5))/2 nearest k is greater than k (Chow-Long).
%F A005652 Numbers k such that 2*{k*phi} - {2k*phi} = 1, where { } denotes fractional part. - _Clark Kimberling_, Jan 01 2007
%F A005652 Positive integers k such that A078588(k) = 1. - _Clark Kimberling_ and _Jianing Song_, Sep 10 2019
%t A005652 f[n_] := Block[{k = Floor[n/GoldenRatio]}, If[n - k*GoldenRatio > (k + 1)*GoldenRatio - n, 1, 0]]; Select[ Range[131], f[ # ] == 1 &]
%t A005652 r = (1 + Sqrt[5])/2; z = 300;
%t A005652 t = Table[Floor[2 n*r] - 2 Floor[n*r], {n, 1, z}] (* {A078588(n) : n > 0} *)
%t A005652 Flatten[Position[t, 0]] (* A005653 *)
%t A005652 Flatten[Position[t, 1]] (* this sequence *)
%t A005652 (* _Clark Kimberling_ and _Jianing Song_, Sep 10 2019 *)
%t A005652 r = GoldenRatio;
%t A005652 t = Table[If[FractionalPart[n*r] < 1/2, 0, 1 ], {n, 1, 120}] (* {A078588(n) : n > 0} *)
%t A005652 Flatten[Position[t, 0]] (* A005653 *)
%t A005652 Flatten[Position[t, 1]] (* this sequence *)
%t A005652 (* _Clark Kimberling_ and _Jianing Song_, Sep 12 2019 *)
%Y A005652 Complement of A005653.
%Y A005652 Equals A279934 - 1.
%Y A005652 See A078588 for further comments.
%K A005652 nonn,easy,nice
%O A005652 1,2
%A A005652 _N. J. A. Sloane_, _Simon Plouffe_
%E A005652 Extended by _Robert G. Wilson v_, Dec 02 2002
%E A005652 Definition clarified by _Jeffrey Shallit_, Nov 19 2023