${\displaystyle \frac{2x^{11}+x^{10}-10x^9-9x^8+12x^7+17x^6-30x^5+2x^4+28x^3-24x^2+8x-1}{(2x-1)(x^2+2x-1){(x^2+x-1)}^2{(x-1)}^3}}$

1, 1, 2, 6, 21, 73, 238, 724, 2078, 5706, 15161, 39319, 100168, 251846, 627046, ...

${\displaystyle -(2x-1)(x^2+2x-1){(x^2+x-1)}^2{(x-1)}^{3}F\left(x\right)+2x^{11}+x^{10}-10x^9-9x^8+12x^7+17x^6-30x^5+2x^4+28x^3-24x^2+8x-1=0}$

${\displaystyle a\left(0\right)=1}$
${\displaystyle a\left(1\right)=1}$
${\displaystyle a\left(2\right)=2}$
${\displaystyle a\left(3\right)=6}$
${\displaystyle a\left(4\right)=21}$
${\displaystyle a\left(5\right)=73}$
${\displaystyle a\left(6\right)=238}$
${\displaystyle a\left(7\right)=724}$
${\displaystyle a\left(8\right)=2078}$
${\displaystyle a\left(9\right)=5706}$
${\displaystyle a\left(10\right)=15161}$
${\displaystyle a\left(11\right)=39319}$
${\displaystyle a(n+1)=\frac{2n^2}{7}+2a(n+3)-\frac{2a(n+4)}{7}-\frac{10a(n+5)}{7}+\frac{6a(n+6)}{7}-\frac{a(n+7)}{7}-\frac{2a\left(n\right)}{7}-\frac{8n}{7}-2,n\ge 12}$

${\displaystyle \{\begin{array}{cc}2& n=0\\ \frac{((8n+6)\sqrt{5}-20n-10){(-\frac{\sqrt{5}}{2}-\frac{1}{2})}^{-n}}{20}+\\ \frac{((-8n-6)\sqrt{5}-20n-10){(\frac{\sqrt{5}}{2}-\frac{1}{2})}^{-n}}{20}+\\ \frac{(-15\sqrt{2}+40){(-1-\sqrt{2})}^{-n}}{20}+\frac{(15\sqrt{2}+40){(\sqrt{2}-1)}^{-n}}{20}+n^2+2n\\ -32^n+1& \text{otherwise}\end{array}}$