



 
Jonathan, Check your arithmetic. I tried successive differences and found the third difference produced a row of sixes. Penny
If you want to use successive differences, your first difference sequence should be 10, 24, 44, 70, 102  was it? Then repeating two more times you should (in this case) get a constant third difference. If you just want to get the next number, extend the constant row and work back up, adding. If you want a polynomial, constant third differences identify your pattern as cubic. Divide the constant by 3! (=6) to get the cubic coefficient a_{3}. Then subtract 1a_{3}, 8a_{3}, 27a_{3},.. from your original numbers to get a quadratic remainder sequence. Repeat again; if the first differences are constant (yours won't be) the quadratic term is 0. Otherwise (if your subtraction is accurate) your second difference will be constant; divide by 2! (=2) to get the quadratic coerfficient a_{2}. And so on. Once you've identified the pattern as cubic, another way to find coefficients is to solve the linear system
Good Hunting!  


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