[math-fun] FromContinuedFraction@{0, 1, 1, 2, 2, 3, 3, 4, 4, . . .}

Finch, p424 (Euler-Gompertz Constant): "No one knows the exact outcome if we instead repeat each denominator in c1 [ := FromContinuedFraction@{0, 1, 2, 3, 4,. . .} = I_1(2)/I_0(2)] ~ .6977746579640] although numerically we find that c2 = .5851972651050... ". Using my mysteriously wrong-but-correctible matrix-to-CF (non-regular) converter, c2= 1/2 + 1/(12 + 2*ContinuedFractionK[1 - k^2, 1 + 3*k + 2*k^2 + k^3, {k, 2, ∞}]). Julian & I had a way to further convert these to log-derivatives of pFqs, if anyone is really interested. --rwg