A094214 -id:A094214 - cp's OEIS frontend

A045529 a(n+1) = 5a(n)^3 - 3a(n), a(0) = 1.

1, 2, 34, 196418, 37889062373143906, 271964099255182923543922814194423915162591622175362

Offset: 0

Jose Eduardo Blazek, Dec 11 1999

The next term, a(6), has 153 digits. - Harvey P. Dale, Oct 24 2011

Seiichi Manyama, Table of n, a(n) for n = 0..7
Daniel Duverney and Takeshi Kurosawa, Transcendence of infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2002), Article 68.
Zalman Usiskin, Problem B-265, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 11, No. 3 (1973), p. 333; Fibonacci Numbers for Powers of 3, Solution to Problem B-265 by Ralph Garfield and David Zeitlin, ibid., Vol. 12, No. 3 (1974), p. 315.

Cf. (k^n)-th Fibonacci number: A058635 (k=2), this sequence (k=3), A145231 (k=4), A145232 (k=5), A145233 (k=6), A145234 (k=7), A250487 (k=8), A250488 (k=9), A250489 (k=10).

Cf. A000032, A000045, A001622, A001999, A002814, A006267, A094214.

a := proc(n) option remember; if n = 0 then 1 else 5*a(n-1)^3 - 3*a(n-1) end if; end:
seq(a(n), n = 0..5); # Peter Bala, Nov 24 2022

G = (1 + Sqrt[5])/2; Table[Expand[(G^(3^n) - (1 - G)^(3^n))/Sqrt[5]], {n, 1, 7}] (* Artur Jasinski, Oct 05 2008 *)
Table[Round[(4/5)^(1/2)*Cosh[3^n*ArcCosh[((5/4)^(1/2))]]], {n, 1, 4}] (* Artur Jasinski, Oct 05 2008 *)
RecurrenceTable[{a[0]==1,a[n]==5a[n-1]^3-3a[n-1]},a[n],{n,6}] (* Harvey P. Dale, Oct 24 2011 *)
NestList[5#^3-3#&,1,5] (* Harvey P. Dale, Dec 21 2014 *)

A045529(n):=fib(3^n)$
makelist(A045529(n),n,0,10); /* Martin Ettl, Nov 12 2012 */

The first example I know in which a(n) can be expressed as (4/5)^(1/2)*cosh(3^n*arccosh((5/4)^(1/2))).
a(n) = Fibonacci(3^n). - Leroy Quet, Mar 17 2002
a(n+1) = a(n)*A002814(n+1). - Lekraj Beedassy, Jun 16 2003
a(n) = (phi^(3^n) - (1 - phi)^(3^n))/sqrt(5), where phi is the golden ratio (A001622). - Artur Jasinski, Oct 05 2008
a(n) = Product_{k=0..n-1} (Lucas(2*3^k) - 1) (Usiskin, 1973). - Amiram Eldar, Jan 29 2022
From Peter Bala, Nov 24 2022: (Start)
a(2*n+2) == a(2*n) (mod 3^(2*n+1)); a(2*n+3) == a(2*n+1) (mod 3^(2*n+2));
a(2*n+1) + a(2*n) == 0 (mod 3^(2*n+1)).
a(2*n) == 1 (mod 3) and a(2*n+1) == 2 (mod 3).
5*a(n)^2 == 2 (mod 3^(n+1)).
In the ring of 3-adic integers, the sequences {a(2*n)} and {a(2*n+1)} are both Cauchy sequences and converge to the pair of 3-adic roots of the quadratic equation 5*x^2 - 2 = 0. (End)
From Amiram Eldar, Jan 07 2023: (Start)
Product_{n>=1} (1 + 2/(sqrt(5)*a(n)-1)) = phi (A001622).
Product_{n>=1} (1 - 2/(sqrt(5)*a(n)+1)) = 1/phi (A094214).
Both formulas are from Duverney and Kurosawa (2022). (End)

A139339 Decimal expansion of the square root of the golden ratio.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

The hyperbolas x^2 - y^2 = 1 and xy = 1 meet at (c, 1/c) and (-c, -1/c), where c = sqrt(golden ratio); see the Mathematica program for a graph. - Clark Kimberling, Oct 19 2011
An algebraic integer of degree 4. Minimal polynomial: x^4 - x^2 - 1. - Charles R Greathouse IV, Jan 07 2013
Also the limiting value of the ratio of the slopes of the tangents drawn to the function y=sqrt(x) from the abscissa F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, Apr 04 2021
The length of the base of the isosceles triangle of smallest perimeter which circumscribes a unit-diameter semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
The unique real solution to arcsec(x) = arccot(x). - Wolfe Padawer, Apr 14 2023

			1.2720196495140689642524224617374914917156080418400...

B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 45-48.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176. Solution published in Vol. 12, No. 1 (Winter 2000), pp. 61-62.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Index entries for algebraic numbers, degree 4.

Cf. A000045, A001622, A094214, A104457, A098317, A002390; A197762 (related intersection of hyperbolas).

Digits:=100: evalf(sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Sep 11 2018

N[Sqrt[GoldenRatio], 100]
FindRoot[x*Sqrt[-1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
Plot[{Sqrt[-1 + x^2], 1/x}, {x, 0, 3}] (* Clark Kimberling, Oct 19 2011 *)

sqrt((1+sqrt(5))/2) \\ Charles R Greathouse IV, Jan 07 2013

a(n) = sqrtint(10^(2*n-2)*quadgen(5))%10; \\ Chittaranjan Pardeshi, Aug 24 2024

Equals sqrt((1 + sqrt(5))/2).
Equals 1/sqrt(A094214). - Burak Muslu, Apr 04 2021
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A197762.
Equals tan(arccos(1/phi)).
Equals cot(arcsin(1/phi)). (End)
From Gerry Martens, Jul 30 2023: (Start)
Equals 5^(1/4)*cos(arctan(2)/2).
Equals Re(sqrt(1+2*i)) (the imaginary part is A197762). (End)

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

3, 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

Offset: 0

N. J. A. Sloane

Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).
Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012
One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1 = -A019863). - Alonso del Arte, Apr 25 2015
This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - Wolfdieter Lang, Jan 08 2018
Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019
Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			0.30901699437494742410229341718281905886015458990288143106772431135263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

Zak Seidov, Table of n, a(n) for n = 0..999
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.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.
Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007
Equals 1/A134945. - R. J. Mathar, Jan 17 2021
Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024
Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024
This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025

A007570 a(n) = F(F(n)), where F is a Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 21, 233, 10946, 5702887, 139583862445, 1779979416004714189, 555565404224292694404015791808, 2211236406303914545699412969744873993387956988653, 2746979206949941983182302875628764119171817307595766156998135811615145905740557

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

a(20) is approximately 2.830748520089124 * 10^1413, much too large to include even in the b-file. - Alonso del Arte, Apr 30 2020
Let M(0) denote the 2 X 2 identity matrix, and let M(1) = [[0, 1], [1, 1]]. Let M(n) = M(n-2) * M(n-1). Then a(n) is equal to both the (1, 2)-entry and the (2, 1)-entry of M(n). - John M. Campbell, Jul 02 2021
This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 06 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..19 (terms n = 0..17 from T. D. Noe)
Alonso del Arte, Table of n, a(n) for n = 0 .. 24, with digits grouped in hundreds
John M. Campbell, A Matrix-Based Recursion Relation for F_{F_n}, Fib. Quart., Vol. 60, No. 3 (2022), pp. 256-261.
Bakir Farhi, Summation of certain infinite Fibonacci related series, arXiv:1512.09033 [math.NT], 2015. See p. 6, eq. 2.9.
George Ledin, Jr., Problem H-147, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 352; Converging Fractions, Solution to Problem H-147 by David Zeitlin, ibid., Vol. 8, No. 4 (1970), pp. 387-389.
Edward A. Parberry, Two recursion relations for F(F(n)), Fib. Quart., Vol. 15, No. 2 (1977), p. 122 and p. 139.
Martin Stein, Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers, Dissertation, Hannover: Gottfried Wilhelm Leibniz Universität Hannover, 2012.
Chris Street, A Recurrence for the Sequence {F(F(n)), n>=0}.

Cf. A000045, A001622, A005371, A058051, A094214.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Oct 09 2015

F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2]; Table[F[F[n]], {n, 0, 14}]
Fibonacci[Fibonacci[Range[0, 20]]] (* Harvey P. Dale, May 05 2012 *)

a(n)=fibonacci(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

from sympy import fibonacci
def a(n): return fibonacci(fibonacci(n))
print([a(n) for n in range(15)]) # Michael S. Branicky, Feb 02 2022

[fibonacci(fibonacci(n)) for n in range(0, 14)] # Zerinvary Lajos, Nov 30 2009

val fibo: LazyList[BigInt] = (0: BigInt) #:: (1: BigInt) #:: fibo.zip(fibo.tail).map { n => n._1 + n._2 }
val fiboLimited: LazyList[Int] = 0 #:: 1 #:: fiboLimited.zip(fiboLimited.tail).map { n => n._1 + n._2 } // Limited to 32-bit integers because that's the type for LazyList apply()
(0 to 19).map(n => fibo(fiboLimited(n))) // Alonso del Arte, Apr 30 2020

a(n+1)/a(n) ~ phi^(F(n-1)), with phi = (1 + sqrt(5))/2 = A001622. - Carmine Suriano, Jan 24 2011
Sum_{n>=1} 1/a(n) = 3.7520024260... is transcendental (Stein, 2012). - Amiram Eldar, Oct 30 2020
Sum_{n>=1} (-1)^(F(n)+1)*a(n-1)/(a(n)*a(n+1)) = 1/phi (A094214) (Farhi, 2015). - Amiram Eldar, Apr 07 2021
Limit_{n->oo} a(n+1)/a(n)^phi = 5^((phi-1)/2) = 1.6443475285..., where phi is the golden ratio (A001622) (Ledin, 1968) - Amiram Eldar, Feb 02 2022

One more term from Harvey P. Dale, May 05 2012

A101864 Wythoff BB numbers.

Original entry on oeis.org

5, 13, 18, 26, 34, 39, 47, 52, 60, 68, 73, 81, 89, 94, 102, 107, 115, 123, 128, 136, 141, 149, 157, 162, 170, 178, 183, 191, 196, 204, 212, 217, 225, 233, 238, 246, 251, 259, 267, 272, 280, 285, 293, 301, 306, 314, 322, 327, 335, 340, 348, 356, 361, 369, 374, 382, 390, 395

Offset: 1

N. J. A. Sloane, Jan 28 2005

nonn

a(n)-3 are also the positions of 1 in A188436. - Federico Provvedi, Nov 22 2018

The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Jean-Paul Allouche and F. Michel Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018-2019.
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, JIS 11 (2008), Article 08.3.3.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik, Vol. 78, No. 2 (2021), pp. 1-8.
Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), 267-273.

Second row of A101858.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.
Cf. A094214, A188436.

b:=n->floor(n*((1+sqrt(5))/2)^2): seq(b(b(n)),n=1..60); # Muniru A Asiru, Dec 05 2018

b[n_] := Floor[n * GoldenRatio^2]; a[n_] := b[b[n]]; Array[a, 60] (* Amiram Eldar, Nov 22 2018 *)

from sympy import S
for n in range(1,60): print(int(S.GoldenRatio**2*(int(n*S.GoldenRatio**2))), end=', ') # Stefano Spezia, Dec 06 2018

a(n) = B(B(n)), n>=1, with B(k)=A001950(k) (Wythoff B-numbers). a(0)=0 with B(0)=0.

A099971 Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 9, 25, 57, 121, 1145, 3193, 11385, 27769, 60537, 191609, 453753, 978041, 2026617, 10415225, 27192441, 94301305, 228519033, 496954489, 2644438137, 11234372729, 28414241913, 62773980281, 131493457017, 268932410489, 543810317433

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			(sqrt(5)-1)/2 = 0.618033988749894848204586834365638117720309179805762862135... = 0.100111100011011101111001101110010111111101001010011111000001010111110011... in binary.

Cf. A094214, A099969, A099970, A099972, A099973, A099974.

d = 100; l = First[RealDigits[N[(Sqrt[5]-1)/2, d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)

More terms from Ryan Propper, Aug 18 2005

A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).

Original entry on oeis.org

0, 1, 2, 5, 10, 19, 34, 59, 100, 167, 276, 453, 740, 1205, 1958, 3177, 5150, 8343, 13510, 21871, 35400, 57291, 92712, 150025, 242760, 392809, 635594, 1028429, 1664050, 2692507, 4356586, 7049123

Offset: 0

Creighton Dement, Mar 10 2005

A floretion-generated sequence.
Floretion Algebra Multiplication Program, FAMP Code: 1vesrokseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)('i + i' + 'ji' + 'ki' + e) ] RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + p.
Partial sums of Leonardo numbers A001595. - Jonathan Vos Post, Jan 01 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A006355, A001595, A117501, A145912.

List([0..40], n-> 2*Fibonacci(n+2) -(n+2)); # G. C. Greubel, Jul 09 2019

[2*Fibonacci(n+2) -(n+2): n in [0..40]]; // G. C. Greubel, Jul 09 2019

a=0;b=1;Table[c=b+a+n; a=b; b=c, {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[x*(1-x+x^2)/((1-x)^2*(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,2,5},40] (* Harvey P. Dale, Sep 06 2012 *)

my(x='x+O('x^40)); concat(0, Vec(x*(1-x+x^2)/((1-x)^2*(1-x-x^2)))) \\ G. C. Greubel, Sep 26 2017

[2*fibonacci(n+2) -(n+2) for n in (0..40)] # G. C. Greubel, Jul 09 2019

Superseeker results (incomplete): a(2) - 2a(n+1) + a(n) = A006355(n+1) (Number of binary vectors of length n containing no singletons); a(n+1) - a(n) = A001595(n) (2-ranks of difference sets constructed from Segre hyperovals); a(n) + n + 1 = A001595(n+1).
A107909(a(n)) = A000975(n). - Reinhard Zumkeller, May 28 2005
From Ross La Haye, Aug 03 2005: (Start)
a(n) = 2*(Fibonacci(n+2) - 1) - n.
a(n) = Sum_{k=0..n} A101220(n-k, 0, k). (End)
From Gary W. Adamson, Apr 02 2006: (Start)
a(n) = a(n-1) + a(n-2) + n-1.
a(n) = row sums of A117501, starting (1, 2, 5, 10, ...). (End)
a(n) = Sum_{k=0..n} A109754(n-k,k). - Ross La Haye, Apr 12 2006
a(n) = (Sum_{k=0..n} (n-k)*Fibonacci(k-1) + Fibonacci(k)) - n. - Ross La Haye, May 31 2006
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = -2 - n + (-A094214)^n*(1-A010499/5) + (1+A010499/5)/A094214^n.
a(n) = A006355(n+3) - n - 2. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5. - Harvey P. Dale, Sep 06 2012

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

Original entry on oeis.org

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)

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

A296184 Decimal expansion of 2 + phi, with the golden section phi from A001622.

Original entry on oeis.org

3, 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

Offset: 1

Wolfdieter Lang, Jan 08 2018

In a regular pentagon, inscribed in a unit circle this equals twice the largest distance between a vertex and a midpoint of a side.
This is an integer in the quadratic number field Q(sqrt(5)).
Only the first digit differs from A001622.

			3.618033988749894848204586834365638117720309179805762862135448622705260462...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.25, p. 417.

Sumit Kumar Jha, Two complementary relations for the Rogers-Ramanujan continued fraction, arXiv:2112.12081 [math.NT], 2021.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A094214, A104457, A176055, A020837.

2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A332438 (n = 9), A019973 (n = 12).

First@ RealDigits[2 + GoldenRatio, 10, 77] (* Michael De Vlieger, Jan 13 2018 *)

(5 + sqrt(5))/2 \\ Altug Alkan, Mar 19 2018

Equals 2 + A001622 = 1 + A104457 = 3 + A094214.
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!+40*n!^2)/(2*n!^2*3^(2*n+2)).
Equals 5/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Constant c = 2 + 2*cos(2*Pi/10). The linear fractional transformation z -> c - c/z has order 10, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z)))))))))). - Peter Bala, May 09 2024

cp's OEIS Frontend

A045529 a(n+1) = 5*a(n)^3 - 3*a(n), a(0) = 1.

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A139339 Decimal expansion of the square root of the golden ratio.

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A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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A007570 a(n) = F(F(n)), where F is a Fibonacci number.

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A101864 Wythoff BB numbers.

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A099971 Write (sqrt(5)-1)/2 as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A045529 a(n+1) = 5a(n)^3 - 3a(n), a(0) = 1.

A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).