A134973 Decimal expansion of 3/phi = 6/(1 + sqrt(5)).
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, 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, 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
Offset: 1
Examples
1.8541019662496845446137605030969143531609275394172885864063458681157...
Links
- R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
- Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, Vol. 1385, pp. 97-100.
- Index entries for algebraic numbers, degree 2.
Programs
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Mathematica
RealDigits[3/GoldenRatio,10,120][[1]] (* Harvey P. Dale, Apr 01 2018 *)
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PARI
(sqrt(5)-1)*3/2 \\ Charles R Greathouse IV, Jun 26 2011
Formula
Equals A090550 - 4. - R. J. Mathar, Oct 27 2008
Equals Product_{n>=1} (1 + 1/A192222(n)). - Charles R Greathouse IV, Jun 26 2011
Equals 1 + Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1)*5^k). - Amiram Eldar, Jun 06 2021
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