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A065445 Decimal expansion of Product{k=0..inf} (1+1/2^(2k))^(1/2).

%I A065445 #28 Apr 18 2019 17:42:16
%S A065445 1,6,4,6,7,6,0,2,5,8,1,2,1,0,6,5,6,4,8,3,6,6,0,5,1,2,2,2,2,8,2,2,9,8,
%T A065445 4,3,5,6,5,2,3,7,6,7,2,5,7,0,1,0,2,7,4,0,9,0,1,2,4,0,5,3,1,7,5,5,1,7,
%U A065445 2,8,1,6,2,4,3,9,1,4,1,3,7,2,1,6,1,8,8,6,9,3,9,9,9,0,7,6,5,6,4,3,5,6,6,7,9
%N A065445 Decimal expansion of Product{k=0..inf} (1+1/2^(2k))^(1/2).
%C A065445 Scaling constant with CORDIC algorithm, see p.647 of fxtbook (see link below).
%D A065445 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
%H A065445 Harry J. Smith, <a href="/A065445/b065445.txt">Table of n, a(n) for n=1..2000</a>
%H A065445 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 33.2.
%H A065445 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/dig/dig.html">Digital Search Tree Constants</a> [Broken link]
%H A065445 Steven R. Finch, <a href="http://web.archive.org/web/20010413214937/http://www.mathsoft.com:80/asolve/constant/dig/dig.html">Digital Search Tree Constants</a> [From the Wayback machine]
%e A065445 1.646760258121065648366051222282298435652376725701027409...
%p A065445 evalf(product((1+1/2^(2k))^(1/2),k=0..infinity), 120) # _Vaclav Kotesovec_, Sep 20 2014
%t A065445 N[ Product[ Sqrt[ (1 + 1/2^(2k) ) ], {k, 0, Infinity} ], 500 ]
%o A065445 (PARI) { default(realprecision, 2080); x=prodinf(k=0, sqrt(1 + 1/2^(2*k))); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065445.txt", n, " ", d)) } \\ _Harry J. Smith_, Oct 04 2009
%o A065445 (PARI)
%o A065445 pent(z, n)= 1+sum(k=1,n, (-1)^k*(z^(k*(3*k-1)/2) + z^(k*(3*k+1)/2)));
%o A065445 /* == prod(n>=1, 1-z^n) via pentagonal number theorem */
%o A065445 N=30; u=0.25; K=sqrt( 2 * pent(u^2,N)/pent(u,N) )
%o A065445 /* using prod(n>=1, 1+z^2) = prod(n>=1, 1-(z^2)^2)/prod(n>=1, 1-z^n) */
%o A065445 /* gives: 1.6467602581210... */ /* _Joerg Arndt_, Jan 17 2011 */
%Y A065445 Cf. A065045.
%K A065445 nonn,cons
%O A065445 1,2
%A A065445 _N. J. A. Sloane_, Nov 18 2001
%E A065445 More terms from _Robert G. Wilson v_, Nov 19 2001
%E A065445 Terms corrected and terms added by _Harry J. Smith_, Oct 04 2009