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%I #58 Jun 30 2024 22:50:01
%S 1,4,8,11,15,19,22,26,29,33,37,40,44,48,51,55,58,62,66,69,73,76,80,84,
%T 87,91,95,98,102,105,109,113,116,120,124,127,131,134,138,142,145,149,
%U 152,156,160,163,167,171,174,178,181,185,189,192,196,199,203
%N Integers k for which the base-phi representation of k includes 1.
%C Conjecture: L(2k-1) and L(2k)+1 are terms of this sequence for all positive integers k, where L=A000032 (Lucas numbers).
%C Proof of this conjecture: this follows directly from the well known formula L(2k)=phi^{2k}+phi^{-2k}, and the recursion L(2k+1)=L(2k)+L(2k-1). - _Michel Dekking_, Jun 25 2019
%C Conjecture: If D is the difference sequence, then D-3 is the infinite Fibonacci word A096270. If so, then A214971 can be generated as in Program 3 of the Mathematica section. - _Peter J. C. Moses_, Oct 19 2012
%C Conjecture: A very simple formula for this sequence seems to be a(n) = ceiling((n-1)*phi) + 2*(n-1) for n>1; thus, see the related sequence A004956. - _Thomas Baruchel_, May 14 2018
%C Moses' conjecture is equivalent to Baruchel's conjecture: Baruchel's conjecture expresses that this sequence is a generalized Beatty sequence, and since A096270 equals the Fibonacci word A005614 with an initial zero, this follows directly from Lemma 8 in Allouche and Dekking. - _Michel Dekking_, May 04 2019
%C The conjectures by Baruchel and Moses are proved in my paper 'Base phi representations and golden mean beta-expansions'. - _Michel Dekking_, Jun 25 2019
%C a(n) equals A198270(n-1) for 0<n<15, and a(n) equals either A198270(n-1) or A198270(n-1)+1 for all n<90, after which the two sequences very slowly diverge from each other. - _Greg Dresden_, Aug 15 2020
%H Clark Kimberling, <a href="/A214971/b214971.txt">Table of n, a(n) for n = 1..2000</a>
%H J.-P. Allouche and F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%H Michel Dekking, <a href="https://arxiv.org/abs/1906.08437">Base phi representations and golden mean beta-expansions</a>, arXiv:1906.08437 [math.NT], 2019.
%F a(n) = floor((n-1)*phi) + 2*n - 1. - _Primoz Pirnat_, Jun 09 2024
%e 1 = 1,
%e 4 = r^2 + 1 + 1/r^2,
%e 8 = r^4 + 1 + 1/r^4,
%e 11 = r^4 + r^1 + 1 + 1/r^2 + 1/r^4.
%e where r = phi = (1 + sqrt(5))/2 = the golden ratio.
%t (* 1st program *)
%t r = GoldenRatio; f[x_] := Floor[Log[r, x]];
%t t[n_] := RealDigits[n, r, 1000]
%t p[n_] := Flatten[Position[t[n][[1]], 1]]
%t Table[{n, f[n] + 1 - p[n]}, {n, 1, 47}] (* {n, exponents of r in base phi repr of n} *)
%t m[n_] := If[MemberQ[f[n] + 1 - p[n], 0], 1, 0]
%t u = Table[m[n], {n, 1, 900}]
%t Flatten[Position[u, 1]] (* A214971 *)
%t (* 2nd program *)
%t A214971 = Map[#[[1]] &, Cases[Table[{n, Last[#] - Flatten[Position[First[#], 1]] &[RealDigits[n, GoldenRatio, 1000]]}, {n, 1, 5000}], {_, {___, 0, ___}}]] (* _Peter J. C. Moses_, Oct 19 2012 *)
%t (* 3rd program; see Comments *)
%t Accumulate[Flatten[{1, Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 1, 1}}] &, {0}, 8] + 3}]] (* _Peter J. C. Moses_, Oct 19 2012 *)
%Y Cf. A055778, A214969, A214970, A000032, A096270.
%K nonn,base,easy
%O 1,2
%A _Clark Kimberling_, Oct 17 2012
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