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A048651
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Decimal expansion of Product_{k >= 1} (1-1/2^k).
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38
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2, 8, 8, 7, 8, 8, 0, 9, 5, 0, 8, 6, 6, 0, 2, 4, 2, 1, 2, 7, 8, 8, 9, 9, 7, 2, 1, 9, 2, 9, 2, 3, 0, 7, 8, 0, 0, 8, 8, 9, 1, 1, 9, 0, 4, 8, 4, 0, 6, 8, 5, 7, 8, 4, 1, 1, 4, 7, 4, 1, 0, 6, 6, 1, 8, 4, 9, 0, 2, 2, 4, 0, 9, 0, 6, 8, 4, 7, 0, 1, 2, 5, 7, 0, 2, 4, 2, 8, 4, 3, 1, 9, 3, 3, 4, 8, 0, 7, 8, 2
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OFFSET
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0,1
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COMMENTS
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This is the probability that a large random binary matrix is nonsingular (cf. A002884).
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,20000
S. R. Finch, Digital Search Tree Constants
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, J. Integer Seq., Vol 9 (2006), Article 06.2.1, PDF version.
Eric Weisstein's World of Mathematics, Tree Searching
Eric Weisstein's World of Mathematics, Infinite Product
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FORMULA
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exp(-sum{k>0, sigma_1(k)/k*2^(-k)})=exp(-sum{k>0, A000203(k)/k*2^(-k)}) - Hieronymus Fischer, Jul 28 2007
lim inf product{0<=k<=floor(log_2(n)), floor(n/2^k)*2^k/n} for n-->oo. - Hieronymus Fischer, Aug 13 2007
lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n= ))) for n-->oo. - Hieronymus Fischer, Aug 13 2007
lim inf A098844(n)/n^(1+floor(log_2(n)))*2^A000217(floor(log_2(n)) for n-->oo. - Hieronymus Fischer, Aug 13 2007
lim inf A098844(n)/(n+1)^((1+log_2(n+1))/2) for n-->oo. - Hieronymus Fischer, Aug 13 2007
(1/2)*exp(-sum{n>0, 2^(-n)*sum{k|n, 1/(k*2^k))}}). - Hieronymus Fischer, Aug 13 2007
A048651=limit of A177510(n)/A000079(n-1) as n-->infinity (conjecture). - Mats Granvik, Mar 27 2011.
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EXAMPLE
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(1/2) (3/4) (7/8) (15/16) ... = 0.288788095086602421278899721929230780088911904840685784114741...
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MATHEMATICA
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RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
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PROG
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(PARI) { default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); } [From Harry J. Smith, May 07 2009]
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CROSSREFS
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Cf. A002884, A005329, A048652, A098844, A067080, A100220, A132019, A132020, A132026, A132038.
Sequence in context: A020769 A105388 A178247 * A138300 A137575 A183393
Adjacent sequences: A048648 A048649 A048650 * A048652 A048653 A048654
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KEYWORD
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nonn,cons
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrected by Hieronymus Fischer, Jul 28 2007
Fixed my PARI program, had -n. Deleted old PARI program Harry J. Smith, May 19 2009
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STATUS
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approved
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