A391708 - OEIS

A391708

Decimal expansion of the sum of the reciprocals of the positive Pell numbers.

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, section 3.7, pp. 91-101.

LINKS

Table of n, a(n) for n=1..105.

A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.

R. S. Melham, Lambert Series and Elliptic Functions and Certain Reciprocal Sums, The Fibonacci Quarterly, Vol. 37, No. 3 (1999), pp. 208-212.

FORMULA

Equals Sum_{k>=1} 1/A000129(k).

Equals A391706 + A391707.

Equals 2 * A391706 + A391709 = 2 * A391707 - A391709.

EXAMPLE

1.84220304982752858079237158327980838900527021185437...

MATHEMATICA

L[q_] := (Log[1 - q] + QPolyGamma[1, q])/Log[q]; RealDigits[EllipticTheta[2, (Sqrt[2] - 1)^2]^2/Sqrt[2] + 2*Sqrt[2] *(L[3 - 2*Sqrt[2]] - L[17 - 12*Sqrt[2]]), 10, 120][[1]]

PROG

(PARI) L(x) = suminf(k=1, x^k/(1-x^k));

theta_2(q) = {my(x = -I*log(q)/Pi); 2*eta(2*x, 1)^2/eta(x, 1); }

2*sqrt(2) * (L(3-2*sqrt(2)) - L(17-12*sqrt(2))) + theta_2((sqrt(2)-1)^2)^2/sqrt(2)

CROSSREFS

Cf. A000129, A079586, A093540, A391706, A391707, A391709.

Sequence in context: A080828 A131916 A131606 * A033328 A097529 A114321

Adjacent sequences: A391705 A391706 A391707 * A391709 A391710 A391711

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Dec 18 2025

STATUS

approved