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A360928
Decimal expansion of Sum_{i>=0} 1/(phi^(4*i+2) - 1) where phi = (1+sqrt(5))/2 is the golden ratio.
2
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
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
KEYWORD
cons,nonn
AUTHOR
Kevin Ryde, Feb 25 2023
STATUS
approved