A176015 Decimal expansion of (5 + 3*sqrt(5))/10.

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

Offset: 1

Author

Klaus Brockhaus, Apr 06 2010

Keywords

• cons
• nonn

Comments

Continued fraction expansion of (5 + 3*sqrt(5))/10 is A010686.

The horizontal distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the vertical distance is A244847). - Amiram Eldar, May 18 2021

Examples

			(5 + 3*sqrt(5))/10 = 1.17082039324993690892...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..1000 [a(1000) corrected by _Georg Fischer_, Apr 02 2020]
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A000032, A000045, A001622, A002163 (decimal expansion of sqrt(5)), A010686 (repeat 1, 5), A090550, A134976.

Cf. A010499 (decimal expansion of 3*sqrt(5)).

Programs

Magma

SetDefaultRealField(RealField(105)); n:=(5+3*Sqrt(5))/10; Reverse(Intseq(Floor(10^104*n))); // Arkadiusz Wesolowski, Jan 07 2018

Maple

Digits := 1000:  (5+3*sqrt(5.0))/10; # Muniru A Asiru, Jan 22 2018

Mathematica

RealDigits[(5 + 3 Sqrt[5])/10, 10, 1001][[1]] (* Georg Fischer, Apr 02 2020 *)

PARI

(5 + 3*sqrt(5))/10 \\ Michel Marcus, Apr 20 2020

Formula

Equals (A134976 + 8)/10. - R. J. Mathar, Apr 12 2010

From Arkadiusz Wesolowski, Jan 07 2018: (Start)

Equals A001622^2 / sqrt(5).

Equals lim_{n -> infinity} A000045(n+2) / A001622^n. (End)

Equals 1/A090550 + 1. - Michel Marcus, Apr 20 2020

Minimal polynomial is 5x^2 - 5x - 1 (this number is an algebraic number but not an algebraic integer). - Alonso del Arte, Apr 20 2020

Equals lim_{k->oo} Fibonacci(k+2)/Lucas(k). - Amiram Eldar, Feb 06 2022