A094214 -id:A094214 - cp's OEIS frontend

A139340 Decimal expansion of the cube root of the golden ratio. That is, the decimal expansion of ((1+sqrt(5))/2)^(1/3).

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

Larger of the real roots of x^6 - x^3 - 1. - Charles R Greathouse IV, Apr 14 2014

			1.1739849967053285...

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 6

Cf. A001622, A094214, A104457, A098317, A002390, A139339.

phi := (sqrt(5)+1)/2 ; evalf(root[3](phi)) ; # R. J. Mathar, Oct 16 2015

RealDigits[N[GoldenRatio^(1/3),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)

polrootsreal(x^6-x^3-1)[2] \\ Charles R Greathouse IV, Apr 14 2014

A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.

Original entry on oeis.org

1, 2, 2, 3, 4, 32, 1065, 2038968, 5977146319204, 36314862033946243071181679, 1028280647188781709727717632740627249617427013751977, 958046899855070460620234639622630375078362220775180051610386376308132568342498992099474472596860400289

Offset: 1

Clark Kimberling, Apr 07 2016

Let x > 0, and let r = (r(k)) be a sequence of positive irrational numbers.  Let a(1) be the least positive integer m such that r(1)/m < x, and inductively let a(n) be the least positive integer m such that r(1)/a(1) + ... + r(n-1)/a(n-1) + r(n)/m < x.  The sequence (a(n)) is the (r,x)-greedy sequence.  We are interested in choices of r and x for which the series r(1)/a(1) + ... + r(n)/a(n) + ... converges to x.  (The same algorithm is used to generate sequences listed at A269993.)
Guide to related sequences:
  x     r(k)
 1/tau^k             A270744
 k/tau^k             A270745
 2/e^k               A270746
 4/Pi^k              A270747
 2/log(k+1)          A270748
 k/log(k+1)          A270749
 1/(k*log(k+1))      A270750
 1/(k*tau)           A270751
 1/(k*e)             A270752
 1/(k*sqrt(2))       A270916

			a(1) = ceiling(r(1)) = ceiling(1/tau) = ceiling(0.618...) = 1;
a(2) = ceiling(r(2)/(1 - r(1)/1)) = 2;
a(3) = ceiling(r(3)/(1 - r(1)/1 - r(2)/2)) = 2.
The first 6 terms of the series r(1)/a(1) + ... + r(n)/a(n) + ... are 0.618..., 0.809..., 0.927..., 0.975..., 0.998..., 0.999967... .

Cf. A001620, A270745, A094214, A269993.

$MaxExtraPrecision = Infinity; z = 13;
r[k_] := N[1/GoldenRatio^k, 1000]; f[x_, 0] = x;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
 x = 1; Table[n[x, k], {k, 1, z}]
N[Sum[r[k]/n[x, k], {k, 1, 13}], 200]

a(n) = ceiling(r(n)/s(n)), where s(n) = 1 - r(1)/a(1) - r(2)/a(2) - ... - r(n-1)/a(n-1).

r(1)/a(1) + ... + r(n)/a(n) + ... = 1.

A134861 Wythoff BAA numbers.

Original entry on oeis.org

2, 10, 15, 23, 31, 36, 44, 49, 57, 65, 70, 78, 86, 91, 99, 104, 112, 120, 125, 133, 138, 146, 154, 159, 167, 175, 180, 188, 193, 201, 209, 214, 222, 230, 235, 243, 248, 256, 264, 269, 277, 282, 290, 298, 303, 311, 319, 324, 332, 337, 345, 353, 358, 366, 371

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAA = A+2B-3.
Also numbers with suffix string 0010, when written in Zeckendorf representation (with leading zero's for the first term). - A.H.M. Smeets, Mar 20 2024
The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

A.H.M. Smeets, Table of n, a(n) for n = 1..20000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, A134862, A134863, A035338, A134864, A035513.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

A[n_] := Floor[n * GoldenRatio]; B[n_] := Floor[n * GoldenRatio^2]; a[n_] := B[A[A[n]]]; Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

from sympy import floor
from mpmath import phi
def A(n): return floor(n*phi)
def B(n): return floor(n*phi**2)
def a(n): return B(A(A(n))) # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A134861(n): return 3*((n+isqrt(5*n**2)>>1)-1)+(n<<1) # Chai Wah Wu, Aug 10 2022

a(n) = B(A(A(n))), n>=1, with A=A000201, the lower Wythoff sequence and B=A001950, the upper Wythoff sequence.

A134864 Wythoff BBB numbers.

Original entry on oeis.org

13, 34, 47, 68, 89, 102, 123, 136, 157, 178, 191, 212, 233, 246, 267, 280, 301, 322, 335, 356, 369, 390, 411, 424, 445, 466, 479, 500, 513, 534, 555, 568, 589, 610, 623, 644, 657, 678, 699, 712, 733, 746, 767, 788, 801, 822, 843, 856, 877, 890, 911, 932, 945

Offset: 1

Clark Kimberling, Nov 14 2007

nonn

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB = 3A+5B.

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Clark Kimberling, Complementary equations and Wythoff Sequences, Journal of Integer Sequences 11 (2008), Article 08.3.3.

Cf. A000201, A001950, A003622, A003623, A035336, A094214, A101864, A134859, A035337, A134860, A134861, A134862, A035338, A134863, A035513.

Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.

a:=n->floor(n*((1+sqrt(5))/2)^2): [a(a(a(n)))$n=1..55]; # Muniru A Asiru, Nov 24 2018

Nest[Quotient[#(3+Sqrt@5),2]&,#,3]&/@Range@100 (* Federico Provvedi, Nov 24 2018 *)
b[n_]:=Floor[n GoldenRatio^2]; a[n_]:=b[b[b[n]]]; Array[a, 60] (* Vincenzo Librandi, Nov 24 2018 *)

from sympy import floor
from mpmath import phi
def B(n): return floor(n*phi**2)
def a(n): return B(B(B(n))) # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A134864(n): return (m:=5*n)+(((n+isqrt(n*m))&-2)<<2) # Chai Wah Wu, Aug 10 2022

a(n) = B(B(B(n))), n>=1, with B=A001950, the upper Wythoff sequence.

A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

Original entry on oeis.org

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

Offset: 1

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

The constant k in A277266 such that A277266(n) ~ k*n.

			1/(1*1) + 1/(1*2) + 1/(2*3) + 1/(3*5) + ... = 1 + 1/2 + 1/6 + 1/15 + ... = 1.77387758328513234380...

Cf. A000045, A000796, A001622, A001654, A002163, A002390, A079586, A094214, A265290, A277266.

RealDigits[ Sum[1/(Fibonacci[k]*Fibonacci[k + 1]), {k, 265}], 10, 111][[1]]

suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1))) \\ Michel Marcus, Feb 19 2019

Equals Sum_{n>=1} 1/(Fibonacci(n)*Fibonacci(n+1)).
Equals lim_{n->infinity} A277266(n)/n.
Equals 2 * (Sum_{k>=1} 1/(phi^k * F(k))) - 1/phi = 2 * A265290 - A094214, where phi is the golden ratio (A001622) and F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Equals 3/2 + 10*c*Integral_{x=0..infinity} f(x) dx, where c = sqrt(5)/log(phi) = A002163/A002390, phi = (1+sqrt(5))/2 = A001622, and f(x) = sin(x)/((exp(Pi*x/(2*log(phi)))-1)*(7-2*cos(x))*(3+2*cos(x))). - Gleb Koloskov, Sep 12 2021

More terms from Alois P. Heinz, Aug 06 2017

A139342 Decimal expansion of e^(-(1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 14 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.19828815286220623226788895660486467084208489250129...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341.

evalf(exp(1)^(-(1+sqrt(5))/2),100); # Wesley Ivan Hurt, Feb 09 2017

RealDigits[E^-GoldenRatio,10,120][[1]] (* Harvey P. Dale, Dec 30 2020 *)

exp(-(sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals exp(-A001622).

Equals 1/A139341. - Amiram Eldar, Feb 08 2022

Leading zero removed by R. J. Mathar, Feb 05 2009

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A337669 Decimal expansion of Product_{n>=3} (1-1/Fibonacci(n)).

Original entry on oeis.org

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

Offset: 0

Michel Marcus, Sep 15 2020

			0.18978914361786603634948251429...

Daniel Duverney and Yohei Tachiya, Algebraic independence of certain infinite products involving the Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 97, No. 5 (2021), pp. 29-31; arXiv preprint, arXiv:2009.06250 [math.NT], 2020.
Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Wikipedia, Dedekind eta function.
Wikipedia, Theta function.

Cf. A000045, A001622, A094214, A337668.

With[{b = 1/GoldenRatio}, RealDigits[(Sqrt[5]/6)*b^(-5/4) * EllipticTheta[2, 0, b] * EllipticTheta[3, 0, b] * EllipticTheta[4, 0, b]/EllipticTheta[4, 0, b^4], 10, 100][[1]]] (* Amiram Eldar, May 27 2021 *)

PARI
```
prodinf(n=3, 1-1/fibonacci(n))
```

Equals (sqrt(5)/6) * b^(-5/4) * theta_2(b) * theta_3(b) * theta_4(b)/theta_4(b^4), where theta_i are the Jacobi theta functions and b = 1/phi = A094214 (Duverney and Tachiya, 2021). - Amiram Eldar, May 27 2021

Equals (sqrt(5) * phi^(5/4) / 3) * eta(tau_0)^3 * eta(4*tau_0) / eta(2*tau_0)^2, where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022). - Amiram Eldar, Mar 26 2024

More terms from Jinyuan Wang, Sep 19 2020

A084242 Least k, 1 <= k <= n, such that the number of elements of the continued fraction for n/k is maximum.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 11, 9, 10, 11, 11, 11, 13, 13, 14, 13, 14, 15, 17, 17, 18, 19, 18, 23, 19, 21, 22, 22, 23, 22, 25, 29, 23, 26, 25, 27, 26, 27, 29, 31, 30, 29, 28, 33, 33, 31, 34, 41, 32, 36, 33, 37, 33, 35, 37, 39, 47, 37, 41, 42, 40, 41, 41, 53, 45, 43

Offset: 1

Benoit Cloitre, Jun 21 2003

nonn

Also, for n > 1, the smallest number k such that the Euclidean algorithm for (n,k) requires the maximum number of steps, A034883(n). - T. D. Noe, Mar 24 2011

John Metcalf, Table of n, a(n) for n = 1..10000

Cf. A000045, A071677, A094214.

a(n)=if(n<0,0,s=1; while(abs(vecmax(vector(n,i,length(contfrac(n/i))))-length(contfrac(n/s)))>0,s++); s)

For k > 1, a(F(k)) = F(k-1) where F(k) denotes the k-th Fibonacci number.

Probably, lim_{n->oo} (1/n)*Sum_{k=1..n} a(k) = 1/phi = A094214.

A337668 Decimal expansion of Product_{n>=1} (1+1/Fibonacci(n)).

Original entry on oeis.org

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

Offset: 2

Michel Marcus, Sep 15 2020

			13.15096665778418436761243337...

Daniel Duverney and Yohei Tachiya, Algebraic independence of certain infinite products involving the Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 97, No. 5 (2021), pp. 29-31; arXiv preprint, arXiv:2009.06250 [math.NT], 2020.
Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Wikipedia, Theta function.

Cf. A000045, A001622, A094214, A337669.

With[{b = 1/GoldenRatio}, RealDigits[2*b^(-5/4)*EllipticTheta[2, 0, b]/EllipticTheta[4, 0, b^4], 10, 100][[1]]] (* Amiram Eldar, May 27 2021 *)

PARI
```
prodinf(n=1, 1+1/fibonacci(n))
```

Equals 2 * b^(-5/4) * theta_2(b)/theta_4(b^4), where theta_i are the Jacobi theta functions and b = 1/phi = A094214 (Duverney and Tachiya, 2021). - Amiram Eldar, May 27 2021

Equals 4 * phi^(5/4) * eta(4*tau_0) / eta(tau_0), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022). - Amiram Eldar, Mar 26 2024

More terms from Jinyuan Wang, Sep 19 2020

cp's OEIS Frontend

A139340 Decimal expansion of the cube root of the golden ratio. That is, the decimal expansion of ((1+sqrt(5))/2)^(1/3).

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A270744 (r,1)-greedy sequence, where r(k) = 1/tau^k, where tau = golden ratio.

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A134861 Wythoff BAA numbers.

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A134864 Wythoff BBB numbers.

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A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

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A139342 Decimal expansion of e^(-(1+sqrt(5))/2).

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A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

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