Find a sequence

Question

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.

Answer 1

Let's build such function: $$ f(x) = \sum_{n=1}^\infty F_nx^n, $$ where $F_n$ are Fibonacci numbers.

Then $$ f(x) = x+\sum_{n=2}^\infty (F_{n-1}+F_{n-2})x^n = x + \sum_{k=1}^\infty F_kx^{k+1}+\sum_{m=0}^\infty F_mx^{m+2} \\= x+xf(x)+x^2f(x), $$ or much wide: $$ f(x) = x+x^2+2x^3+3x^4+5x^5+8x^6+13x^7+\cdots \\ = x \;\;+\;\; (x^2+x^3+2x^4+3x^5+5x^6+8x^7+\cdots )\\ \qquad\qquad\quad\;\; +( x^3+x^4+2x^5+3x^6+5x^7+\cdots) \\ =x+xf(x)+x^2f(x), $$

hence $$ (1-x-x^2)f(x)=x, $$

$$ f(x)=\dfrac{x}{1-x-x^2}. $$