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%I #11 Dec 02 2024 02:09:43
%S 0,4,8,4,8,0,8,0,1,4,4,9,4,6,3,6,3,2,7,0,5,7,2,4,9,3,3,8,8,2,4,7,6,5,
%T 5,6,3,3,3,0,5,6,0,0,6,6,9,5,2,3,7,1,3,9,7,7,1,6,6,5,5,9,9,8,3,8,6,6,
%U 2,0,4,8,2,0,5,4,0,2,2,5,4,2,7,6,2,5,8,8,8,8,8,7,3,1,1,3,3,9,2,4,7,7
%N Decimal expansion of p_2, a probability associated with continuant polynomials.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's Constant, p. 161.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Continuant_(mathematics)">Continuant polynomial</a>.
%F p_2 = Sum_{i >= 1}(sum_{j >= 1} 1/((i*j + 1)^2*(i*j + i + 1)^2)).
%F p_2 = Sum_{n >= 0} (-1)^n*(n + 1)*zeta(n + 4)*(zeta(n + 2) - 1).
%e 0.04848080144946363270572493388247655633305600669523713977...
%t digits = 101; s = NSum[(-1)^n*(n + 1)*Zeta[n + 4]*(Zeta[n + 2] - 1), {n, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 10]; p2 = -5 + 2*Pi^2/3 - 2*Zeta[3] + 2*s; Join[{0}, RealDigits[p2, 10, digits] // First]
%Y Cf. A074903, A145426.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Sep 12 2014