A242600 Decimal expansion of -dilog(phi) = -polylog(2, 1-phi) with phi = (1 + sqrt(5))/2.

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

Offset

1,1

Comments

-dilog(phi) = -polylog(2, 1-phi) = -Sum_{k>=1} ((1-phi)^k/k^2) = Pi^2/15 - (log(phi-1)^2)/2. See the Abramowitz-Stegun link, p. 1004, eqs. 27.7.3 - 27.7.6 with x = phi-1, solving for -dilog(x+1) = -f(1+x), using log(2-phi) = 2*log(phi-1).

This solution for -Sum)k>=1} (-2*sin(Pi/10)^k/k^2) should also have been mentioned in the Jolley reference pp. 66-69 under (360).

References

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

Links

Table of n, a(n) for n=1..107.

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) = -polylog(2, 1-phi) = -sum(((1-phi)^k)/k^2, k=1..infinity) = Pi^2/15 - (log(phi-1)^2)/2 = 0.542191216450693..., with the golden section phi = (1 + sqrt(5))/2.

Mathematica

RealDigits[PolyLog[2, 1 - GoldenRatio], 10, 120][[1]] (* Amiram Eldar, May 30 2023 *)

Crossrefs

Cf. A001622, A076788 (polylog(2,1/2)), A152115, A242599.

Sequence in context: A109430 A085917 A180130 * A102593 A090462 A246966

Adjacent sequences: A242597 A242598 A242599 * A242601 A242602 A242603

Keyword

nonn,cons

Author

Wolfdieter Lang, Jun 16 2014

Status

approved