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A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k). 72
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the probability that a large random binary matrix is nonsingular (cf. A002884).

REFERENCES

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

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

S. R. Finch, Digital Search Tree Constants

Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Eric Weisstein's World of Mathematics, Tree Searching

Eric Weisstein's World of Mathematics, Infinite Product

FORMULA

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_{k=0..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

Product_{k >= 1} (1-1/2^k) = (1/2; 1/2)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015

exp(Sum_{n>=1}(1/n/(1 - 2^n))) (According to Mathematica). - Mats Granvik, Sep 07 2016

EXAMPLE

(1/2) (3/4) (7/8) (15/16) ... = 0.288788095086602421278899721929230780088911904840685784114741...

MATHEMATICA

RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)

RealDigits[QPochhammer[1/2], 10, 100][[1]] (* Jean-Fran├žois Alcover, Nov 18 2015 *)

PROG

(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)); } \\ Harry J. Smith, May 07 2009

CROSSREFS

Cf. A002884, A005329, A048652, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584.

Sequence in context: A020769 A105388 A178247 * A243596 A256849 A138300

Adjacent sequences:  A048648 A048649 A048650 * A048652 A048653 A048654

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected by Hieronymus Fischer, Jul 28 2007

STATUS

approved

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Last modified December 5 19:19 EST 2016. Contains 278770 sequences.