

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
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OFFSET

0,1


COMMENTS

"In his Book 'The Theory of Poker,' David Sklansky coined the phrase 'Fundamental Theorem of Poker,' a tongueincheek 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.  JeanFrançois Alcover, Sep 02 2014, after Steven Finch
This constant is also the largest argument of the arctangent function in the Viètelike 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 Viètelike formula for Pi can be written as Pi/8 = arctan(sqrt(2  sqrt(2))/sqrt(2 + sqrt(2))) = arctan(sqrt(2)  1), (after AbrarovQuine, 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: NoLimit 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 Viètelike 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



