A094874 -id:A094874 - 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

A081012 a(n) = Fibonacci(4n+1) - 2, or Fibonacci(2n+2)*Lucas(2n-1).

Original entry on oeis.org

3, 32, 231, 1595, 10944, 75023, 514227, 3524576, 24157815, 165580139, 1134903168, 7778742047, 53316291171, 365435296160, 2504730781959, 17167680177563, 117669030460992, 806515533049391, 5527939700884755, 37889062373143904

Offset: 1

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A094874.

List([1..30], n-> Fibonacci(4*n+1) -2); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+1)-2: n in [1..30]]; // Vincenzo Librandi, Apr 20 2011

with(combinat) for n from 0 to 25 do printf(`%d,`,fibonacci(4*n+1)-2) od # James Sellers, Mar 03 2003

Fibonacci[4*Range[30]+1] -2 (* G. C. Greubel, Jul 14 2019 *)

vector(30, n, fibonacci(4*n+1)-2) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+1)-2 for n in (1..30)] # G. C. Greubel, Jul 14 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: x*(3+8*x-x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 22 2012
a(n) = 7*a(n-1) - a(n-2) + 10, n>=3. - R. J. Mathar, Nov 07 2015
Product_{n>=1} (1 + 1/a(n)) = (5-sqrt(5))/2 = A094874. - Amiram Eldar, Nov 28 2024

A109866 9's complement of the digits of the golden ratio phi (A001622): 9.999999999999... - 1.6180339887... = 8.3819660112501051517954131656334...

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jul 09 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A079585, A094874, A132338.

With[{nn=120},PadRight[{},nn,9]-RealDigits[GoldenRatio,10,nn][[1]]] (* Harvey P. Dale, Jun 25 2018 *)

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... .

A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 01 2024

			0.93548928378863903321291906615298281...

Cf. A019845, A060294, A086089, A089491, A094874, A112628, A182007, A258403.

RealDigits[5 Sqrt[3 - GoldenRatio]/(2 Pi), 10, 99][[1]]

Equals Product_{k>=1} (1 - 1/(5*k)^2).

Equals A258403/Pi. - Hugo Pfoertner, Apr 01 2024

A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 12 2021

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

			1.12256994144896343110486287949381696894803120580270...

Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
Mathematics Stack Exchange, Area of a five pointed star, 2014.
Wikipedia, Pentagram.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).
Equals 5/(cot(Pi/5) + cot(Pi/10)).
Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).
Equals (5/2)*sqrt((25 -11*sqrt(5))/2).
Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

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

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A081012 a(n) = Fibonacci(4n+1) - 2, or Fibonacci(2n+2)*Lucas(2n-1).

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A109866 9's complement of the digits of the golden ratio phi (A001622): 9.999999999999... - 1.6180339887... = 8.3819660112501051517954131656334...

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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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A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

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A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

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