Is there a way to generate formula using trigonometric functions from periodic sequences? One example is sequence A153130 and formula discovered by Leonid Bedratyuk: $$a\left(n\right)=\ -\frac{\cos\left(\pi\cdot n\right)}{2}-3\cdot\cos\left(\pi\cdot\frac{n}{3}\right)-3^{\frac{1}{2}}\sin\left(\frac{\pi\cdot n}{3}\right)+\frac{9}{2}$$
Is there general principles to write down such equations?
If one has the recurrence relation $$a_n=a_{n-k}$$ i.e. periodic with period $k$ then the general solution is of the form $$a_n=\sum_{j=0}^{k-1}C_j \exp{\left(\frac{2j\pi ni}{k}\right)}$$ For some constants $\{C_0,C_1,\dots C_{k-1}\}\subset\mathbb{C}$ which depend on the initial terms (this follows from the characteristic equation $r^k=1$). This can be rewritten in terms of trigonometric functions by using Euler's identity $$a_n=\sum_{j=0}^{k-1}C_j \left(\cos{\left(\frac{2j\pi n}{k}\right)}+i\sin{\left(\frac{2j\pi n}{k}\right)}\right)$$ It may be the case that the terms simplify somewhat to give formulas such as those given by Leonid. One could simplify the resulting formula using some trigonometric identities on a case-by-case basis.