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

1,2

Links

Table of n, a(n) for n=1..94.

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.

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

Example

1.8541019662496845446137605030969143531609275394172885864063458681157...

Mathematica

RealDigits[3/GoldenRatio, 10, 120][[1]] (* Harvey P. Dale, Apr 01 2018 *)

Prog

(PARI) (sqrt(5)-1)*3/2 \\ Charles R Greathouse IV, Jun 26 2011

Crossrefs

Cf. A000032, A001622 (golden ratio), A090550 (5/x = x-5), A192222 (Fibonacci(2^n + 1)).

Sequence in context: A019609 A334502 A093341 * A030437 A200290 A273959

Adjacent sequences: A134970 A134971 A134972 * A134974 A134975 A134976

Keyword

cons,nonn,easy

Author

Omar E. Pol, Nov 15 2007

Status

approved