Category:Recurrence, linear, order 03, (3,-3,1)

Main article page: Recurrence, linear, order 03, (3,-3,1)

These integer sequences obey the relation

${\displaystyle \sum _{i=0}^{3}(-1)^{i}{\binom {3}{i}}\,a(n-i)=a(n)-3\,a(n-1)+3\,a(n-2)-a(n-3)=0,\quad n\geq 3;\;a(n)=a_{n},\quad 0\leq n\leq 2,\,}$

and are thus given by the recurrence

${\displaystyle a(n)=\sum _{i=1}^{3}(-1)^{i+1}{\binom {3}{i}}\,a(n-i)=3\,a(n-1)-3\,a(n-2)+a(n-3),\quad n\geq 3;\;a(n)=a_{n},\quad 0\leq n\leq 2.\,}$

These integer sequences can also be described by the quadratic polynomials

${\displaystyle a(n)=\left({\frac {a_{2}-2\,a_{1}+a_{0}}{2}}\right)n^{2}+\left({\frac {4\,a_{1}-a_{2}-3\,a_{0}}{2}}\right)n+a_{0},\quad n\geq 0.\,}$