# Formula for pattern data

How to derive formula for the series of numbers as below:
(a) 4,8,15,16,23,42
(b) 4,17, 23,38,41,46

haruspex
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The sequences are so jerky that you really need longer samples for the question even to be meaningful. Yes, you could make up formulae to generate these, but they would likely contain five or more arbitrary constants, so you'd have no confidence they mean anything.

uart
Hi Ryan. Do the numbers represent anything in particular? Do you have a reason to use such a formula, like interpolation or extrapolation?

There are many ways to fit those points to a curve, but whether or not any particular formula has any significant meaning is another question.

For the first one for example, you can write:

$$T_k = - \frac{9}{40} k^5 + \frac{25}{8} k^4 - \frac{117}{8} k^3 + \frac{215}{8} k^2 - \frac{223}{20} k + 4 \, \, \, \, \, : \,k = 0,1,2,3 \ldots$$

uart
$$T_k = \frac{1}{217} \left( 454 \, T_{k-3} - 63 \, T_{k-2} + 144 \, T_{k-1} \right)$$