Closed-Form for RD(f(n)): Finding k(n)

[Big-Daddy]

I have 3 functions in n which are rounded down to the nearest integer, and I need a closed-form way of writing each function. n will be a positive integer in all cases.

(Let the notation RD(f(n)) denote rounding-down the value of f(n) for a certain n, to the nearest integer.) What I have found with each function is that f(n)=RD(f(n)) for all even values of n, and RD(f(n))=f(n)-x where x is a fraction which varies from function to function but, for a given function, is constant across all values of n. For the first function, x=0.5, for the second, x=0.75, for the third, x=0.25.

Knowing this, how can I rewrite RD(f(n)) as a closed-form function, k(n), of n and f(n)? The key lies in finding some g(n) so that g(n)=-1 if n is odd and g(n)=0 if n is even; then I could write k(n)=RD(f(n))=f(n)+g(n)*x.

 

[HallsofIvy]

Can't you use the "floor" function, f(x)= \lfloor x \rfloor which is defined as "the largest integer less than or equal to x"?

 

[Big-Daddy]

Can't you use the "floor" function, f(x)= \lfloor x \rfloor which is defined as "the largest integer less than or equal to x"?

No, that's the very definition of my problem. I need a k(x) such that k(x) = RD(f(x)) = \lfloor f(x) \rfloor to use your notation.