Padovan sequence

Construct a recurrence relation with initial terms $a_0=1$, $a_1=0$, $a_2=0$ and $a_n=a_{n-3}+a_{n-2}$ for $n>2$. The first few terms of the sequence defined by this recurrence relation are: 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151 (listed in A000931 of Sloane’s OEIS). This is the Padovan sequence, named after mathematician Richard Padovan. Its generating function is

$$G(a(n);x)=\frac{1-x^2}{1-x^2-x^3}$$

.

It has been observed that in taking seven consecutive terms of this sequence, the sum of the squares of the first, third and seventh terms is equal to the sum of the squares of the second, fourth, fifth and sixth terms.

The $n$th Padovan number asymptotically matches the $n$th power of the plastic constant.