A020762 -id:A020762 - cp's OEIS frontend

A144984 Denominators of an Egyptian fraction for 1/sqrt(5) (A020762).

3, 9, 362, 148807, 432181530536, 615828580117398011389583, 385329014801969222669766835659574445455872858297

Offset: 1

Artur Jasinski, Sep 28 2008

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A020762, A069139, A006487, A006526, A006525, A006524, A001466, A110820, A117116, A118323, A118324, A118325, A144835, A132480-A132574, A069261, A144984-A145003.

a = {}; k = N[1/Sqrt[5], 1000]; Do[s = Ceiling[1/k]; AppendTo[a, s]; k = k - 1/s, {n, 1, 10}]; a

A010476 Decimal expansion of square root of 20.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {2, 8} repeated. - Harry J. Smith, Jun 03 2009

			4.472135954999579392818347337462552470881236719223051448541794490821041....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Except for offset, same as A020762.
Cf. A040015 (continued fraction). - Harry J. Smith, Jun 03 2009
Cf. A002163 (decimal expansion of square root of 5).

RealDigits[N[Sqrt[20], 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(20); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010476.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

sqrt(20) = 2*sqrt(5). - Alonso del Arte, Jun 26 2015

Equals Sum_{k>=0} binomial(2*k,k) * k/5^k. - Amiram Eldar, Aug 03 2020

A344212 Decimal expansion of 1 + 1/sqrt(5).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 11 2021

Decimal expansion of the midradius of a rhombic triacontahedron with unit edge length.

Essentially the same sequence of digits as A176453, A134974, A020762 and A010476. - R. J. Mathar, May 16 2021

			1.447213595499957939281834733746255247088123671922305...

Wikipedia, Rhombic triacontahedron.
Index entries for algebraic numbers, degree 2.

Cf. A019952 (rhombic triacontahedron inscribed sphere radius).
Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344172 (rhombic triacontahedron volume).
Cf. A010476, A020762, A081005, A134974, A176453, A187798, A242671.

RealDigits[1 + 1/Sqrt[5], 10, 100][[1]] // Flatten

5^-.5+1 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(5*x^2-10*x+4)[2] \\ Charles R Greathouse IV, Feb 04 2025

From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A242671 = 1/A187798.
Equals Product_{k>=0} (1 + 1/A081005(k)). (End)

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

A134974 Decimal expansion of 4(-1 + phi) = 4A094214, where the golden ratio phi = A001622.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

This equals the dimensionless q-entropy (Tsallis entropy) of the set of 5 probabilities {p_i = 1/5, i = 1..5} for q = 1/2, which is S/k = -(1 - 5*(1/5)^(1/2))/(1 - 1/2) (k is the Boltzmann constant). See the Wikipedia link. - Wolfdieter Lang, Dec 06 2018

This constant - 2 = 2*sqrt(5) - 4 is the area of a regular pentagram formed by connecting the vertices of a unit-area regular pentagon. - Amiram Eldar, Nov 12 2021

			2.47213595499957939281834733746255247...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Wikipedia, Tsallis entropy.

Cf. A001622, A010476, A020762, A094214, A134972.

evalf[100](8/(1+sqrt(5))); # Muniru A Asiru, Dec 19 2018

RealDigits[4/GoldenRatio,10,120][[1]] (* Harvey P. Dale, Oct 30 2016 *)

2*(sqrt(5)-1) \\ or: digits( % \1e-35). - M. F. Hasler, Dec 14 2018

Equals 4*(-1 + phi) = 4*A094214, where phi = A001622. This is an integer in the field Q(sqrt(5)).
Equals 4/phi = 8/(1 + sqrt(5)).
Equals 10*A020762-2 = A010476-2. - R. J. Mathar, Oct 27 2008
Equals 2*(sqrt(5) - 1) = 2*A134972. - M. F. Hasler, Dec 14 2018

More terms from Harvey P. Dale, Oct 30 2016

Edited by Wolfdieter Lang, Dec 14 2018

A322159 Decimal expansion of 1 - 1/sqrt(5).

Original entry on oeis.org

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

Offset: 0

Tristan Cam, Nov 29 2018

Continued fraction: [0;1,1,4,4,4...].

Least root of the polynomial: 5x^2 - 10x + 4.

			0.552786404500042060718165266253744752911876328077...

Cf. A001622, A002163, A020762, A081010, A244847, A296182.

evalf[110](1-1/sqrt(5)); # Muniru A Asiru, Dec 01 2018

RealDigits[1-1/Sqrt[5], 10, 100][[1]] (* Amiram Eldar, Nov 29 2018 *)

Equals 1 - 1/A002163.
Equals 1/(1 - cos(4*Pi/5)) = (1/2)*csc(2*Pi/5)^2.
Also equal to 2/(phi*sqrt(5)) = 2/(A001622*A002163).
Equals 1 - A020762. - Andrew Howroyd, Nov 30 2018
From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A244847 = 1/A296182.
Equals Product_{k>=0} (1 - 1/A081010(k)). (End)

A020772 Decimal expansion of 1/sqrt(15).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(15) = 0.258198889747161125678617693318826640722194780352772721772504917740898872796... [Vladimir Joseph Stephan Orlovsky, May 30 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[N[1/Sqrt[15],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals 1/A010472 = A020760 * A020762. - R. J. Mathar, Nov 19 2024

A176453 Decimal expansion of 4+2*sqrt(5).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of 4+2*sqrt(5) is A010698 preceded by 8.
a(n) = A010476(n) = A020762(n-1) = A134974(n) for n > 1.
Rajan (2010) claims the variance of a discrete distribution generated by the linear convolution of Fibonacci sequence with itself, saturates to a constant of value 8.4721359. [From Jonathan Vos Post, May 10 2010]

			4+2*sqrt(5) = 8.47213595499957939281...

Arulalan Rajan, Jamadagni, Vittal Rao, Ashok Rao, Convolutions Induced Discrete Probability Distributions and a New Fibonacci Constant, May 6, 2010. [From _Jonathan Vos Post_, May 10 2010]

Cf. A002163 (decimal expansion of sqrt(5)), A010476 (decimal expansion of sqrt(20)), A020762 (decimal expansion of 1/sqrt(5)), A134974 (decimal expansion of 8/(1+sqrt(5))), A010698 (repeat 2, 8).

RealDigits[4+2Sqrt[5],10,120][[1]] (* Harvey P. Dale, Sep 08 2018 *)

A380862 Decimal expansion of the largest acute angles, in radians, in a deltoidal hexecontahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 06 2025

A deltoidal hexecontahedron face is a kite with one smallest acute angle (A380861), two largest acute angles (this constant) and one obtuse angle (A380863).

			1.517985377460215463602191357386072448171233382527...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Deltoidal hexecontahedron.

Cf. A020762, A379385, A379386, A379387, A379388, A379389, A380861, A380863.

First[RealDigits[ArcCos[1/2 - 1/Sqrt[5]], 10, 100]]

Equals arccos(1/2 - 1/sqrt(5)) = arccos(1/2 - A020762).

Equals (2*Pi - A380861 - A380863)/2.

A023118 Signature sequence of 1/sqrt(5) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
C. Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A020762.

signseq[n_] := Take[ Transpose[ Sort[ Flatten[ Table[{i + j*n, i}, {i, Max[15, 15n]}, {j, Max[15, 15/n]}], 1], #1[[1]] < #2[[1]] &]][[2]], 105]; signseq[1/Sqrt[5]] (* Robert G. Wilson v, Sep 20 2004 *)

More terms from Robert G. Wilson v, Sep 20 2004

cp's OEIS Frontend

A144984 Denominators of an Egyptian fraction for 1/sqrt(5) (A020762).

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A010476 Decimal expansion of square root of 20.

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A344212 Decimal expansion of 1 + 1/sqrt(5).

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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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A134974 Decimal expansion of 4*(-1 + phi) = 4*A094214, where the golden ratio phi = A001622.

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A322159 Decimal expansion of 1 - 1/sqrt(5).

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A020772 Decimal expansion of 1/sqrt(15).

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A134974 Decimal expansion of 4(-1 + phi) = 4A094214, where the golden ratio phi = A001622.