2 : BesselI[0, 2] - 1 ; n 2 n : Log[2] ; n 2 : π 2 /6 ; (3n - 1)(3n - 2): π √ 3 /9 ; 3 - 16n + 16n 2 : π /8 ; n n!

The sequence of odd numbers gives the continued fraction for Coth[1] ; the sequence of even numbers for BesselI[0, 1]/BesselI[1, 1] . In general, continued fractions whose n th term is a n + b correspond to numbers given by BesselI[b/a, 2/a]/BesselI[b/a + 1, 2/a] . … As discovered by Jeffrey Shallit in 1979, numbers of the form Sum[1/k 2 i , {i, 0, ∞ }] that have nonzero digits in base k only at positions 2 i turn out to have continued fractions with terms of limited size, and with a nested structure that can be found using a substitution system according to {0, k - 1, k + 2, k, k, k - 2, k, k + 2, k - 2, k} 〚 Nest[Flatten[{{1, 2}, {3, 4}, {5, 6}, {7, 8}, {5, 6}, {3, 4}, {9, 10}, {7, 8}, {9, 10}, {3, 4}} 〚 # 〛 ]&, 1, n] 〛 The continued fractions for square roots are always periodic; for higher roots they never appear to show any significant regularities.