A124118 - OEIS

%I #32 Jun 22 2020 01:51:44

%S 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,

%T 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,

%U 7,7,4,7,0,0,8,3,4,2,8,5,5,1,9,5,9,0,9

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

%D 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.

%H D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, <a href="http://repositories.cdlib.org/lbnl/LBNL-53654/">On the binary expansions of algebraic numbers</a>, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518. LBNL-53854.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/seq2real/a00/A004018.txt">Constants derived from sums of A004018</a> [broken link?].

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

%F 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

%F Equals 1 + 4 * Sum_{k>=0} (-1)^k/(2^(2*k+1) - 1). - _Amiram Eldar_, Jun 22 2020

%e 4.532372014258974100827957178...

%t 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 *)

%t RealDigits[1 + 4*Sum[(-1)^n/(2^(2*n + 1) - 1), {n, 0, 200}], 10, 100][[1]] (* _Amiram Eldar_, Jun 22 2020 *)

%Y Cf. A004018.

%K cons,nonn

%O 1,1

%A _R. J. Mathar_, Nov 25 2006