A195907 -id:A195907

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A002161

Decimal expansion of square root of Pi.
(Formerly M4332 N1814)

+10
76

1, 7, 7, 2, 4, 5, 3, 8, 5, 0, 9, 0, 5, 5, 1, 6, 0, 2, 7, 2, 9, 8, 1, 6, 7, 4, 8, 3, 3, 4, 1, 1, 4, 5, 1, 8, 2, 7, 9, 7, 5, 4, 9, 4, 5, 6, 1, 2, 2, 3, 8, 7, 1, 2, 8, 2, 1, 3, 8, 0, 7, 7, 8, 9, 8, 5, 2, 9, 1, 1, 2, 8, 4, 5, 9, 1, 0, 3, 2, 1, 8, 1, 3, 7, 4, 9, 5, 0, 6, 5, 6, 7, 3, 8, 5, 4, 4, 6, 6, 5

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also Gamma(1/2). - Franklin T. Adams-Watters, Apr 07 2006

The integral of the Gaussian function exp(-x^2) over the real line. - Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008

Also equals the average distance between two points in two dimensions where coordinates are independent normally distributed random variables with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014, after Steven Finch

Also diameter of a sphere whose surface area equals Pi^2. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018

REFERENCES

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 190.

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

Donald Knuth, Why pi?, Christmas Tree lecture, Dec 06 2010 (video).

Index entries for sequences related to the number Pi.

Index entries for transcendental numbers.

FORMULA

Equals (1/2) * Sum_{n>=0} ((-1)^n * (4*n+1) * (1/8)^(n+1) * (2^(n+1))^3 * Gamma(n+1/2)^3 / Gamma(n+1)^3). - Alexander R. Povolotsky, Mar 25 2013

Equals Integral_{x=0..1} 1/sqrt(-log(x)) dx. - Jean-François Alcover, Apr 29 2013

Equals Sum_{k>=0} (k+1/2)!/(k+2)!. - Amiram Eldar, Jun 19 2023

Equals Integral_{x=0..oo} exp(-x)/sqrt(x) dx. - Michal Paulovic, Sep 24 2023

EXAMPLE

1.7724538509055160272981674833411451827975494561223871282138...

MAPLE

evalf(sqrt(Pi), 120); # Muniru A Asiru, Nov 11 2018

MATHEMATICA

RealDigits[N[Sqrt[Pi], 120]][[1]] (* Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008 *)

PROG

(PARI) default(realprecision, 20080); x=sqrt(Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002161.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009

(Magma) R:= RealField(100); Sqrt(Pi(R)); // G. C. Greubel, Mar 10 2018

CROSSREFS

Cf. A000796, A002388, A073006, A195907, A203145.

Cf. decimal expansions of Gamma(1/k): A073005 (k=3), A068466 (k=4), A175380 (k=5), A175379 (k=6), A220086 (k=7), A203142 (k=8).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Franklin T. Adams-Watters, Apr 07 2006

STATUS

approved

A218792

Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

+10
7

1, 2, 7, 1, 3, 4, 1, 5, 2, 2, 1, 8, 9, 0, 1, 5, 2, 2, 5, 2, 2, 2, 3, 8, 2, 5, 7, 8, 7, 9, 0, 9, 3, 5, 6, 2, 4, 9, 7, 6, 8, 6, 4, 9, 8, 7, 7, 1, 7, 6, 2, 6, 7, 0, 1, 1, 6, 4, 4, 1, 0, 8, 0, 1, 6, 9, 7, 4, 7, 7, 5, 8, 5, 6, 6, 5, 5, 3, 0, 7, 5, 0, 6, 2, 3, 9, 3

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

Eric Weisstein's World of Mathematics, Dedekind Eta Function

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

FORMULA

Equals Jacobi theta_{3}(0,exp(-2)). - G. C. Greubel, Feb 01 2017

Equals eta(2*i/Pi)^5 / (eta(i/Pi)*eta(4*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - Jianing Song, Oct 14 2021

EXAMPLE

1.2713415221890152252223825787909356249768649877176...

For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

MATHEMATICA

RealDigits[Sum[E^(-2*k^2), {k, -Infinity, Infinity}], 10, 200][[1]]

RealDigits[EllipticTheta[3, 0, 1/E^2], 10, 200][[1]] (* Vaclav Kotesovec, Sep 22 2013 *)

PROG

(PARI) 1 + 2*suminf(n=1, exp(-2*n^2)) \\ Charles R Greathouse IV, Jun 06 2016

(PARI) (eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021

CROSSREFS

Cf. A195907, A069998, A167009, A167010, A229052.

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Nov 05 2012

STATUS

approved

A348333

Decimal expansion of Sum_{n=-oo..oo} 1/(n^2)!.

+10
0

3, 0, 8, 3, 3, 3, 8, 8, 4, 4, 7, 9, 7, 2, 7, 3, 7, 2, 0, 0, 5, 8, 1, 1, 1, 7, 2, 1, 4, 4, 2, 1, 3, 2, 5, 8, 2, 3, 5, 0, 7, 5, 9, 4, 9, 6, 0, 1, 5, 3, 2, 7, 0, 2, 6, 6, 3, 0, 9, 7, 8, 2, 9, 1, 3, 4, 9, 6, 9, 3, 6, 0, 4, 0, 0, 9, 5, 2, 3, 7, 9, 6, 7, 9, 8, 9, 5, 0, 4, 2

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This constant is irrational. The proof is similar to the one that e is irrational. Is this constant transcendental?

LINKS

Table of n, a(n) for n=1..90.

EXAMPLE

3.083338844797273...

MATHEMATICA

RealDigits[N[1 + 2 *Sum[1/(n^2)!, {n, 1, Infinity}], 90]][[1]] (* Amiram Eldar, Oct 14 2021 *)

PROG

(PARI) 1 + 2*suminf(n=1, 1/(n^2)!)

CROSSREFS

Cf. A195907, A001113 (e = Sum_{n>=0} 1/n!).

KEYWORD

nonn,cons

AUTHOR

Jianing Song, Oct 13 2021

STATUS

approved

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