A079585 -id:A079585 - cp's OEIS frontend

A094874 Decimal expansion of (5-sqrt(5))/2.

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

Offset: 1

N. J. A. Sloane, Jun 14 2004

Also the limiting ratio of Lucas(n)/Fibonacci(n+1), or Fibonacci(n-1)/Fibonacci(n+1) + 1. - Alexander Adamchuk, Oct 10 2007

			1.38196601125010515179541316563436188...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Paul Cooijmans, Odds.
Yiyan Ni, Myron Hlynka, and Percy H. Brill, Urn Models and Fibonacci Series, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Index entries for algebraic numbers, degree 2.

Equals A079585-1.

Cf. A000032, A000045, A081012, A182007, A187426, A187798, A192223, A242671, A244847.

RealDigits[5/2 - Sqrt[5]/2, 10, 100][[1]] (* Alonso del Arte, Jun 26 2018 *)

(5-sqrt(5))/2 \\ Charles R Greathouse IV, Jun 26 2011

Equals (2-phi)*(2+phi) = 2 - 1/phi = 3 - phi = (5-sqrt(5))/2 = (2*sin(Pi/5))^2, where phi is the golden ratio (A001622).
Equals Product_{n > 0} (1 + 1/A192223(n)). - Charles R Greathouse IV, Jun 26 2011
Equals 1 + Sum_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - Michel Marcus, Jun 26 2018; corrected by Michel Marcus, Mar 11 2024
Equals Sum_{k>=0} binomial(2*k,k)/((k+1) * 5^k). - Amiram Eldar, Aug 03 2020
From Amiram Eldar, Nov 28 2024: (Start)
Equals 5*A244847 = 2*A187798 = 1/A242671 = A182007^2 = sqrt(A187426).
Equals Product_{k>=1} (1 + 1/A081012(k)). (End)

A132338 Decimal expansion of 1 - 1/phi.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 07 2007

Density of 1's in Fibonacci word A003849.
Also decimal expansion of Sum_{n>=1} ((-1)^(n+1))*1/phi^n. - Michel Lagneau, Dec 04 2011
The Lambert series evaluated at this point is 0.8828541617125076... [see André-Jeannin]. - R. J. Mathar, Oct 28 2012
Because this equals 2 - phi, this is an integer in the quadratic number field Q(sqrt(5)). (Note that this is also sqrt(5 - 3*phi).) - Wolfdieter Lang, Jan 08 2018
When m >= 1, the equation m*x^m + (m-1)*x^(m-1) + ... + 2*x^2 + x - 1 = 0 has only one positive root, u(m) (say); then lim_{m->oo} u(m) = (3-sqrt(5))/2 (see Aubonnet). - Bernard Schott, May 12 2019
Cosine of the zenith angle at which a string should be cut so that a ball tied to one of its ends, set moving without friction around a vertical circle with the minimum speed in a uniform gravitational field, will then travel through the fixed center of the circle. - Stefano Spezia, Oct 25 2020
Algebraic number of degree 2 with minimal polynomial x^2 - 3*x + 1. The other root is 1 + phi = A104457. - Wolfdieter Lang, Aug 29 2022

			0.38196601125010515179541316563436188...

F. Aubonnet, D. Guinin and A. Ravelli, Oral, Concours d'entrée des Grandes Ecoles Scientifiques, Exercices résolus, "Crus" 1982-83, Bréal, 1983, Exercice 210, 40-42.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
R. André-Jeannin, Lambert series and the summation of reciprocals in certain Fibonacci-Lucas-Type sequences, Fib. Quart. 28 (1990) 223-226.
Yiyan Ni, Myron Hlynka, and Percy H. Brill, Urn Models and Fibonacci Series, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
Index entries for algebraic numbers, degree 2

Cf. A001622, A003849, A094874, A079585, A094214, A104457.

RealDigits[N[1/GoldenRatio^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
RealDigits[1-1/GoldenRatio,10,120][[1]] (* Harvey P. Dale, Mar 30 2024 *)

(3-sqrt(5))/2 \\ Michel Marcus, Oct 26 2020

Equals 1 - 1/phi = 2 - phi, with phi from A001622.
Equals A094874 - 1, or A079585 - 2, or the square of A094214.
Equals (5-sqrt(5))^2/20 = 1/phi^2 = 1/A104457. - Joost Gielen, Sep 28 2013 [corrected by Joerg Arndt, Sep 29 2013]
Equals (3-sqrt(5))/2. - Bernard Schott, May 12 2019
Equals Sum_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - Michel Marcus, Jun 26 2018

A020837 Decimal expansion of 1/sqrt(80) = sqrt(5)/20.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Multiplied by 100, this is sqrt(125). - Alonso del Arte, Jan 06 2013
Multiplied by 10, this is sqrt(5)/2. As such, it appears in the Pythagorean tree as the ratio of the distance between 2 consecutive square centers divided by the length of the initial square (see CNRS link). - Michel Marcus, Feb 20 2013
The two-dimensional Steinitz constant K_2(0,0), related to sum of vectors, is sqrt(5)/2. - Jean-François Alcover, Jun 04 2014
sqrt(5)/2 is the length of the shortest line segment needed to dissect the unit square into 4 regions with equal areas if all the line segments start at the same vertex of the square. - Wesley Ivan Hurt, May 18 2021
sqrt(5)/2 is the standard deviation of rolling a 4-sided die. - Mohammed Yaseen, Feb 23 2023

			sqrt(5)/20 = 0.111803398874989484820458683436563811772...
sqrt(5)/2  = 1.118033988749894848204586834365638117720...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.13 Steinitz constants, p. 241.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Étienne Ghys and Jos Leys, Un arbre pythagoricien — Images des Mathématiques, CNRS, 2013.
Index entries for algebraic numbers, degree 2

c = (1/10)*(A001622 - 1/2) = (1/10)*(7/2 - A079585) = (A176055 - 1)/10.

RealDigits[1/Sqrt[80], 10, 120][[1]] (* Harvey P. Dale, May 01 2012 *)

sqrt(1/80) \\ Charles R Greathouse IV, Apr 25 2016

Equals 1/sqrt(80) = sqrt(5)/20 = (-1 + 2*phi)/20, with phi from A001622.

Equals 0.1 * Sum_{k>=0} binomial(2*k,k)/20^k. - Amiram Eldar, Aug 04 2022

A343202 Decimal expansion of Sum_{k>=0} 1/(k! * Fibonacci(2^k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 07 2021

The transcendence of this constant was proved independently by Mignotte (1974) and Mahler (1975).

			2.17464539389651955644333792522982188971668174552838...

Maurice Mignotte, Quelques problèmes d'effectivité en théorie des nombres, Thesis, Univ. Paris XIII, Paris, 1974.

Kurt Mahler, On the transcendency of the solutions of a special class of functional equations, Bulletin of the Australian Mathematical Society, Vol. 13, No. 3 (1975), pp. 389-410.

Cf. A000045, A000142, A058635, A079585.

RealDigits[Sum[1/(n!*Fibonacci[2^n]), {n, 0, 20}], 10, 100][[1]]

suminf(k=0, 1/(k!*fibonacci(2^k))) \\ Michel Marcus, Jul 07 2021

A225667 Decimal expansion of 13-5*sqrt(5).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 21 2013

Let d(n) = - 2*F(n) + h(2 + F(n+1), 1 + F(n+2)), where h = harmonic mean, F = A000045 (Fibonacci numbers). Then floor(d(n)) = 2F(n) + 1 for n>1, and limit(d(n)) = 13 - 5*sqrt(5).

Apart from leading digits the same as A132338, A109866, A094874 and A079585. - R. J. Mathar, Jul 30 2013

			13-5*sqrt(5) = 1.819660112501051517954131656343618822797...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000045, A226765.

f[n_] := Fibonacci[n]; h[n_] := HarmonicMean[{2 + f[n + 1], 1 + f[n + 2]}]; x = Limit[-2 f[n] + h[n], n -> Infinity] (* "proof" *)
d = RealDigits[x, 10, 120][[1]] (* A225667 *)

A338304 Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 21 2020

Erdős and Graham (1980) asked whether this constant is irrational or transcendental.
Badea (1987) proved that it is irrational, and André-Jeannin (1991) proved that it is not a quadratic irrational in Q(sqrt(5)), in contrast to the corresponding sum with Fibonacci numbers, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585).
Bundschuh and Pethö (1987) proved that it is transcendental.

			1.49792038099906271987068555399285960807207719857085...

Richard André-Jeannin, A note on the irrationality of certain Lucas infinite series, The Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 132-136.
Catalin Badea, The irrationality of certain infinite series, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.
Peter Bundschuh and Attila Pethö, Zur transzendenz gewisser Reihen, Monatshefte für Mathematik, Vol. 104, No. 3 (1987), pp. 199-223, alternative link.
Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, pp. 64-65.
Index entries for transcendental numbers

Cf. A000045, A000032, A001566, A079585, A338305.

RealDigits[Sum[1/LucasL[2^n], {n, 0, 10}], 10, 100][[1]]

Equals 1 + Sum_{k>=0} 1/A001566(k).

A109866 9's complement of the digits of the golden ratio phi (A001622): 9.999999999999... - 1.6180339887... = 8.3819660112501051517954131656334...

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jul 09 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A079585, A094874, A132338.

With[{nn=120},PadRight[{},nn,9]-RealDigits[GoldenRatio,10,nn][[1]]] (* Harvey P. Dale, Jun 25 2018 *)

A338305 Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 21 2020

Erdős and Graham (1980) asked whether this constant is irrational or transcendental.
Badea (1987) proved that it is irrational.
Becker and Töpper (1994) proved that it is transcendental.
Note that a similar sum, Sum_{k>=0} 1/F(2^k) = (7-sqrt(5))/2 (A079585), is quadratic rational in Q(sqrt(5)).

			1.73003822250424324230412356649689901034795500481030...

Catalin Badea, The irrationality of certain infinite series, Glasgow Mathematical Journal, Vol. 29, No. 2 (1987), pp. 221-228.
Paul-Georg Becker and Thomas Töpper, Transcendency results for sums of reciprocals of linear recurrences, Mathematische Nachrichten, Vol. 168, No. 1 (1994), pp. 5-17.
Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, p. 64-65.
Index entries for transcendental numbers

Cf. A000045, A079585, A192222, A338304.

RealDigits[Sum[1/Fibonacci[2^n + 1], {n, 0, 10}], 10, 100][[1]]

suminf(k=0, 1/fibonacci(2^k+1)) \\ Michel Marcus, Oct 21 2020

Equals Sum_{k>=0} 1/A192222(k).

A371647 Decimal expansion of Sum_{k>=0} 1/Fibonacci(5^k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 31 2024

This constant is a transcendental number (Nyblom, 2001).

			1.20001332889036987670776409546835505643055506881380...

M. A. Nyblom, A Theorem on Transcendence of Infinite Series II, Journal of Number Theory, Vol. 91, No. 1 (2001), pp. 71-80.
Index entries for transcendental numbers.

Cf. A000045, A145232, A371648.

Similar constants: A079585, A079586, A153386, A153387, A371649.

RealDigits[Sum[1/Fibonacci[5^k], {k, 0, 10}], 10, 120][[1]]

PARI
```
suminf(k = 0, 1/fibonacci(5^k))
```

Equals Sum_{k>=0} 1/A145232(k).

A378968 Decimal expansion of log(4 - phi), where phi = A001622.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 12 2024

			0.86792620183470792805547663269019599656918062403358...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.6.8, p. 311.

Cf. A001622, A079585.

RealDigits[Log[4-GoldenRatio],10,100][[1]]

Equals log(A079585).

cp's OEIS Frontend

A094874 Decimal expansion of (5-sqrt(5))/2.

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A132338 Decimal expansion of 1 - 1/phi.

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A020837 Decimal expansion of 1/sqrt(80) = sqrt(5)/20.

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A343202 Decimal expansion of Sum_{k>=0} 1/(k! * Fibonacci(2^k)).

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A225667 Decimal expansion of 13-5*sqrt(5).

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A338304 Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032).

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A109866 9's complement of the digits of the golden ratio phi (A001622): 9.999999999999... - 1.6180339887... = 8.3819660112501051517954131656334...

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