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A197374 Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z). 1
5, 2, 9, 9, 9, 1, 6, 2, 5, 0, 8, 5, 6, 3, 4, 9, 8, 7, 1, 9, 4, 1, 0, 6, 8, 4, 9, 8, 9, 4, 5, 3, 1, 6, 1, 0, 7, 7, 1, 5, 6, 0, 5, 6, 1, 4, 6, 0, 7, 6, 7, 2, 5, 9, 0, 3, 8, 0, 7, 1, 5, 7, 2, 5, 5, 0, 6, 3, 5, 9, 0, 0, 5, 1, 8, 4, 3, 2, 3, 7, 4, 0, 8, 1, 6, 4, 6, 0, 9, 8, 0, 0, 0, 0, 1, 5, 0, 7, 6, 1, 6, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Pi(3) = 5.29991 62508 56349 87194 ... is the real period of the doubly-periodic Dixonian elliptic functions sm(z) (A104133) and cm(z) (A104134): sm(z+Pi(3)) = sm(z); cm(z+Pi(3)) = cm(z). The other period equals Pi(3)*w, where w = exp(2*I*Pi/3).

REFERENCES

A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy = 1, Quarterly J. Pure Appl. Math. 24 (1890), 167-233.

LINKS

Table of n, a(n) for n=1..102.

E. van Fossen Conrad and P. Flajolet The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion, arXiv:math/0507268v1 [math.CO], Sem. Lothar. Combin. 54 (2005/06), Art. B54g, 44 pp.

FORMULA

Pi(3) = 3*int {0..1} 1/(1-t^3)^(2/3) dt = B(1/3,1/3) = Gamma(1/3)^2/Gamma(2/3) = sqrt(3)/(2*Pi)*Gamma(1/3)^3.

EXAMPLE

5.299916250856349...

MATHEMATICA

Sqrt[3]/(2*Pi)*Gamma[1/3]^3 // N[#, 103]& // RealDigits // First (* Jean-Fran├žois Alcover, Jan 21 2013 *)

PROG

(PARI) sqrt(3)/(2*Pi)*gamma(1/3)^3 \\ Charles R Greathouse IV, Mar 04, 2012

CROSSREFS

Cf. A104133, A104134.

Sequence in context: A255899 A019841 A064582 * A119946 A276610 A065270

Adjacent sequences:  A197371 A197372 A197373 * A197375 A197376 A197377

KEYWORD

easy,nonn,cons

AUTHOR

Peter Bala, Mar 04 2012

STATUS

approved

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Last modified March 4 23:02 EST 2021. Contains 341812 sequences. (Running on oeis4.)