A214971 Integers k for which the base-phi representation of k includes 1.

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

Offset: 1

Author

Clark Kimberling, Oct 17 2012

Keywords

• nonn
• base
• easy

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 0A198270(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

Examples

			1 = 1,
4 = r^2 + 1 + 1/r^2,
8 = r^4 + 1 + 1/r^4,
11 = r^4 + r^1 + 1 + 1/r^2 + 1/r^4.
where r = phi = (1 + sqrt(5))/2 = the golden ratio.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..2000
• J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
• Michel Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

Crossrefs

Cf. A055778, A214969, A214970, A000032, A096270.

Programs

Mathematica

(* 1st program *)
r = GoldenRatio; f[x_] := Floor[Log[r, x]];
t[n_] := RealDigits[n, r, 1000]
p[n_] := Flatten[Position[t[n][[1]], 1]]
Table[{n, f[n] + 1 - p[n]}, {n, 1, 47}] (* {n, exponents of r in base phi repr of n} *)
m[n_] := If[MemberQ[f[n] + 1 - p[n], 0], 1, 0]
u = Table[m[n], {n, 1, 900}]
Flatten[Position[u, 1]]  (* A214971 *)
(* 2nd program *)
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 *)
(* 3rd program; see Comments *)
Accumulate[Flatten[{1, Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 1, 1}}] &, {0}, 8] + 3}]]  (* Peter J. C. Moses, Oct 19 2012 *)

Python

from math import isqrt
def A214971(n): return (n<<1)-1+(n-1+isqrt(5*(n-1)**2)>>1) # Chai Wah Wu, May 22 2025

Formula

a(n) = floor((n-1)*phi) + 2*n - 1. - Primoz Pirnat, Jun 09 2024