A226765 -id:A226765 - cp's OEIS frontend

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

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

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

A225667 Decimal expansion of 13-5*sqrt(5).

Original entry on oeis.org

1, 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

Offset: 1

Clark Kimberling, Jul 21 2013

Let d(n) = - 2*F(n) + h(2 + F(n+1), 1 + F(n+2)), where h = harmonic mean, F = A000045 (Fibonacci numbers). Then floor(d(n)) = 2F(n) + 1 for n>1, and limit(d(n)) = 13 - 5*sqrt(5).

Apart from leading digits the same as A132338, A109866, A094874 and A079585. - R. J. Mathar, Jul 30 2013

			13-5*sqrt(5) = 1.819660112501051517954131656343618822797...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000045, A226765.

f[n_] := Fibonacci[n]; h[n_] := HarmonicMean[{2 + f[n + 1], 1 + f[n + 2]}]; x = Limit[-2 f[n] + h[n], n -> Infinity] (* "proof" *)
d = RealDigits[x, 10, 120][[1]] (* A225667 *)

A322259 Decimal expansion of exp(-9 + 5*phi), where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Dec 01 2018

			0.40259263632247824757446721584399016437464148244440...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Don Redmond, Infinite products and Fibonacci numbers, Fib. Quart., Vol. 32, No. 3 (1994), pp. 234-239.
Index entries for transcendental numbers

Cf. A000032, A000045, A001622, A008683, A226765, A322258.

SetDefaultRealField(RealField(100)); Exp(-(13-5*Sqrt(5))/2); // G. C. Greubel, Dec 16 2018

evalf[100](exp(-9+5*(1+sqrt(5))/2)); # Muniru A Asiru, Dec 06 2018

RealDigits[Exp[-9+5*GoldenRatio], 10, 120][[1]]

exp(-(13-5*sqrt(5))/2) \\ Michel Marcus, Dec 02 2018

numerical_approx(exp(-(9-5*golden_ratio)), digits=100) # G. C. Greubel, Dec 16 2018

Equals Product_{k>=1} (L(k)/(sqrt(5)*F(k)))^(mu(k)/k), where L(k) and F(k) are the Lucas and Fibonacci numbers, and mu(k) is the Moebius function.

Equals exp(-A226765).

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

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

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A225667 Decimal expansion of 13-5*sqrt(5).

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A322259 Decimal expansion of exp(-9 + 5*phi), where phi is the golden ratio.

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