sequences and series - Seeking for a formula or an expression to generate non-repeatative random number ..

When we hear random permutations, we bring in our intuition about permutations, and try to give a method which could generate a complicated permutation. Thus, I think we didn't pay enough attention to your examples like n*3 mod N, which for most situations would not be an acceptable way of generating random numbers. The only problem is what to do if N is divisible by 3. As far as I can tell, divisibility by 10 is irrelevant, so I'm not sure why you mentioned it.

You say you don't want to write a program, just a simple formula in Excel. This is reasonable, and even something which makes sense mathematically: There are a few operations available in Excel formulas such as addition, exponentiation, factorial, conditional evaluation based on whether a statement is true or false (characteristic functions), etc. Can one create a formula with fixed complexity which takes in n and N, and which is a permutation of {1,...,N] for a fixed N? Trivially returning n works, but can one produce a permutation other than (+-n+k mod N)+1?

I suggest creating a formula which is equivalent to the following:

If N is not divisible by 71, return (71*n mod N) + 1.
Otherwise N is divisible by 71. Permute the last digit base 71: return a + (3*b mod 71) +1
where n-1 = a + b and a is divisible by 71 and $0 le b lt 93$, i.e.,
b = n-1 mod 71.
a = n-1 - (n-1 mod 71).

IF(MOD(N,71)!=0,n-(MOD(n-1,71)) + MOD(3*(MOD(n-1,71),71),MOD(71*n,N)+1).

(Debugging left to the reader.)

This would be lousy as a random permutation, but it may be acceptable for some purposes.

A better random permutation might be based on f(n), where f reverses the lowest binary digits of n if n is at most than the greatest power of 2 less than N, and does nothing if n is greater. Try f(N+1-f(n)). This can be done using the DEC2BIN and StrReverse functions, but you need a little Excel expertise to use those.

Once you have a few ways to generate random permutations, you can compose them, and even using unsatisfactory random permutations like adding floor(sqrt(N)) can improve the appearance of the resulting permutation.