A188593 - OEIS

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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

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

EXAMPLE

1.902113032590307144232878666758764286811397268251...

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 *)

PROG

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

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

CROSSREFS

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

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

Cf. A195693, A195723, A296184.

Sequence in context: A221507 A370347 A089481 * A065421 A198556 A261169

Adjacent sequences: A188590 A188591 A188592 * A188594 A188595 A188596

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Apr 04 2011

STATUS

approved