cp's OEIS Frontend

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

%I A188593 #78 Jun 12 2025 08:33:03
%S A188593 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,
%T A188593 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,
%U A188593 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
%N A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.
%C A188593 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).
%C A188593 Largest root of x^4 - 5x^2 + 5. - _Charles R Greathouse IV_, May 07 2011
%C A188593 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
%C A188593 Edge length of a pentagram (regular star pentagon) with unit circumradius. - _Stanislav Sykora_, May 07 2014
%C A188593 The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - _Mohammed Yaseen_, Nov 04 2020
%H A188593 Chai Wah Wu, <a href="/A188593/b188593.txt">Table of n, a(n) for n = 1..10001</a>
%H A188593 Michael Penn, <a href="https://www.youtube.com/watch?v=ZdXfgmxPfHI">On the fifth root of the identity matrix</a>, YouTube video, 2022.
%H A188593 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRectangle.html">Golden Rectangle</a>.
%H A188593 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pentagram.html">Pentagram</a>.
%H A188593 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A188593 Equals 2*A019881. - _Mohammed Yaseen_, Nov 04 2020
%F A188593 Equals csc(A195693) = sec(A195723). - _Amiram Eldar_, May 28 2021
%F A188593 Equals i^(1/5) + i^(-1/5). - _Gary W. Adamson_, Jul 08 2022
%F A188593 Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - _Wolfdieter Lang_, Aug 28 2022
%F A188593 Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - _Antonio Graciá Llorente_, Feb 24 2024
%F A188593 Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - _Amiram Eldar_, Nov 23 2024
%e A188593 1.902113032590307144232878666758764286811397268251...
%t A188593 r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
%t A188593 RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* _Harvey P. Dale_, Oct 27 2023 *)
%o A188593 (PARI) sqrt((5+sqrt(5))/2)
%o A188593 (Magma) SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // _G. C. Greubel_, Nov 02 2018
%Y A188593 Cf. A001622 (decimal expansion of the golden ratio), A019881.
%Y A188593 Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
%Y A188593 Cf. A090771, A195693, A195723, A296184.
%K A188593 nonn,cons,easy
%O A188593 1,2
%A A188593 _Clark Kimberling_, Apr 04 2011