A391706 Decimal expansion of the sum of the reciprocals of the positive even-indexed Pell numbers.

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

Offset

0,1

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=0..104.

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.

TuAnh Gia Nguyen, Fibonacci and Lucas series with elliptic functions, Master Thesis, San Jose State University, 2005, p. 96, eq. (3.41).

Formula

Equals Sum_{k>=1} 1/A000129(2*k) = Sum_{k>=1} 1/A001542(k).

Equals A391708 - A391707 = A391707 - A391709.

Equals 2*sqrt(2) * (L(3-2*sqrt(2)) - L(17-12*sqrt(2))), where L(q) = Sum_{k>=1} q^k/(1-q^k) = (log(1-q) + psi_q(1))/log(q), and psi_q(z) is the q-digamma function.

Example

0.60057764229914422383586211493888695113761166233013...

Mathematica

L[q_] := (Log[1 - q] + QPolyGamma[1, q])/Log[q]; RealDigits[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));
2*sqrt(2) * (L(3-2*sqrt(2)) - L(17-12*sqrt(2)))

Crossrefs

Cf. A000129, A001542, A153386, A153415, A391707, A391708, A391709.

Sequence in context: A114760 A261508 A397231 * A344182 A217219 A113555

Adjacent sequences: A391703 A391704 A391705 * A391707 A391708 A391709

Keyword

nonn,cons

Author

Amiram Eldar, Dec 18 2025

Status

approved