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 A214971 Integers k for which the base-phi representation of k includes 1. 9
 1, 4, 8, 11, 15, 19, 22, 26, 29, 33, 37, 40, 44, 48, 51, 55, 58, 62, 66, 69, 73, 76, 80, 84, 87, 91, 95, 98, 102, 105, 109, 113, 116, 120, 124, 127, 131, 134, 138, 142, 145, 149, 152, 156, 160, 163, 167, 171, 174, 178, 181, 185, 189, 192, 196, 199, 203 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: L(2k-1) and L(2k)+1 are terms of this sequence for all positive integers k, where L=A000032 (Lucas numbers). 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 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 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 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 The conjectures by Baruchel and Moses are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 25 2019 a(n) equals A198270(n-1) for 0 {0, 1}, 1 -> {0, 1, 1}}] &, {0}, 8] + 3}]] (* Peter J. C. Moses, Oct 19 2012 *) CROSSREFS Cf. A055778, A214969, A214970, A000032, A096270. Sequence in context: A248232 A047346 A198270 * A081840 A311050 A311051 Adjacent sequences: A214968 A214969 A214970 * A214972 A214973 A214974 KEYWORD nonn,base,easy AUTHOR Clark Kimberling, Oct 17 2012 STATUS approved

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Last modified September 15 10:23 EDT 2024. Contains 375932 sequences. (Running on oeis4.)