A098317 -id:A098317 - cp's OEIS frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

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

			5.04316564336002865131188218928542471032359017541384...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139342.

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

RealDigits[Exp[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 08 2022 *)

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

From Amiram Eldar, Feb 08 2022: (Start)
Equals exp(A001622).
Equals 1/A139342. (End)

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

Original entry on oeis.org

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

A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

Original entry on oeis.org

1, 5, 2, 13, 5, 29, 10, 53, 17, 85, 26, 125, 37, 173, 50, 229, 65, 293, 82, 365, 101, 445, 122, 533, 145, 629, 170, 733, 197, 845, 226, 965, 257, 1093, 290, 1229, 325, 1373, 362, 1525, 401, 1685, 442, 1853, 485, 2029, 530, 2213, 577, 2405, 626, 2605, 677

Offset: 0

Paul Curtz, Aug 15 2015

Using (n+sqrt(4+n^2))/2, after the integer 1 for n=0, the reduced metallic means are b(1) = (1+sqrt(5))/2, b(2) = 1+sqrt(2), b(3) = (3+sqrt(13))/2, b(4) = 2+sqrt(5), b(5) = (5+sqrt(29))/2, b(6) = 3+sqrt(10), b(7) = (7+sqrt(53))/2, b(8) = 4+sqrt(17), b(9) = (9+sqrt(85))/2, b(10) = 5+sqrt(26), b(11) = (11+sqrt(125))/2 = (11+5*sqrt(5))/2, ... . The last value yields the radicals in a(n) or A013946.
b(2) = 2.41, b(3) = 3.30, b(4) = 4.24, b(5) = 5.19 are "good" approximations of fractal dimensions corresponding to dimensions 3, 4, 5, 6: 2.48, 3.38, 4.33 and 5.45 based on models. See "Arbres DLA dans les espaces de dimension supérieure: la théorie des peaux entropiques" in Queiros-Condé et al. link. DLA: beginning of the title of the Witten et al. link.
Consider the symmetric array of the half extended Rydberg-Ritz spectrum of the hydrogen atom:
   0,    1/0,     1/0,     1/0,     1/0,      1/0,      1/0,     1/0, ...
-1/0,      0,     3/4,     8/9,   15/16,    24/25,    35/36,   48/49, ...
-1/0,   -3/4,       0,    5/36,    3/16,   21/100,      2/9,  45/196, ...
-1/0,   -8/9,   -5/36,       0,   7/144,   16/225,     1/12,  40/441, ...
-1/0, -15/16,   -3/16,  -7/144,       0,    9/400,    5/144,  33/784, ...
-1/0, -24/25, -21/100, -16/225,  -9/400,        0,   11/900, 24/1225, ...
-1/0, -35/36,    -2/9,   -1/12,  -5/144,  -11/900,        0, 13/1764, ...
-1/0, -48/49, -45/196, -40/441, -33/784, -24/1225, -13/1764,       0, ... .
The numerators are almost A165795(n).
Successive rows: A000007(n)/A057427(n), A005563(n-1)/A000290(n), A061037(n)/A061038(n), A061039(n)/A061040(n), A061041(n)/A061042(n), A061043(n)/A061044(n), A061045(n)/A061046(n), A061047(n)/A061048(n), A061049(n)/A061050(n).
A144433(n) or A195161(n+1) are the numerators of the second upper diagonal (denominators: A171522(n)).
c(n+1) = a(n) + a(n+1) = 6, 7, 15, 18, 34, 39, 63, 70, 102, 111, ... .
c(n+3) - c(n+1) = 9, 11, 19, 21, 29, 31, ... = A090771(n+2).
The final digit of a(n) is neither 4 nor 8. - Paul Curtz, Jan 30 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Diogo Queiros-Condé, Jean Chaline, and Jacques Dubois, Le monde des fractales La Nature trans-échelles, 478p., ellipses, Paris, 2015, page 220.
T. A. Witten, Jr. and L. M. Sander, Diffusion-Limited Aggregation, a Kinetic Critical Phenomenom, Phys. Rev. Lett., Vol. 47 (Nov 09 1981), pp. 1400-1403.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000007, A000290, A001622, A002522, A005563, A010685, A010698, A013946, A014176, A057427, A061035-A061050, A078370, A087475, A090771, A098316, A098317, A098318, A144433, A168077, A171621, A176398, A176439, A176458, A176522, A176537, A195161.

Cf. A069894.

[Numerator(1+n^2/4): n in [0..60]]; // Vincenzo Librandi, Aug 15 2015

A261327:=n->numer((4 + n^2)/4); seq(A261327(n), n=0..60); # Wesley Ivan Hurt, Aug 15 2015

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 5, 2, 13, 5, 29}, 60] (* Vincenzo Librandi, Aug 15 2015 *)
a[n_] := (n^2 + 4) / 4^Mod[n + 1, 2]; Table[a[n], {n, 0, 52}] (* Peter Luschny, Mar 18 2022 *)

vector(60, n, n--; numerator(1+n^2/4)) \\ Michel Marcus, Aug 15 2015

Vec((1+5*x-x^2-2*x^3+2*x^4+5*x^5)/(1-x^2)^3 + O(x^60)) \\ Colin Barker, Aug 15 2015

a(n)=if(n%2,n^2+4,(n/2)^2+1) \\ Charles R Greathouse IV, Oct 16 2015

[(n*n+4)//4**((n+1)%2) for n in range(60)] # Gennady Eremin, Mar 18 2022

[numerator(1+n^2/4) for n in (0..60)] # G. C. Greubel, Feb 09 2019

a(n) = numerator(1 + n^2/4). (Previous name.) See A010685 (denominators).
a(2*k) = 1 + k^2.
a(2*k+1) = 5 + 4*k*(k+1).
a(2*k+1) = 4*a(2*k) + 4*k + 1.
a(4*k+2) = A069894(k). - Paul Curtz, Jan 30 2019
a(-n) = a(n).
a(n+2) = a(n) + A144433(n) (or A195161(n+1)).
a(n) = A168077(n) + period 2: repeat 1, 4.
a(n) = A171621(n) + period 2: repeat 2, 8.
From Colin Barker, Aug 15 2015: (Start)
a(n) = (5 - 3*(-1)^n)*(4 + n^2)/8.
a(n) = n^2/4 + 1 for n even;
a(n) = n^2 + 4 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
G.f.: (1 + 5*x - x^2 - 2*x^3 + 2*x^4 + 5*x^5)/ (1 - x^2)^3. (End)
E.g.f.: (5/8)*(x^2 + x + 4)*exp(x) - (3/8)*(x^2 - x + 4)*exp(-x). - Robert Israel, Aug 18 2015
Sum_{n>=0} 1/a(n) = (4*coth(Pi)+tanh(Pi))*Pi/8 + 1/2. - Amiram Eldar, Mar 22 2022

New name by Peter Luschny, Mar 18 2022

A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

Original entry on oeis.org

1, 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, 9, 6, 9, 5

Offset: 1

Omar E. Pol, Nov 15 2007

Convergents are 4/2, 8/8, 32/24, 96/80, 320/256, 1024/832, 3328/2688, 10752/8704, 34816/28160, 112640/91136, 364544/294912, 1179648/954368, 3817472/3088384, 12353536/9994240, ... = A209084/A063727. - Seiichi Kirikami, Mar 14 2012

2*(-1 + phi) is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Feb 16 2016

			1.236067977499789696...

Michael Penn, How large is the blue ▲, YouTube video, 2021.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019863, A063727, A209084, A033887.

RealDigits[ N[4/(1+Sqrt[5]), 150] ] [ [1] ] (* Seiichi Kirikami, Mar 14 2012 *)

4/(1+sqrt(5)) \\ Altug Alkan, Apr 11 2016

Equals A134945 - 2 = A002163 - 1 = A098317 - 3. - R. J. Mathar, Oct 27 2008
2*(-1 + A001622). - Wolfdieter Lang, Feb 17 2016
Equals the harmonic mean of 1 and phi, 2*phi/(1+phi). - Stanislav Sykora, Apr 11 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(n!^2*3^(2*n+2)).
Equals -1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019863. - R. J. Mathar, Jan 17 2021
Equals 2*sin(Pi/5)/sin(2*Pi/5) = hypergeom([1/5, 3/5], [7/5], 1) = hypergeom([-1/5, -3/5], [3/5], 1). - Peter Bala, Mar 04 2022

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

A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 11, 16, 22, 32, 46, 65, 92, 130, 185, 262, 371, 526, 745, 1056, 1496, 2119, 3001, 4251, 6021, 8528, 12080, 17110, 24236, 34328, 48622, 68869, 97547, 138166, 195700, 277191, 392616, 556104, 787670, 1115663, 1580234, 2238256, 3170284

Offset: 0

Kival Ngaokrajang, Apr 11 2014

a(n) is the perimeter (rounded down) of a decaflake after n iterations, let a(0) = 1. The total number of sides is 10*A000400(n). The total number of holes is A002275(n). 2 + sqrt(5) = A098317.

Kival Ngaokrajang, Illustration of decaflake for n = 0..3
Wikipedia, n-flake

Cf. A000400, A002275, A098317, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).

A240734:=n->floor(6^n/(2+sqrt(5))^n); seq(A240734(n), n=0..50); # Wesley Ivan Hurt, Apr 12 2014

Table[Floor[6^n/(2 + Sqrt[5])^n], {n, 0, 50}] (* Wesley Ivan Hurt, Apr 12 2014 *)

{a(n)=floor(6^n/(2+sqrt(5))^n)}
       for (n=0, 100, print1(a(n), ", "))

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

A134945 Decimal expansion of 1 + sqrt(5).

Original entry on oeis.org

3, 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

Offset: 1

Omar E. Pol, Nov 15 2007

If "index" equals (0,2) then this sequence is the decimal expansion of (golden ratio divided by 5 = phi/5 = (1 + sqrt(5))/10). Example: 0.323606797...
Apart from the leading digit the same as A134972, A098317 and A002163. - R. J. Mathar, Aug 06 2013
Length of the longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020
Abscissa of the first superstable point of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			3.2360679774997896964...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Index entries for algebraic numbers, degree 2.

Cf. A000045, A002163, A098317, A134972.

RealDigits[1 + Sqrt[5], 10, 100][[1]] (* Michael De Vlieger, Nov 13 2020 *)

PARI
```
1 + sqrt(5) \\ Altug Alkan, Mar 19 2018
```

From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!+8*n!^2)/(n!^2*3^(2*n+2)).
Equals 1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019827. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} (1 + 1/Fibonacci(2*k)). - Amiram Eldar, May 27 2021

More terms from Jinyuan Wang, Mar 30 2020

A244593 Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jul 01 2014

Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014

The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 2017

			11.09016994374947424102293417182819058860154589902881431067724311352630...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

RealDigits[GoldenRatio^5, 10, 103] // First

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A139345 Decimal expansion of sine of the golden ratio. That is, the decimal expansion of sin((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

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

			0.99888450909488479883326824263012904463865119212705...

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176; Solution, ibid., Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139350.

RealDigits[Sin[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

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

Equals sin(A001622).

Equals 1/A139350. - Amiram Eldar, Feb 07 2022

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

cp's OEIS Frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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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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A261327 a(n) = (n^2 + 4) / 4^((n + 1) mod 2).

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A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

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

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A240734 a(n) = floor(6^n/(2+sqrt(5))^n).

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