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A188582 Decimal expansion of sqrt(2) - 1. 3
4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8, 8, 0, 1, 6, 8, 8, 7, 2, 4, 2, 0, 9, 6, 9, 8, 0, 7, 8, 5, 6, 9, 6, 7, 1, 8, 7, 5, 3, 7, 6, 9, 4, 8, 0, 7, 3, 1, 7, 6, 6, 7, 9, 7, 3, 7, 9, 9, 0, 7, 3, 2, 4, 7, 8, 4, 6, 2, 1, 0, 7, 0, 3, 8, 8, 5, 0, 3, 8, 7, 5, 3, 4, 3, 2, 7, 6, 4, 1, 5, 7, 2, 7, 3, 5, 0, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

"In his Book 'The Theory of Poker,' David Sklansky coined the phrase 'Fundamental Theorem of Poker,' a tongue-in-cheek reference to the Fundamental Theorem of Algebra and Fundamental Theorem of Calculus from introductory texts on those two subjects. The constant [sqrt(2) - 1] appears so often in poker analysis that we will in the same vein go so far as to call it 'the golden mean of poker,' and we call it 'r' for short. We will see this value in a number of important results throughout this book." [Chen and Ankenman]

If a triangle has sides whose lengths form an harmonic progression in the ratio 1/(1 - d):1:1/(1 + d) then the triangle inequality condition requires that d be in the range 1 - sqrt(2) < d < sqrt(2) - 1. - Frank M Jackson, Oct 01 2013

This constant is the 6th smallest radius r<1 for which a compact packing of the plane exists, with disks of radius 1 and r. - Jean-François Alcover, Sep 02 2014, after Steven Finch

This constant is also the largest argument of the arctangent function in the Viète-like formula for Pi given by Pi/2^{k + 1} = arctan(sqrt(2 - a_{k - 1})/a_k), where the index k >= 2 and the nested radicals are defined by recurrence using the relations a_k = sqrt(2 + a_{k - 1}), a_1 = sqrt(2). When k = 2 the argument of the arctangent function sqrt(2 - a_1)/a_2 = sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2)) = sqrt(2) - 1 is largest. Consequently, at k = 2 the Viète-like formula for Pi can be written as Pi/8 = arctan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) = arctan(sqrt(2) - 1), (after Abrarov-Quine, see the article). - Sanjar Abrarov, Jan 07 2017

REFERENCES

Bill Chen and Jerrod Ankenman, The Mathematics of Poker, Chpt 14 - You Don't Have To Guess: No-Limit Bet Sizing, p. 153, ConJelCo, LLC, Pittsburgh PA 2006.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 396 and 486.

LINKS

Table of n, a(n) for n=0..103.

S. M. Abrarov and B. M. Quine, A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals, figshare, 4509014, (2017).

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.

FORMULA

sqrt(2)-1 = exp(asinh(cos(Pi))) = exp(asinh(-1)). - Geoffrey Caveney, Apr 23 2014

EXAMPLE

= 0.414213562373095048801688724209698078569671875376948073...

MATHEMATICA

RealDigits[ Sqrt[2] - 1, 10, 111][[1]]

CROSSREFS

Cf. A002193, A014176, A020807, A120731.

Sequence in context: A156896 A002193 A020807 * A230077 A279605 A055190

Adjacent sequences:  A188579 A188580 A188581 * A188583 A188584 A188585

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Apr 04 2011

STATUS

approved

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Last modified June 23 09:21 EDT 2017. Contains 288660 sequences.