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A134973 Decimal expansion of 3/phi = 6/(1 + sqrt(5)).

%I A134973 #47 Feb 11 2025 14:55:37
%S A134973 1,8,5,4,1,0,1,9,6,6,2,4,9,6,8,4,5,4,4,6,1,3,7,6,0,5,0,3,0,9,6,9,1,4,
%T A134973 3,5,3,1,6,0,9,2,7,5,3,9,4,1,7,2,8,8,5,8,6,4,0,6,3,4,5,8,6,8,1,1,5,7,
%U A134973 8,1,3,8,8,4,5,6,7,0,7,3,4,9,1,2,1,6,2,1,6,1,2,5,6,8
%N A134973 Decimal expansion of 3/phi = 6/(1 + sqrt(5)).
%H A134973 R. S. Melham and A. G. Shannon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-1/melham2.pdf">Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers</a>, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
%H A134973 Jonathan Sondow, <a href="http://arxiv.org/abs/1106.4246">Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers</a>, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, Vol. 1385, pp. 97-100.
%H A134973 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A134973 Equals A090550 - 4. - _R. J. Mathar_, Oct 27 2008
%F A134973 Equals Product_{n>=1} (1 + 1/A192222(n)). - _Charles R Greathouse IV_, Jun 26 2011
%F A134973 Equals 1 + Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1)*5^k). - _Amiram Eldar_, Jun 06 2021
%F A134973 Equals Product_{k>=1} (Lucas(3*k)^2 + 5*(-1)^(k+1))/(Lucas(3*k)^2 + 5*(-1)^k) (Melham and Shannon, 1995). - _Amiram Eldar_, Jan 15 2022
%e A134973 1.8541019662496845446137605030969143531609275394172885864063458681157...
%t A134973 RealDigits[3/GoldenRatio,10,120][[1]] (* _Harvey P. Dale_, Apr 01 2018 *)
%o A134973 (PARI) (sqrt(5)-1)*3/2 \\ _Charles R Greathouse IV_, Jun 26 2011
%Y A134973 Cf. A000032, A001622 (golden ratio), A090550 (5/x = x-5), A192222 (Fibonacci(2^n + 1)).
%K A134973 cons,nonn,easy
%O A134973 1,2
%A A134973 _Omar E. Pol_, Nov 15 2007