A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

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

Offset: 1

Author

N. J. A. Sloane, Nov 18 2001

Keywords

• nonn
• cons

Examples

			1.1373387363441965966969133683013475838308493098...
		

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]
• Eric Weisstein's World of Mathematics, Tree Searching.

Crossrefs

Cf. A060867, A065442, A065608, A066766, A067616.

Programs

Mathematica

RealDigits[NSum[1/(2^k - 1)^2, {k, 1, Infinity}, PrecisionGoal -> 40, AccuracyGoal -> 40, WorkingPrecision -> 500, NSumTerms -> 50, NSumExtraTerms -> 50]][[1]] (* Peter Bertok (peter(AT)bertok.com), Dec 04 2001 *)
RealDigits[(Log[2] QPolyGamma[0, 1, 1/2] + QPolyGamma[1, 1, 1/2])/Log[2]^2 - 1, 10, 20][[1]] (* Eric W. Weisstein, Jun 02 2025 *)

PARI

{ default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)^2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065443.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Formula

Equals Sum_{n>=1} 1/A060867(n).

From Amiram Eldar, Oct 16 2022: (Start)

Equals Sum_{k>=1} k/(2^(k+1)-1).

Equals A066766 - A065442. (End)

Equals Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 evaluated at q = 1/2 (see A065608). - Peter Bala, Oct 16 2022

Extensions

More terms from Peter Bertok (peter(AT)bertok.com), Dec 04 2001