[efdb]

In addition to this question I would like to know if you can in general say/proof that for every sequence which has some relation between the successive numbers there is a rational number whose decimal expansion is the same as the sequence.

[magnio]

For a linearly recursive sequence x_0, x_1, x_(n+2) = ax_(n+1) + bx_n, the general formula for the terms is

x_n = cα^n + dβ^n,

where α, β are the roots of the quadratic x²−ax−b; c, d are solutions to the system

c + d = x_0 cα + dβ = x_1.

If the series Σ x_n⋅10^n converges then its value is

10c/(10−α) + 10d/(10−β) = ((100−10a)x_0 + 10x_1)/(100 − 10a − b).

If a, b, x_0, x_1 are all rational then the above series converges to a rational number, too. This is the case for the Fibonacci sequence, with a=b=1, x_0=0 and x_1=1.

[stabbles]

A standard technique to find explicit expressions for the `n`th number of a recursion is through generation functions; see e.g. https://en.wikipedia.org/wiki/Generating_function. You can plugin x = 10^-1 there. Not sure if the result is always a rational number.