There are infinitely many polynomials of order 6 that will give these as the first six numbers and any one of these could be "the" rule. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.
The simplest solution, based on a polynomial of order 5 is:
U(n) = (32*n^5 - 440*n^4 + 2360*n^3 - 6010*n^2 + 7133*n - 3070)/5.
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4567
That series is the cubes of the counting numbers.
It is not a rule as such; those number are the first 10 prime numbers.
One rule for this pattern is to add twice the previous value added 4 + 1 = 5 5 + 2×1 = 5 + 2 = 7 7 + 2×2 = 7 + 4 = 11 11 + 2×4 = 11 + 8 = 19 Continuing the next numbers would be: 19 + 2×8 = 19 + 16 = 35 35 + 2×16 = 35 + 32 = 67 ...
a recursive pattern is when you always use the next term in the pattern... for example 4,(x2+1) 9,(x2+1) 19,(x2+1) 39,(x2+1) 79,(x2+1) 159