login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A188582 Decimal expansion of sqrt(2) - 1. 7
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 a 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

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

Equals tan(Pi/8) = A182168 / A144981. - Bernard Schott, Apr 12 2022

EXAMPLE

= 0.414213562373095048801688724209698078569671875376948073...

MATHEMATICA

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

PROG

(PARI) sqrt(2) - 1 \\ G. C. Greubel, Jan 31 2018

(MAGMA) Sqrt(2) - 1; // G. C. Greubel, Jan 31 2018

CROSSREFS

Cf. A002193, A014176, A020807, A120731, A182168 (sin(Pi/8)), A144981 (cos(Pi/8)).

Sequence in context: A156896 A002193 A020807 * A230077 A055190 A155781

Adjacent sequences:  A188579 A188580 A188581 * A188583 A188584 A188585

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Apr 04 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 18 22:37 EDT 2022. Contains 353826 sequences. (Running on oeis4.)