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%I #32 May 27 2023 04:18:28
%S 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,
%T 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,
%U 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
%N Decimal expansion of dilog(phi-1) = polylog(2, 2-phi) with phi = (1 + sqrt(5))/2.
%C 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).
%C 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).
%D L. B. W. Jolley, Summation of Series, Dover, 1961.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F 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.
%e 0.42640880616209618209...
%t RealDigits[PolyLog[2, 2 - GoldenRatio], 10, 120][[1]] (* _Amiram Eldar_, May 27 2023 *)
%o (PARI) polylog(2, 2 - (1+sqrt(5))/2) \\ _Gheorghe Coserea_, Sep 30 2018
%o (PARI) sumpos(k=1, (1 - 2*sin(Pi/10))^k/k^2) \\ _Gheorghe Coserea_, Sep 30 2018
%Y Cf. A152115, A076788 (dilog(1/2)), A242600.
%K nonn,cons
%O 0,1
%A _Wolfdieter Lang_, Jun 16 2014
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