Gosper probably thought all this out decades ago, but Henry Baker's recent post discussed rational approximations to 2pi constrained to converge from above. Classical continued fractions converge to their target in alternating fashion, with the first convergent being below the target, the second closer but above, the third closer still but below, and so on. If the target is T0, we have C0 = floor(T0), and the reciprocal of the error is T1 = 1/(T0 - C0). Then we continue to calculate, with Cn = floor(Tn), and T[n+1] = 1/(Tn - Cn). But suppose we use ceiling instead of floor? That is, with target S0, set D0 = ceiling(S0), S1 = 1/(D0 - T0), Dn = ceiling(Sn), and S[n+1] = 1/(Dn - Sn). The resulting series of convergent rationals stays above the target, converging on it monotonically downward. The representation, expanded out, looks like S0 = D0 - 1/(D1 - 1/(D2 - 1/(...))), with subtraction instead of addition in the denominators. Until somebody comes up with a better name, I will call this a "contratinued fraction". Henry's first sequence, the one that OEIS doesn't know, is the sequence of numerators of the convergents of the contratinued fraction for 2pi. (Are these "contravergents"?). The coefficients of the contratinued fraction for 2pi are 7, 2, 2, 3, 9, 2, 148. In general, for an arbitrary positive real target, these contraefficients are always at least 2, except the leading one which can be 1. Probably there is fairly simple machinery for converting a continued fraction directly to a contratinued fraction, but I am not seeing it immediately.