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

1,5

Comments

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

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]

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

Example

1.10091819083620073637985510165438004320345439787328165635989...

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 *)

Prog

(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

Crossrefs

Equals 9*A073668.

Cf. A067617 (continued fraction).

Sequence in context: A339605 A371856 A182494 * A010169 A113657 A098783

Adjacent sequences: A065441 A065442 A065443 * A065445 A065446 A065447

Keyword

nonn,cons

Author

N. J. A. Sloane, Nov 18 2001

Extensions

More terms from John W. Layman, Nov 19 2001

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

Status

approved