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

Author

Kevin Ryde, Feb 25 2023

Keywords

• cons
• nonn

Examples

			0.68663856568126063939655676567056596...
		

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)).

Crossrefs

Cf. A001622 (phi), A153386.

Programs

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 *)

PARI

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

Formula

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

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