A214970 -id:A214970 - cp's OEIS frontend

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

Clark Kimberling, Oct 17 2012

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

			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.

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.

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

(* 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 *)

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

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

A214969 Decimal representation of Sum{d(i)3^i: i=0,1,...}, where Sum{d(i)2^i: i=0,1,...} is the base 2 representation of sqrt(2).

Original entry on oeis.org

1, 1, 5, 2, 7, 2, 1, 2, 8, 3, 5, 4, 0, 5, 8, 2, 9, 0, 6, 8, 0, 8, 3, 0, 3, 3, 0, 1, 9, 9, 0, 9, 6, 4, 3, 5, 6, 8, 0, 1, 4, 2, 5, 7, 5, 7, 6, 5, 6, 3, 7, 6, 1, 8, 5, 5, 2, 7, 1, 1, 2, 9, 2, 6, 0, 1, 1, 1, 8, 1, 8, 5, 1, 4, 3, 4, 2, 0, 2, 4, 8, 4, 5, 3, 6, 4, 6, 8, 7, 2, 7, 6, 6, 5, 7, 6, 7, 6, 0

Offset: 1

Clark Kimberling, Sep 01 2012

This constant can be used to illustrate a fractal-type "change-of-base function". Suppose that b>1 and c>1, and for x>=0 given by the greedy algorithm as x = sum{d(i)*b^i}, define f(x) = sum{d(i)*c^i}. The self-similarity of the graph of y = f(x) is given by the equation f(x/b) = (1/c)*f(x). If bc, then f is not monotonic on any open interval. The self-similarity is illustrated graphically by the second Mathematica program, for which b=2 and c=3.

			1.1527212835405829068083033019909643568...
= 1 + 1/3^2 + 1/3^3 + 1/3^5 + ... obtained from
sqrt(2) = 1 + 1/2^2 + 1/2^3 + 1/2^5 + ... .

Clark Kimberling, Fractal change-of-base functions, Advances and Applications in Mathematical Sciences, 12 (2013), 255-261.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A214970

f[x_, b_, c_, d_] := FromDigits[RealDigits[x, b, d], c]
N[f[Sqrt[2], 2, 3, 500], 120]
RealDigits[%]  (* A214969 *)
(* second program:  self-similar (fractal) graphs *)
f[x_, b_, c_, digits_] := FromDigits[RealDigits[x, b, digits], c]
Plot[f[x, 2, 3, 150], {x, 0, 1}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, 1/2}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, (1/2)^2}, PlotPoints -> 300]
Plot[f[x, 2, 3, 150], {x, 0, (1/2)^3}, PlotPoints -> 300]

A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

Original entry on oeis.org

3, 5, 7, 10, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 34, 36, 39, 41, 43, 45, 47, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 97, 99, 101, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 126, 128, 130, 133, 135, 137, 139, 141, 144

Offset: 1

Michel Dekking, May 06 2019

nonn

Let b = A214970 be the sequence of the integers k for which the base phi representation includes 1, and let c be the sequence of integers k for which the base phi representation includes phi.
Note that a, b and c form a complementary triple (since consecutive digits 11 do not occur in a base phi representation).
Conjecture (Moses 2012/Baruchel 2018): b is the generalized Beatty sequence b(n) = floor(n*phi) + 2*n + 1.
Conjecture (Kimberling 2012): c = A054770 = A000201(n) + 2*n - 1 = floor(n*phi) + 2*n - 1.
One can prove that the Moses/Baruchel conjecture and the Kimberling conjecture are equivalent.
Conjecture: (a(n)) is a union of two generalized Beatty sequences v and w, given by v(n) = floor(n*phi) + 2*n = A003231(n), and w(n) = 3*floor(n*phi) + n + 1 = A190509(n) + 1.
This conjecture is compatible with the Moses/Baruchel/Kimberling conjecture.
These three conjectures are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 26 2019

			3 = phi^2 + phi^{-2}, 5 = phi^3 + phi^{-1} + phi^{-4}.

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
George Bergman, A Number System with an Irrational Base, Mathematics Magazine, Vol. 31, No. 2 (Nov. - Dec., 1957), pp. 98-110.
M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

Cf. A214970, A054770, A000201.

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

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A214969 Decimal representation of Sum{d(i)3^i: i=0,1,...}, where Sum{d(i)2^i: i=0,1,...} is the base 2 representation of sqrt(2).

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A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

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cp's OEIS Frontend

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

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Programs

Mathematica

Python

Formula

A214969 Decimal representation of Sum{d(i)*3^i: i=0,1,...}, where Sum{d(i)*2^i: i=0,1,...} is the base 2 representation of sqrt(2).

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Keywords

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Examples

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Programs

Mathematica

A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A214969 Decimal representation of Sum{d(i)3^i: i=0,1,...}, where Sum{d(i)2^i: i=0,1,...} is the base 2 representation of sqrt(2).