A094885 -id:A094885 - cp's OEIS frontend

A159823 Continued fraction for phi*e A094885, where phi = (1 + sqrt(5))/2.

4, 2, 1, 1, 22, 1, 1, 4, 5, 2, 2, 1, 1, 15, 1, 12, 2, 2, 6, 10, 6, 1, 11, 3, 1, 3, 33, 1, 1, 1, 2, 2, 1, 4, 1, 2, 3, 3, 8, 1, 1, 1, 1, 2, 1, 3, 32, 3, 1, 1, 2, 2, 1, 5, 10, 1, 1, 1, 2, 2, 1, 1, 1, 4, 2, 2, 20, 2, 1, 2, 1, 1, 3, 1, 1, 2, 5, 1, 9, 1, 23, 1, 291, 1, 3, 2, 9, 7, 1, 1, 3, 10, 5, 2, 1, 13, 3, 7

Offset: 0

Harry J. Smith, Apr 27 2009

			phi*e = 4.398272389447946395... = 4 + 1/(2 + 1/(1 + 1/(1 + 1/(22 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

ContinuedFraction((1+Sqrt(5))*Exp(1)/2) // G. C. Greubel, May 19 2018

ContinuedFraction[E*GoldenRatio, 100] (* G. C. Greubel, May 19 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); phi=(1+sqrt(5))/2; x=contfrac(phi*exp(1)); for (n=1, 20001, write("b159823.txt", n-1, " ", x[n])); }

A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

Original entry on oeis.org

2, 6, 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, 7, 4, 8

Offset: 1

Eric W. Weisstein, Mar 08 2005

Only first term differs from the decimal expansion of phi.
Zelo extends work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace. - Jonathan Vos Post, Mar 02 2009 (Cf. last sentence in the Zelo reference. - Joerg Arndt, Jan 04 2014)
Hawkes asks: "What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?". - Charles R Greathouse IV, Dec 11 2012
This is the case n=10 in (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1+2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
An algebraic integer of degree 2, with minimal polynomial x^2 - 3x + 1. - Charles R Greathouse IV, Nov 12 2014 [The other root is 2 - phi = A132338 - Wolfdieter Lang, Aug 29 2022]
To eight digits: 5*(((Pi+1)/e)-1) = 2.61803395481182... - Dan Graham, Nov 21 2017
The ratio diagonal/side of the second smallest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
phi^2/10 is the moment of inertia of a solid regular icosahedron with a unit mass and a unit edge length (see A341906). - Amiram Eldar, Jun 08 2021

			2.6180339887498948482045868343656381177203091798...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.17.1, p. 153.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Murray Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, Vol. 4, No. 2 (1966), pp. 157-162.
John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Damien Roy, Diophantine Approximation in Small Degree, arXiv:math/0303150 [math.NT], 2003.
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions.
Wikipedia, Perron number.
Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, arXiv:0903.0086 [math.NT], 2009.
Index entries for algebraic numbers, degree 2.
Index entries for sequences related to moment of inertia.

Cf. A001622, A094214, A094885, A132338, A229780, A239798, A341906.
Cf. A001519, A001906, A032908, A049310.
2 + 2*cos(2*Pi/n): A116425 (n = 7), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // G. C. Greubel, Nov 23 2018

RealDigits[N[GoldenRatio+1,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

(3+sqrt(5))/2 \\ Charles R Greathouse IV, Aug 21 2012

numerical_approx(golden_ratio^2, digits=100) # G. C. Greubel, Nov 23 2018

Equals 2 + A094214 = 1 + A001622. - R. J. Mathar, May 19 2008
Satisfies these three equations: x-sqrt(x)-1 = 0; x-1/sqrt(x)-2 = 0; x^2-3*x+1 = 0. - Richard R. Forberg, Oct 11 2014
Equals the nested radical sqrt(phi^2+sqrt(phi^4+sqrt(phi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (5*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+1)).
Equals 3/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Equals 1/A132338 = 2*A239798 = 5*A229780. - Mohammed Yaseen, Nov 04 2020
Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
c^n = phi * A001906(n) + A001519(n), where c = phi^2. - Gary W. Adamson, Sep 08 2023
Equals lim_{n->oo} S(n, 3)/S(n-1, 3) with the S-Chebyshev polynomials (see A049310), S(3, n) = A000045(2*(n+1)) = A001906(n+1). - Wolfdieter Lang, Nov 15 2023
From Peter Bala, May 08 2024: (Start)
Constant c = 2 + 2*cos(2*Pi/5).
The linear fractional transformation z -> c - c/z has order 5, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/z)))). (End)
Equals Product_{k>=1} (1 + 1/A032908(k)). - Amiram Eldar, Nov 28 2024

A094886 Decimal expansion of phi*Pi, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 15 2004

The area of a golden ellipse with a semi-major axis phi and a minor semi-axis 1. - Amiram Eldar, Jul 05 2020

phi*Pi = area of the region having boundaries y = 0, x = Pi/2, and y = (tan x)^(4/5). - Clark Kimberling, Oct 25 2020

			5.0832036923152598158...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for transcendental numbers

Cf. A000796, A001622, A094881, A094885.

First@ RealDigits[N[GoldenRatio Pi, 120]] (* Michael De Vlieger, May 24 2016 *)

{ default(realprecision, 20080); phi=(1+sqrt(5))/2; x=phi*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b094886.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

Pi*(1+sqrt(5))/2 \\ Michel Marcus, May 25 2016

Equals the nested radical sqrt(Pi^2+sqrt(Pi^4+sqrt(Pi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016

Equals Integral_{x=0..Pi/2} tan(x)^(4/5) dx. - Clark Kimberling, Nov 18 2020

A131563 Decimal expansion of ePiphi, where phi = (sqrt(5) + 1)/2.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Aug 27 2007, Dec 17 2008

			e*Pi*phi = 13.817580227...

Harry J. Smith, Table of n, a(n) for n = 2..20000

Decimal expansion of e: A001113. Decimal expansion of Pi: A000796. Decimal expansion of phi: A001622. e*Pi: A019609. Pi*phi: A094886. e*phi: A094885.

Maple
```
exp(1)*Pi*(1+sqrt(5))/2;
```

phi=(5^(1/2)+1)/2;RealDigits[N[Pi*E*phi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)
RealDigits[E*Pi*GoldenRatio,10,120][[1]] (* Harvey P. Dale, Nov 02 2020 *)

{ default(realprecision, 20080); phi = (1 + sqrt(5))/2; x=exp(1)*Pi*phi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b131563.txt", n, " ", d)); } \\ Harry J. Smith, Apr 26 2009

More terms from N. J. A. Sloane, Dec 19 2008

A131566 Decimal expansion of (ePiphi)^2.

Original entry on oeis.org

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

Offset: 3

Omar E. Pol, Aug 27 2007

phi = (5^(1/2) + 1)/2 = (1 + sqrt(5))/2.

			190.925523334...

Harry J. Smith, Table of n, a(n) for n = 3..20000

Cf. A001113, A000796, A001622, A019609, A094886, A094885.

phi=(5^(1/2)+1)/2;RealDigits[N[(Pi*E*phi)^2,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)
RealDigits[(E*Pi*GoldenRatio)^2,10,120][[1]] (* Harvey P. Dale, May 01 2017 *)

{ default(realprecision, 20080); phi = (1 + sqrt(5))/2; x=(exp(1)*Pi*phi)^2/100; for (n=3, 20000, d=floor(x); x=(x-d)*10; write("b131566.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

More terms from Harry J. Smith, Apr 26 2009

Fixed my PARI program, had -n Harry J. Smith, May 19 2009

A273580 Decimal expansion of the infinite nested radical sqrt(F_0 + sqrt(F_1 + sqrt(F_3 + ...))), where F_k are the Fermat numbers A000215.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, May 25 2016

The convergence of this expression follows from Vijayaraghavan's theorem, for which it represents an extreme example.

Two PARI programs to compute this constant are listed below. The first one is a brute-force implementation of the definition and allows the computation of only 13 digits before exceeding current PARI capabilities. The second one implements the following 'trick' inspired by a comment in A094885: Let us try to compute first x = a/sqrt(2). We have x = (1/sqrt(2))sqrt(3+ sqrt(5+ sqrt(17+ ... ))) = sqrt(3/2+ (1/2)sqrt(5+ sqrt(17+ ... ))) = sqrt(3/2+ sqrt(5/4+ (1/4)sqrt(17+ ... ))) = sqrt(3/2+ sqrt(5/4+ sqrt(17/16+ ... ))) = sqrt(c_0+sqrt(c_1+sqrt(c_3+...))), where c_n = (2^(2^n)+1)/2^(2^n) = 1+d_n, with d_n = 2^(-2^n). This nested radical is easy to manage to any precision. However, evaluating it up to N terms, its convergence with increasing N is no better than that of the original algorithm. To speed it up, one must notice that, since the c_n converge rapidly to 1, and since the nested radical sqrt(1+sqrt(1+...)) evaluates to the golden ratio phi (A001622), the latter is the natural best stand-in for the neglected part (terms from N+1 to infinity). With this modification, i.e., 'seeding' the iterations with phi instead of 0, the convergence becomes extremely fast (the number of valid digits more than doubles upon incrementing N by 1).

			2.5295433262203984303103791288597533351935371244593834178657187113967...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A000215, A001622, A094885.

/* This function crashes PARI beyond N=28: */
s(N)={my(r=0.0);for(k=1,N,r=sqrt(2^(2.0^(N-k))+1+r));return(r)}
/* N is the number of terms to include in the evaluation. It turns out that the starting digits s(28) shares with s(27) are only 13 */

/* This alternative can easily generate millions of digits: */
d=vector(30);d[1]=0.5;for(n=2,#d,d[n]=d[n-1]^2);
S(N)={my(r=(1+sqrt(5))/2);for(k=1,N,r=sqrt(1+d[N-k+1]+r));return(r*sqrt(2))}
/* S(12) exceeds 1200 stable digits, S(20) goes over 150000. For the b-file, the first 2000 digits of S(13) were used, computed with the realprecision of 2100 digits */

Equals sqrt(2)*sqrt(1+1/2+sqrt(1+1/4+sqrt(1+1/16+sqrt(1+1/256+ ... )))).

A131567 Decimal expansion of 1/((ePiphi)^2).

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 27 2007

phi=(5^(1/2) + 1)/2.

			0.005237644...

Cf. A001113, A000796, A001622, A019609, A094886, A094885.

Join[{0,0},RealDigits[1/(E*Pi*GoldenRatio)^2,10,120][[1]]] (* Harvey P. Dale, Sep 28 2015 *)

More terms from Harvey P. Dale, Sep 28 2015

A277115 Decimal expansion of e*phi/Pi, where phi = (sqrt(5) + 1)/2.

Original entry on oeis.org

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

Offset: 1

Paulo Romero Zanconato Pinto, Sep 30 2016

An approximation to gamma(2) = 7/5. See A274981. - Omar E. Pol, Sep 30 2016

			1.400013583690484856298613502999790260381986690253106429917...

J. DePompeo (pers. comm., Mar. 29, 2004)

Eric Weisstein's World of Mathematics, Almost Integer

Cf. A000796, A001113, A001622, A094885, A274981.

First@ RealDigits[N[E GoldenRatio/Pi, 120]] (* Michael De Vlieger, Sep 30 2016 *)

PARI
```
exp(1)/Pi*(1+sqrt(5))/2;
```

Equals A001113 * A001622 / A000796.

cp's OEIS Frontend

A159823 Continued fraction for phi*e A094885, where phi = (1 + sqrt(5))/2.

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A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

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A094886 Decimal expansion of phi*Pi, where phi = (1+sqrt(5))/2.

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A131563 Decimal expansion of e*Pi*phi, where phi = (sqrt(5) + 1)/2.

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A131566 Decimal expansion of (e*Pi*phi)^2.

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A273580 Decimal expansion of the infinite nested radical sqrt(F_0 + sqrt(F_1 + sqrt(F_3 + ...))), where F_k are the Fermat numbers A000215.

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A131567 Decimal expansion of 1/((e*Pi*phi)^2).

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A131563 Decimal expansion of ePiphi, where phi = (sqrt(5) + 1)/2.

A131566 Decimal expansion of (ePiphi)^2.

A131567 Decimal expansion of 1/((ePiphi)^2).