We could assume that the series is definable in terms of a polynomial expression and take successive differences:

#{:
(color(black)("number"),color(black)("1st order"),color(black)("2nd order"),color(black)("3rd order"),color(black)("4th order")),
(,color(black)("difference"),color(black)("difference"),color(black)("difference"),color(black)("difference")),
(4,,,,),
(,5,,,),
(9,,12,,),
(,17,,50,),
(26,,62,,228),
(,79,,278,),
(105,,340,,),
(,419,,,),
(524,,,,)
:}#

If we assume that the series stabilizes with a #4^(th)# order difference of #228#,

we can generate the next line by back-filling from right to left:

#{:
(color(black)("number"),color(black)("1st order"),color(black)("2nd order"),color(black)("3rd order"),color(black)("4th order")),
(,color(black)("difference"),color(black)("difference"),color(black)("difference"),color(black)("difference")),
(4,,,,),
(,5,,,),
(9,,12,,),
(,17,,50,),
(26,,62,,228),
(,79,,278,),
(105,,340,,color(red)(228)),
(,419,,color(red)(506),),
(524,,color(red)(846),,),
(,color(red)(1265),,,),
(color(red)(1789),,,,)
:}#