A360928 Decimal expansion of Sum_{i>=0} 1/(phi^(4*i+2) - 1) where phi = (1+sqrt(5))/2 is the golden ratio.

6, 8, 6, 6, 3, 8, 5, 6, 5, 6, 8, 1, 2, 6, 0, 6, 3, 9, 3, 9, 6, 5, 5, 6, 7, 6, 5, 6, 7, 0, 5, 6, 5, 9, 6, 1, 0, 1, 8, 6, 9, 0, 3, 1, 2, 3, 8, 2, 1, 8, 1, 6, 1, 6, 4, 9, 8, 1, 2, 5, 0, 3, 3, 1, 2, 9, 4, 3, 5, 1, 0, 5, 3, 3, 3, 5, 5, 3, 2, 5, 3, 8, 2, 1, 4, 9, 1, 8, 7, 5, 5, 2, 8, 4, 8, 8, 4, 8, 0, 9, 1, 5, 7, 1, 9

Offset

0,1

Links

Kevin Ryde, Table of n, a(n) for n = 0..10000

W. E. Greig, Sums of Fibonacci Reciprocals, The Fibonacci Quarterly, Vol. 15, No. 1, February 1977, pp. 46-48 (see equation (15)).

Formula

Equals Sum_{j>=1} phi^(2*j)/(phi^(4*j) - 1) [Greig, equation (15)].

Equals A153386 / sqrt(5) [Greig, equations (13) and (14)].

Example

0.68663856568126063939655676567056596...

Mathematica

RealDigits[Sum[1/(GoldenRatio^(4*i + 2) - 1), {i, 0, Infinity}], 10, 105][[1]] (* Amiram Eldar, Feb 26 2023 *)
RealDigits[(Log[(5 + 3*Sqrt[5])/2] + QPolyGamma[0, 1/2, (7 + 3*Sqrt[5])/2]) / Log[(7 - 3*Sqrt[5])/2], 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)

Prog

(PARI) sumpos(i=0, 1/(((1+sqrt(5))/2)^(4*i+2) - 1)) \\ Michel Marcus, Feb 26 2023

Crossrefs

Cf. A001622 (phi), A153386.

Sequence in context: A331941 A169685 A339606 * A200326 A088751 A153627

Adjacent sequences: A360925 A360926 A360927 * A360929 A360930 A360931

Keyword

cons,nonn

Author

Kevin Ryde, Feb 25 2023

Status

approved