Re:The Remarkable Number 1/89 (2004)
by efdb on Thursday October 29, 2020 @09:49AM
(#24930433)
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.
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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.
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.