cp's OEIS Frontend

A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

%I A134972 #78 Feb 03 2025 12:46:49
%S A134972 1,2,3,6,0,6,7,9,7,7,4,9,9,7,8,9,6,9,6,4,0,9,1,7,3,6,6,8,7,3,1,2,7,6,
%T A134972 2,3,5,4,4,0,6,1,8,3,5,9,6,1,1,5,2,5,7,2,4,2,7,0,8,9,7,2,4,5,4,1,0,5,
%U A134972 2,0,9,2,5,6,3,7,8,0,4,8,9,9,4,1,4,4,1,4,4,0,8,3,7,8,7,8,2,2,7,4,9,6,9,5
%N A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).
%C A134972 Convergents are 4/2, 8/8, 32/24, 96/80, 320/256, 1024/832, 3328/2688, 10752/8704, 34816/28160, 112640/91136, 364544/294912, 1179648/954368, 3817472/3088384, 12353536/9994240, ... = A209084/A063727. - _Seiichi Kirikami_, Mar 14 2012
%C A134972 2*(-1 + phi) is an integer in the quadratic number field Q(sqrt(5)). - _Wolfdieter Lang_, Feb 16 2016
%H A134972 Michael Penn, <a href="https://www.youtube.com/watch?v=OFgF6_HCuuU">How large is the blue ▲</a>, YouTube video, 2021.
%H A134972 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A134972 Equals A134945 - 2 = A002163 - 1 = A098317 - 3. - _R. J. Mathar_, Oct 27 2008
%F A134972 2*(-1 + A001622). - _Wolfdieter Lang_, Feb 17 2016
%F A134972 Equals the harmonic mean of 1 and phi, 2*phi/(1+phi). - _Stanislav Sykora_, Apr 11 2016
%F A134972 From _Christian Katzmann_, Mar 19 2018: (Start)
%F A134972 Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(n!^2*3^(2*n+2)).
%F A134972 Equals -1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
%F A134972 Equals 1/A019863. - _R. J. Mathar_, Jan 17 2021
%F A134972 Equals 2*sin(Pi/5)/sin(2*Pi/5) = hypergeom([1/5, 3/5], [7/5], 1) = hypergeom([-1/5, -3/5], [3/5], 1). - _Peter Bala_, Mar 04 2022
%e A134972 1.236067977499789696...
%t A134972 RealDigits[ N[4/(1+Sqrt[5]), 150] ] [ [1] ] (* _Seiichi Kirikami_, Mar 14 2012 *)
%o A134972 (PARI) 4/(1+sqrt(5)) \\ _Altug Alkan_, Apr 11 2016
%Y A134972 Cf. A001622, A019863, A063727, A209084, A033887.
%K A134972 nonn,easy,cons
%O A134972 1,2
%A A134972 _Omar E. Pol_, Nov 15 2007