A360928 - OEIS

%I #19 Mar 01 2023 05:36:36

%S 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,

%T 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,

%U 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

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

%H Kevin Ryde, <a href="/A360928/b360928.txt">Table of n, a(n) for n = 0..10000</a>

%H W. E. Greig, <a href="https://www.fq.math.ca/Scanned/15-1/greig2.pdf">Sums of Fibonacci Reciprocals</a>, The Fibonacci Quarterly, Vol. 15, No. 1, February 1977, pp. 46-48 (see equation (15)).

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

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

%e 0.68663856568126063939655676567056596...

%t RealDigits[Sum[1/(GoldenRatio^(4*i + 2) - 1), {i, 0, Infinity}], 10, 105][[1]] (* _Amiram Eldar_, Feb 26 2023 *)

%t 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 *)

%o (PARI) sumpos(i=0, 1/(((1+sqrt(5))/2)^(4*i+2) - 1)) \\ _Michel Marcus_, Feb 26 2023

%Y Cf. A001622 (phi), A153386.

%K cons,nonn

%O 0,1

%A _Kevin Ryde_, Feb 25 2023