As long as we have 11 periodic values, we could plug them, say, into

**Lagrange Interpolating Polynomial** to produce integer values of our choice like [0,1,2,3,4,5,6,7,8,9,10].

Tangent seems like a good choice for a periodic function. We could plug tan(N*pi/11 - 21pi/22) as "x" variable into our Lagrange Polynomial, [tan(-21pi/22), tan(-19pi/22),...,tan(21pi/22)] as x_{k} points and [0,1,...,10] as y_{k} points. Subtract that from N, divide by 11, and voila. I am not constructing the actual formula. The main thing I seem to remember about the Lagrange Interpolating Polynomial from the time when my daughter took all those AP math classes at high school, that it's big.

Anyone, be my guest and work out the details for half of my "best answer" credit.