Knowing what Excel can do for you

If $f (x) = x^7 - p x^6 + q x^5 - r x^4 + s x^3 - t x^2 + u x - 5027$ such that:

$f (1) = 1$

$f (2) = 1$

$f (3) = 2$

$f (4) = 3$

$f (5) = 5$

$f (6) = 8$

$f (7) = 13$

Given that $p, q, r, s, t$ and $u$ are positive rational numbers,

then $find$ the $sum$ of $denominators$ of $p, q, r, s, t$ and $u$ ,

where all of them are $'false'$ fractions of numerator > denominator,

which have been simplified with GCD (numerator, denominator) = 1.

GCD: Greatest Common Divisor (for positive integers), for example GCD (24, 10) = 2. $False$ $fraction$ is an expression commonly used by Chinese communities.

Inspired by original question set by Dev Sharma.