Construct a function pertaining to the OEIS sequence A131229 (Numbers congruent to {1,7} mod 10)

Question

OEIS sequence A131229 ("Numbers congruent to {1,7} mod 10") begins $\{1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51,...\}$.

I want a function $f(x)$, specifically such that $f(\frac{1}{2}) =\frac{7}{2}$, $f(\frac{3}{2}) =\frac{11}{2}$, $f(\frac{5}{2}) =\frac{17}{2}$, $f(\frac{7}{2}) =\frac{21}{2},\ldots$.

The first formula given in the OEIS for A131229 is $a(n) = -3 + sum(5-(-1)^k, k=0..n$).

Answer 1

So you want $$\eqalign{ f\left( 2k + \frac{1}{2} \right) &= \frac{7}{2} + 5k\cr f\left(2k + \frac{3}{2}\right) &= \frac{11}{2} + 5k\cr}$$ Try $$f(x) = \dfrac{5}{2} x + 2 + \frac{1}{4} \sin(\pi x) $$