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A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle. 9
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
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
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)).
Sequence in context: A221507 A370347 A089481 * A065421 A198556 A261169
KEYWORD
nonn,cons,easy,changed
AUTHOR
Clark Kimberling, Apr 04 2011
STATUS
approved

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Last modified February 29 15:18 EST 2024. Contains 370425 sequences. (Running on oeis4.)