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

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Nov 18 2001

Keywords

• nonn
• cons

Comments

Scaling constant with CORDIC algorithm, see p.647 of fxtbook (see link below).

Examples

			1.646760258121065648366051222282298435652376725701027409...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Links

• Harry J. Smith, Table of n, a(n) for n=1..2000
• Joerg Arndt, Matters Computational (The Fxtbook), section 33.2.
• Steven R. Finch, Digital Search Tree Constants [Broken link]
• Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]

Crossrefs

Cf. A065045.

Programs

Maple

evalf(product((1+1/2^(2k))^(1/2),k=0..infinity), 120) # Vaclav Kotesovec, Sep 20 2014

Mathematica

N[ Product[ Sqrt[ (1 + 1/2^(2k) ) ], {k, 0, Infinity} ], 500 ]

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

PARI

pent(z, n)= 1+sum(k=1,n, (-1)^k*(z^(k*(3*k-1)/2) + z^(k*(3*k+1)/2)));
/* == prod(n>=1, 1-z^n) via pentagonal number theorem */
N=30; u=0.25; K=sqrt( 2 * pent(u^2,N)/pent(u,N) )
/* using prod(n>=1, 1+z^2) = prod(n>=1, 1-(z^2)^2)/prod(n>=1, 1-z^n) */
/* gives: 1.6467602581210... */ /* Joerg Arndt, Jan 17 2011 */

Extensions

More terms from Robert G. Wilson v, Nov 19 2001

Terms corrected and terms added by Harry J. Smith, Oct 04 2009