A242599 Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.

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

Offset

0,1

Comments

dilog(phi-1) = polylog(2, 2-phi) = sum((2-phi)^k/k^2 , k =1 ..infinity) = sum((1 - 2*sin(Pi/10))^(2*k)/k^2, k=1..infinity) = Pi^2/15 - (log(phi-1))^2 = Pi^2/15 - (2/5)*log(phi-1)*(log(2-phi) + log(phi-1)/2).

See the Jolley reference pp. 66-69, (360)(e), and the Abramowitz-Stegun link, p. 1004, eqs. 27.7.3 - 27.7.6 with x = phi-1, solving for dilog(x) = f(x).

References

L. B. W. Jolley, Summation of Series, Dover, 1961.

Links

Table of n, a(n) for n=0..106.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

dilog(phi-1) = polylog(2, 2-phi) = Sum_{k>=1} (2-phi)^k/k^2 = Sum_{k>=1} (1 - 2*sin(Pi/10))^k/k^2.

Example

0.42640880616209618209...

Maple

phi := (1+sqrt(5))/2 ; dilog(phi-1) ; evalf(%) ; # R. J. Mathar, Jun 10 2024

Mathematica

RealDigits[PolyLog[2, 2 - GoldenRatio], 10, 120][[1]] (* Amiram Eldar, May 27 2023 *)

Prog

(PARI) polylog(2, 2 - (1+sqrt(5))/2) \\ Gheorghe Coserea, Sep 30 2018
(PARI) sumpos(k=1, (1 - 2*sin(Pi/10))^k/k^2) \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

Cf. A152115, A076788 (dilog(1/2)), A242600.

Sequence in context: A209877 A273667 A187109 * A092205 A272101 A059853

Adjacent sequences: A242596 A242597 A242598 * A242600 A242601 A242602

Keyword

nonn,cons

Author

Wolfdieter Lang, Jun 16 2014

Status

approved