The bisection of the Fibonacci series, Sloane A001906 {1, 3, 8, 21, 55, 144,...}, naturally produces approximations to phi^2, by the division of the nth term by its predecessor: a(n)/a(n-1). ; e.g 55/21, 144/55, etc.

WolframAlpha also lists these fractions as convergents to 5pi/6.

In fact, there will always be a small shortfall between the two: (phi)^2 - 5pi/6 is not zero.

For sufficiently large n, how is the shortfall best approximated, in terms of a rational fraction, say 1/x?