A124118 Decimal expansion of Sum_{i>=0} A004018(i)/2^i.

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

Offset: 1

Author

R. J. Mathar, Nov 25 2006

Keywords

• cons
• nonn

Examples

			4.532372014258974100827957178...
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 6 ed., 2008, section 17.10, p. 340.

Links

• D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518. LBNL-53854.
• Simon Plouffe, Constants derived from sums of A004018 [broken link?].

Crossrefs

Cf. A004018.

Programs

Mathematica

Clear[s]; s[n_] := s[n] = RealDigits[ Sum[ SquaresR[2, k]/2^k, {k, 0, n}], 10, 29] // First; s[n=100]; While[s[n] != s[n-100], n = n+100]; s[n] (* Jean-François Alcover, Feb 13 2013 *)
RealDigits[1 + 4*Sum[(-1)^n/(2^(2*n + 1) - 1), {n, 0, 200}], 10, 100][[1]] (* Amiram Eldar, Jun 22 2020 *)

Formula

Sum_{i>=0} A004018(i)/2^i.

Bailey et al. point out the approximation Pi*(1+2*exp(-Pi^2/log(2))^2)/log(2), correct up to 23 decimal places. - Jean-François Alcover, Jun 27 2015

Equals 1 + 4 * Sum_{k>=0} (-1)^k/(2^(2*k+1) - 1). - Amiram Eldar, Jun 22 2020