A065444 Decimal expansion of 9*Sum_{k>=1} 1/(10^k-1).

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

Offset: 1

Author

N. J. A. Sloane, Nov 18 2001

Keywords

• nonn
• cons

Comments

This constant is the infinite sum of the reciprocals of repunits, R_n, with n>0 (A002275). - Enrique Pérez Herrero, Dec 06 2009

Examples

			1.10091819083620073637985510165438004320345439787328165635989...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Links

• Harry J. Smith, Table of n, a(n) for n=1..2000
• Steven R. Finch, Digital Search Tree Constants [Broken link]
• Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Crossrefs

Equals 9*A073668.

Cf. A000042, A002275, A067617 (continued fraction).

Programs

Mathematica

RealDigits[9*N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]
A065444=RealDigits[ Block[{$MaxExtraPrecision = 100}, N[9*Sum[(-1 + 10^i)^-1, {i, 1, Infinity}], 130]]][[1]] (* Enrique Pérez Herrero, Dec 06 2009 *)
First[RealDigits[9 (Log10[10/9] - QPolyGamma[0, 1, 1/10]/Log[10]), 10, 120]] (* Jan Mangaldan, Apr 25 2016 *)

PARI

{ default(realprecision, 2080); x=9*suminf(k=1, 1/(10^k - 1)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065444.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Formula

Equals 9 * Sum_{k>=1} (1+x^k)/(1-x^k) * x^(k^2) where x = 1/10. This allows fast computation for this and similar sequences (involving Sum_{k>=1} x^k/(1-x^k) for some x < 1 ). - Joerg Arndt, Apr 25 2016

Extensions

More terms from John W. Layman, Nov 19 2001

...733 (50th digit) expanded to ...7328165 etc. by Frank Ellermann, Feb 23 2002