Joost Gielen - cp's OEIS frontend

A229760 Decimal expansion of 25 - 10*sqrt(5).

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

Offset: 1

Joost Gielen, Sep 28 2013

Apart from the first digit the same as A187799.

			2.639320225002103035908263312687237645593816403884742757291027545894790...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A187426, A187798, A094874.

RealDigits[25 - Sqrt@ 500, 10, 111][[1]] (* Robert G. Wilson v, Oct 01 2013 *)

25-sqrt(500) \\ Charles R Greathouse IV, Oct 01 2013

A227400 Decimal expansion of 5/(3*phi^2) where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Joost Gielen, Sep 20 2013

An algebraic number of degree 2. - Charles R Greathouse IV, Sep 28 2013

A quadratic number with denominator 2; minimal polynomial 9x^2 -15x + 25. - Charles R Greathouse IV, Apr 21 2016

			0.6366100187501752529923552760572698037994847...

Index entries for algebraic numbers, degree 2

RealDigits[5/(3*GoldenRatio^2), 10, 100][[1]] (* Alonso del Arte, Sep 28 2013 *)

(15-5*sqrt(5))/6 \\ Charles R Greathouse IV, Sep 28 2013

5/(3*phi^2), with phi = (1 + sqrt(5))/2.

A229780 Decimal expansion of (3+sqrt(5))/10.

Original entry on oeis.org

5, 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

Offset: 0

Joost Gielen, Sep 29 2013

sqrt((3+sqrt(5))/10) = sqrt(phi^2/5) = (5+sqrt(5))/10 = (3+sqrt(5))/10 + 2/10 = 0.723606797... .
Essentially the same as A134972, A134945, A098317 and A002163. - R. J. Mathar, Sep 30 2013
Equals one tenth of the limit of (G(n+2)+G(n+1)+G(n-1)+G(n-2))/G(n), where G(n) is any nonzero sequence satisfying the recurrence G(n+1) = G(n) + G(n-1) including A000032 and A000045, as n --> infinity. - Richard R. Forberg, Nov 17 2014
3+sqrt(5) is the perimeter of a golden rectangle with a unit width. - Amiram Eldar, May 18 2021
Constant x such that x = sqrt(x) - 1/5. - Andrea Pinos, Jan 15 2024

			0.5236067977499...

Cf. A094874, A187798.

RealDigits[GoldenRatio^2/5,10,120][[1]] (* Harvey P. Dale, Dec 02 2014 *)

(3+sqrt(5))/10 = (phi/sqrt(5))^2 = phi^2/5 where phi is the golden ratio.

A229759 Decimal expansion of (25-10*sqrt(5))/2.

Original entry on oeis.org

1, 3, 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

Joost Gielen, Sep 28 2013

Essentially the same as A225667 and A132338. - R. J. Mathar, Sep 30 2013

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A187426, A187798, A094874, A187799.

(25-10*sqrt(5))/2 = 25/2 - 5*sqrt(5) = 1.319660... .

A187799 Decimal expansion of 20/phi^2, where phi is the golden ratio. Also (with a different offset), decimal expansion of 3 - sqrt(5).

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Aug 30 2013

			20/phi^2 = 7.6393202250021030359082633...
3 - sqrt(5) = 0.76393202250021030359082633... (with offset 0).

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, The Value of a Series of Reciprocal Fibonacci Numbers, Problem B-1133, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275; Solution published in Vol. 52, No. 3, August 2014, pp. 277-278.

Cf. A187426, A187798, A094874.

5*(Sqrt(5)-1)^2; // Vincenzo Librandi, Feb 24 2015

First@ RealDigits[N[5*(Sqrt[5] - 1)^2, 111]] (* Michael De Vlieger, Feb 25 2015 *)

5*(sqrt(5)-1)^2 \\ Charles R Greathouse IV, Aug 31 2013

10*(3 - sqrt(5)) = 30 - 10*sqrt(5) = (5 - sqrt(5))^2 = 20/phi^2.

2 * Sum_{i > 1} (-1)^i/(F(i)F(i + 1)) = 3 - sqrt(5), where F(i) is the i-th Fibonacci number. This formula comes from John D. Watson, Jr.'s solution to Azarian's Problem B-1133 in the Fibonacci Quarterly. Azarian originally posed the problem as an infinite alternating sum explicitly written out for the first dozen terms or so. See the Azarian links above. - Alonso del Arte, Aug 25 2016

Extended by Charles R Greathouse IV, Aug 31 2013

A187426 Decimal expansion of (3-phi)^2 = 10 - 5*phi where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Aug 30 2013

This is an integer in Q(sqrt(5)). - Wolfdieter Lang, Jan 07 2018

			1.909830...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 2.

Cf. A226765.

Apart from the first digit the same as A187798.

RealDigits[(3-GoldenRatio)^2,10,120][[1]] (* Harvey P. Dale, Jun 30 2017 *)

(15-5*sqrt(5))/2 \\ Charles R Greathouse IV, Aug 31 2013

polrootsreal(x^2-15*x+25)[1] \\ Charles R Greathouse IV, Feb 04 2025

(3-phi)^2 = 5/phi^2 = 10 - 5*phi.
Smaller root of x^2 - 15x + 25 = 0.
Equals 10*A187798-5. - R. J. Mathar, Feb 08 2023

Corrected and extended by Charles R Greathouse IV, Aug 31 2013

A187798 Decimal expansion of (3-phi)/2, where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Joost Gielen, Aug 30 2013

This is the height h of the isosceles triangle in a regular pentagon inscribed in the unit circle formed from a diagonal as base and two adjacent pentagon sides. h = sqrt(sqrt(3-phi)^2 - (sqrt(2 + phi)/2)^2) = sqrt(10 - 5*phi)/2 = (3 - phi)/2. - Wolfdieter Lang, Jan 07 2018

			0.6909830056250525758977065828171809411398454100971185689322756886473697685905...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 2.

Cf. A001622, A081011, A187426, A226765, A094874, A344212.

RealDigits[(3 - GoldenRatio)/2, 10, 111][[1]] (* or *)
RealDigits[(5 - Sqrt[5])/4, 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

(5-sqrt(5))/4 \\ Charles R Greathouse IV, Aug 31 2013

Equals (3-phi)/2 = A094874/2 with phi from A001622.
From Amiram Eldar, Nov 28 2024: (Start)
Equals 1/A344212.
Equals Product_{k>=0} (1 - 1/A081011(k)). (End)

Extended by Charles R Greathouse IV, Aug 31 2013

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User: Joost Gielen

A229760 Decimal expansion of 25 - 10*sqrt(5).

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A227400 Decimal expansion of 5/(3*phi^2) where phi is the golden ratio.

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A229780 Decimal expansion of (3+sqrt(5))/10.

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A229759 Decimal expansion of (25-10*sqrt(5))/2.

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A187799 Decimal expansion of 20/phi^2, where phi is the golden ratio. Also (with a different offset), decimal expansion of 3 - sqrt(5).

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A187426 Decimal expansion of (3-phi)^2 = 10 - 5*phi where phi is the golden ratio.

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A187798 Decimal expansion of (3-phi)/2, where phi is the golden ratio.

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