The following formula fits the series: an=7n4/6-5n3+59n2/6-4n+8, where an is the term for n≥0.
So a0=8, a1=10, a2=18, a3=44, a4=128, a5=338. Therefore, the next term is 338 according to this formula.
By the formula a3 should be (n=3):
7×81/6-5×27+59×9/6-12+8=189/2-135+177/2-4=183-139=44 which matches a3 from the actual series.
Other formulas may apply. More terms would be needed to confirm the derived formula.