A391707 Decimal expansion of the sum of the reciprocals of the odd-indexed Pell numbers.

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

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.

TuAnh Gia Nguyen, Fibonacci and Lucas series with elliptic functions, Master Thesis, San Jose State University, 2005, p. 90, eq. (3.27).

Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Wikipedia, Theta function.

Formula

Equals Sum_{k>=1} 1/A000129(2*k-1) = Sum_{k>=1} 1/A001653(k).

Equals A391708 - A391706 = A391706 + A391709.

Equals theta_2((sqrt(2)-1)^2)^2/sqrt(2), where theta_2 is a Jacobi theta function.

Example

1.24162540752838435695650946834092143786765854952423...

Mathematica

RealDigits[EllipticTheta[2, (Sqrt[2] - 1)^2]^2/Sqrt[2], 10, 120][[1]]

Prog

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

Crossrefs

Cf. A000129, A001653, A153387, A153416, A391706, A391708, A391709.

Sequence in context: A171233 A362004 A096907 * A360555 A249146 A225679

Adjacent sequences: A391704 A391705 A391706 * A391708 A391709 A391710

Keyword

nonn,cons

Author

Amiram Eldar, Dec 18 2025

Status

approved