A321873 Decimal expansion of the sum of reciprocals of repunit numbers base 4, Sum_{k>=1} 3/(4^k - 1).

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

Offset: 1

Author

A.H.M. Smeets, Nov 20 2018

Keywords

• nonn
• cons

Comments

The sums of reciprocal repunit numbers are related to the Lambert series. A special case is the sum of repunit numbers in base 2, which is known as the Erdős-Borwein constant (A065442).

Examples

			1.263293058100271331887972663903139146884324008972346213817762390...
		

Links

• N. Kurakowa and M. Wakyama, On q-analogues of the Euler Constant and Lerch's limit formula, Proc. AMS 132 (4) (2003) 935, constant gamma(4).
• Eric Weisstein's World of Mathematics, Erdős-Borwein Constant
• Eric Weisstein's World of Mathematics, Lambert Series

Crossrefs

Cf. A002450, A065442 (base 2), A321872 (base 3).

Programs

Maple

evalf[130](sum(3/(4^k-1),k=1..infinity)); # Muniru A Asiru, Dec 20 2018

Mathematica

RealDigits[Sum[3/(4^k-1), {k, 1, Infinity}], 10, 120][[1]] (* Amiram Eldar, Nov 21 2018 *)

PARI

suminf(k=1, 3/(4^k-1)) \\ Michel Marcus, Nov 20 2018

Formula

Equals 3*L(1/4) = 3 * A248721, where L is the Lambert series.

Equals 3 * Sum_{k>=1} x^(k^2)*(1+x^k)/(1-x^k) where x = 1/4.