Given any number it is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question and the next is the given number. There are also non-polynomial solutions.
The simplest polynomial of order 4 is
Un = (-11n4 + 138n3 - 601n2 + 1074n - 624)/24 for n = 1, 2, 3, ...
and, accordingly, the next number is -11.
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1+1=2 : 1+2=3 : 2+3=5 : 3+5=8Therefore the next number is 8
If you fit the following cubic t(n) = (-n3 + 9n2 - 20n + 18)/3 where n = 1, 2, 3, ... then the next number is t(6) = 1
The next number is 1004.The rule is Un = (-73n4 + 893n3 - 3803n2 + 6520n - 3504)/3 for n = 1, 2, 3, ...The next number is 1004.The rule is Un = (-73n4 + 893n3 - 3803n2 + 6520n - 3504)/3 for n = 1, 2, 3, ...The next number is 1004.The rule is Un = (-73n4 + 893n3 - 3803n2 + 6520n - 3504)/3 for n = 1, 2, 3, ...The next number is 1004.The rule is Un = (-73n4 + 893n3 - 3803n2 + 6520n - 3504)/3 for n = 1, 2, 3, ...
The next number is 61. The rule is Un = (33n4 - 394n3 + 1599n2 - 2510n + 1296)/24 for n = 1, 2, 3, ...
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