A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

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

Offset: 1

Author

Clark Kimberling, Apr 04 2011

Keywords

• nonn
• cons
• easy

Comments

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).

Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011

This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012

Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014

The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

Examples

			1.902113032590307144232878666758764286811397268251...
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10001
• Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
• Eric Weisstein's World of Mathematics, Golden Rectangle.
• Eric Weisstein's World of Mathematics, Pentagram.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A001622 (decimal expansion of the golden ratio), A019881.

Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).

Cf. A090771, A195693, A195723, A296184.

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

Mathematica

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI

sqrt((5+sqrt(5))/2)

Formula

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020

Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021

Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022

Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022

Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024

Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024