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A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2) = 0.284079... . 0
2, 8, 4, 0, 7, 9, 0, 4, 3, 8, 4, 0, 4, 1, 2, 2, 9, 6, 0, 2, 8, 2, 9, 1, 8, 3, 2, 3, 9, 3, 1, 2, 6, 1, 6, 9, 0, 9, 1, 0, 8, 8, 0, 8, 8, 4, 4, 5, 7, 3, 7, 5, 8, 2, 7, 5, 9, 1, 6, 2, 6, 6, 6, 1, 5, 5, 0, 4, 5, 8, 7, 7, 3, 5, 1, 4, 8, 4, 5, 5, 3, 7, 3, 0, 3, 7, 8, 4, 1, 7, 7, 5, 2, 2, 3, 1, 6, 2, 5, 8, 6, 7, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This constant xi arose in the Davenport-Heilbronn zeta-function Z(s)=sum(k>=1,b(k)/k^s) where b(k) is the 5-periodic sequence with period [1,xi,-xi,0]. Z satisfies a functional equation (like zeta) but does not satisfy RH. Some nontrivial zeros are off the critical line (see reference).

REFERENCES

Bruce C. Berndt, Heng Huat Chan, Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, J. reine angew. Math. 480 (1996) 141-159, eq. (1.1).

H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series I, II, J. London Math. Soc. 11 (1936), 181-185 and ibid. 307-312.

P. Borwein et al., The Riemann Hypothesis, Springer (2009), 136-137.

LINKS

Table of n, a(n) for n=0..103.

FORMULA

xi=(sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1).

(A001622-1)/(2*A019845+1). - R. J. Mathar, Apr 02 2009

EXAMPLE

0.2840790438404122960282...

MATHEMATICA

(Sqrt[5]-1) / (2+Sqrt[10-2*Sqrt[5]]) // RealDigits[#, 10, 104]& // First (* Jean-Fran├žois Alcover, Mar 04 2013 *)

PROG

(PARI) xi=(cos(Pi/5)-1/2)/(sin(Pi/5)+1/2)

CROSSREFS

Cf. A158241.

Sequence in context: A076588 A068565 A092042 * A021356 A030345 A264818

Adjacent sequences:  A158931 A158932 A158933 * A158935 A158936 A158937

KEYWORD

cons,nonn

AUTHOR

Benoit Cloitre, Mar 31 2009

STATUS

approved

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Last modified September 16 12:28 EDT 2019. Contains 327098 sequences. (Running on oeis4.)