Are these statements always true?

Question

I haven't found an answer in my books. Although the question seems very simple, I want to ask.

Are these statements always true?

a) For any infinity non-negative integer sequence, if there is an exist $n-$th term closed form expression formula, for this sequence, we have always a recurrence formula.

b) For any infinity non-negative integer sequence, if there is an exist recurrence formula,for this sequence, we have always $n-$th term closed form expression formula.

c) For any infinity non-negative integer sequence, if there is not an exist recurrence formula,for this sequence, we don't have an any $n-$th term closed form expression formula.

Thank you very much.

Answer 1

"Closed form expression" is rather ambiguous, but...

(a), (c): trivially yes, because $a_{n+1} - a_n$ is a difference of two closed form expressions.

(b): like in the analogous case of calculation of primitives, we can have a recurrence expression without closed form solution.

Quote from $A = B$:

The following sequences cannot be expressed in closed form. That is to say, in each case the sequence cannot be exhibited as a sum of a fixed (independent of $n$) number of hypergeometric terms:

$\bullet$ The sum of the cubes of the binomial coefficients of order $n$, i.e. $\sum\binom{n}k^3$

$\bullet$ The number of derangements (fixed-point free permutations) of $n$ letters.

$\bullet$ The central trinomial coefficient, i.e., the coefficient of $x^n$ in the expansion of $(1 + x + x^2)^n$.

$\dots$