Search: a346435 -id:a346435
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A068985
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Decimal expansion of 1/e.
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+10
99
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3, 6, 7, 8, 7, 9, 4, 4, 1, 1, 7, 1, 4, 4, 2, 3, 2, 1, 5, 9, 5, 5, 2, 3, 7, 7, 0, 1, 6, 1, 4, 6, 0, 8, 6, 7, 4, 4, 5, 8, 1, 1, 1, 3, 1, 0, 3, 1, 7, 6, 7, 8, 3, 4, 5, 0, 7, 8, 3, 6, 8, 0, 1, 6, 9, 7, 4, 6, 1, 4, 9, 5, 7, 4, 4, 8, 9, 9, 8, 0, 3, 3, 5, 7, 1, 4, 7, 2, 7, 4, 3, 4, 5, 9, 1, 9, 6, 4, 3, 7, 4, 6, 6, 2, 7
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0,1
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COMMENTS
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From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets their own hat.
Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is a tree. See linked file. - Washington Bomfim, Nov 01 2010
Also, -1/e is the global minimum of x*log(x) at x = 1/e and the global minimum of x*e^x at x = -1. - Rick L. Shepherd, Jan 11 2014
Also, the asymptotic probability of success in the secretary problem (also known as the sultan's dowry problem). - Andrey Zabolotskiy, Sep 14 2019
The asymptotic density of numbers with an odd number of trailing zeros in their factorial base representation (A232745). - Amiram Eldar, Feb 26 2021
For large range size s where numbers are chosen randomly r times, the probability when r = s that a number is randomly chosen exactly 1 time. Also the chance that a number was not chosen at all. The general case for the probability of being chosen n times is (r/s)^n / (n! * e^(r/s)). - Mark Andreas, Oct 25 2022
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REFERENCES
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Anders Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
John Harris, Jeffry L. Hirst, and Michael Mossinghoff, Combinatorics and Graph Theory, Springer Science & Business Media, 2009, p. 161.
L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (103) on page 20.
Traian Lalescu, Problem 579, Gazeta Matematică, Vol. 6 (1900-1901), p. 148.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
Manfred R. Schroeder, Number Theory in Science and Communication, Springer Science & Business Media, 2008, ch. 9.5 Derangements.
Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.
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LINKS
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James Grime and Brady Haran, Derangements, Numberphile video, 2017.
Eric Weisstein's World of Mathematics, e.
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FORMULA
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Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]
Equals 1 - Sum_{i >= 1} (-1)^(i - 1)/i!. [Michon]
Equals j_1(i)/i = cos(i) + i*sin(i), where j_1(z) is the spherical Bessel function of the first kind and i = sqrt(-1). - Stanislav Sykora, Jan 11 2017
The series representation 1/e = Sum_{k >= 0} (-1)^k/k! is the case n = 0 of the following series acceleration formulas:
1/e = n!*Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*binomial(-x,k) are the row polynomials of A094816. (End)
1/e = 1 - Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). - Peter Bala, Nov 13 2019
Equals Integral_{x=0..1} x * sinh(x) dx. - Amiram Eldar, Aug 14 2020
Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - Amiram Eldar, Mar 29 2022
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EXAMPLE
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1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.
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MATHEMATICA
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PROG
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(PARI)
default(realprecision, 110);
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CROSSREFS
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Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A049470
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Decimal expansion of cos(1).
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+10
59
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5, 4, 0, 3, 0, 2, 3, 0, 5, 8, 6, 8, 1, 3, 9, 7, 1, 7, 4, 0, 0, 9, 3, 6, 6, 0, 7, 4, 4, 2, 9, 7, 6, 6, 0, 3, 7, 3, 2, 3, 1, 0, 4, 2, 0, 6, 1, 7, 9, 2, 2, 2, 2, 7, 6, 7, 0, 0, 9, 7, 2, 5, 5, 3, 8, 1, 1, 0, 0, 3, 9, 4, 7, 7, 4, 4, 7, 1, 7, 6, 4, 5, 1, 7, 9, 5, 1, 8, 5, 6, 0, 8, 7, 1, 8, 3, 0, 8, 9
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OFFSET
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0,1
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COMMENTS
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Also, decimal expansion of the real part of e^i. - Bruno Berselli, Feb 08 2013
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LINKS
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FORMULA
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Continued fraction representation: cos(1) = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). See A275651 for proof. Cf. A073743. - Peter Bala, Sep 02 2016
Equals (4*(cos(1/4)^4 + sin(1/4)^4) - 3).
Equals (16*(cos(1/4)^6 + sin(1/4)^6) - 10)/6. (End)
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EXAMPLE
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0.5403023058681397...
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MAPLE
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MATHEMATICA
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RealDigits[Cos[1], 10, 110] [[1]]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A346441
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(3*k)!.
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+10
17
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8, 3, 4, 7, 1, 9, 4, 6, 8, 5, 7, 7, 2, 1, 0, 9, 6, 2, 2, 1, 9, 2, 8, 3, 2, 3, 9, 2, 0, 8, 3, 3, 0, 0, 7, 0, 8, 4, 0, 3, 7, 9, 0, 5, 1, 9, 9, 8, 2, 6, 9, 7, 6, 7, 6, 2, 7, 6, 9, 5, 1, 0, 7, 9, 5, 2, 5, 9, 2, 7, 8, 4, 3, 6, 8, 7, 2, 2, 2, 2, 3, 8, 9, 7, 3, 0, 0
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OFFSET
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0,1
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FORMULA
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Equals 1/(3*e) + 2*sqrt(e)*cos(sqrt(3)/2)/3. - Amiram Eldar, Jul 18 2021
Continued fraction: 1/(1 + 1/(5 + 6/(119 + 120/(503 + ... + P(n-1)/((P(n) - 1) + ... ))))), where P(n) = (3*n)*(3*n - 1)*(3*n - 2) for n >= 1. See Bowman and Mc Laughlin, Corollary 10, p. 341 with m = 1, who also show that the constant is irrational. - Peter Bala, Feb 21 2024
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EXAMPLE
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0.8347194685772109622192832392...
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MATHEMATICA
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RealDigits[Sum[(-1)^k/(3*k)!, {k, 0, Infinity}], 10, 100][[1]] (* Amiram Eldar, Jul 18 2021 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A196498
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(10*k)!.
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+10
10
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9, 9, 9, 9, 9, 9, 7, 2, 4, 4, 2, 6, 8, 0, 7, 7, 6, 0, 5, 5, 2, 1, 2, 5, 2, 3, 6, 7, 5, 8, 0, 2, 0, 4, 7, 6, 0, 0, 1, 2, 6, 3, 7, 2, 0, 3, 6, 6, 0, 0, 3, 5, 6, 2, 1, 1, 9, 7, 3, 3, 1, 6, 3, 7, 2, 8, 9, 9, 3, 3, 6, 5, 8, 4, 7, 2, 1, 1, 6, 8, 9, 6, 7, 4, 0, 0, 2, 7, 4, 8, 2, 1, 1, 9, 7, 3, 8, 4, 2, 5, 9, 3, 0, 1, 0
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EXAMPLE
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0.99999972442680776055212523675802047...
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MATHEMATICA
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RealDigits[ HypergeometricPFQ[{}, {1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10}, -10^-10], 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)
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EXTENSIONS
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STATUS
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approved
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A346437
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(7*k)!.
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+10
10
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9, 9, 9, 8, 0, 1, 5, 8, 7, 3, 1, 3, 0, 5, 8, 0, 4, 7, 1, 6, 5, 4, 5, 8, 3, 7, 0, 9, 5, 5, 3, 2, 7, 6, 2, 7, 5, 7, 2, 1, 0, 9, 1, 8, 0, 5, 7, 4, 8, 8, 0, 9, 5, 6, 1, 4, 9, 7, 1, 2, 9, 4, 1, 3, 9, 4, 0, 9, 3, 6, 7, 6, 4, 4, 6, 9, 8, 5, 8, 1, 1, 0, 5, 7, 8, 7, 7
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OFFSET
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0,1
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LINKS
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EXAMPLE
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0.99980158731305804716545837...
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MATHEMATICA
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RealDigits[HypergeometricPFQ[{}, {1/7, 2/7, 3/7, 4/7, 5/7, 6/7}, -1/7^7], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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approved
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A346440
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(4*k)!.
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+10
10
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9, 5, 8, 3, 5, 8, 1, 3, 2, 8, 3, 3, 0, 0, 7, 0, 1, 6, 2, 1, 0, 4, 0, 4, 4, 6, 0, 2, 5, 5, 6, 7, 4, 9, 9, 5, 4, 2, 3, 5, 5, 6, 7, 9, 4, 7, 0, 1, 8, 1, 0, 1, 6, 9, 5, 6, 1, 6, 2, 3, 1, 9, 0, 0, 2, 1, 2, 2, 3, 2, 0, 4, 2, 8, 0, 7, 9, 0, 1, 3, 3, 2, 1, 3, 2, 6, 8
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OFFSET
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0,1
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LINKS
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FORMULA
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Equals cos(sqrt(2)/2)*cosh(sqrt(2)/2). - Amiram Eldar, Jul 18 2021
Continued fraction: 1/(1 + 1/(23 + 24/(1679 + ... + P(n-1)/((P(n) - 1) + ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - Peter Bala, Feb 21 2024
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EXAMPLE
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0.95835813283300701621040446...
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MATHEMATICA
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RealDigits[Sum[(-1)^k/(4*k)!, {k, 0, Infinity}], 10, 100][[1]] (* Amiram Eldar, Jul 18 2021 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A346436
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(8*k)!.
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+10
9
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9, 9, 9, 9, 7, 5, 1, 9, 8, 4, 1, 2, 7, 4, 6, 2, 0, 7, 4, 7, 1, 7, 3, 4, 9, 6, 0, 5, 2, 8, 1, 0, 1, 7, 0, 2, 4, 5, 5, 3, 6, 5, 5, 7, 9, 9, 9, 3, 1, 8, 7, 5, 5, 6, 0, 5, 7, 6, 5, 2, 4, 3, 8, 2, 0, 7, 9, 2, 3, 4, 9, 7, 5, 6, 4, 5, 0, 4, 8, 1, 1, 7, 6, 6, 1, 7, 2
(list;
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OFFSET
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0,1
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LINKS
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EXAMPLE
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0.9999751984127462074717349605281...
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MATHEMATICA
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RealDigits[HypergeometricPFQ[{}, {1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8}, -1/2^24], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A346438
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(6*k)!.
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+10
9
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9, 9, 8, 6, 1, 1, 1, 1, 3, 1, 9, 8, 7, 8, 6, 6, 5, 3, 7, 0, 5, 8, 5, 2, 9, 3, 4, 9, 0, 7, 4, 2, 2, 8, 4, 7, 1, 9, 8, 3, 3, 3, 7, 6, 2, 8, 2, 0, 0, 4, 5, 7, 6, 4, 5, 1, 6, 5, 3, 6, 1, 5, 2, 6, 4, 9, 5, 4, 7, 6, 4, 6, 5, 6, 3, 8, 4, 0, 6, 8, 6, 7, 6, 5, 4, 3, 4
(list;
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graph;
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OFFSET
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0,1
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LINKS
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FORMULA
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Equals (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3. - Amiram Eldar, Jun 04 2023
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EXAMPLE
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0.9986111131987866537058529349...
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MATHEMATICA
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RealDigits[(Cos[1] + 2*Cos[1/2]*Cosh[Sqrt[3]/2])/3, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A346439
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Decimal expansion of the constant Sum_{k>=0} (-1)^k/(5*k)!.
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+10
9
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9, 9, 1, 6, 6, 6, 9, 4, 2, 2, 3, 9, 0, 9, 4, 1, 9, 0, 5, 6, 3, 4, 2, 2, 9, 0, 8, 4, 5, 3, 9, 8, 6, 2, 0, 5, 3, 1, 7, 5, 9, 1, 5, 2, 5, 0, 6, 7, 8, 0, 8, 3, 9, 3, 3, 5, 8, 1, 3, 5, 9, 3, 9, 3, 7, 7, 8, 5, 4, 7, 5, 0, 2, 8, 2, 5, 5, 9, 2, 0, 8, 1, 8, 6, 3, 8, 9
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OFFSET
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0,1
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LINKS
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EXAMPLE
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0.9916669422390941905634229...
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MATHEMATICA
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RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, -1/5^5], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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