# Formula for pattern data

1. Aug 24, 2012

### ryan8200

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

2. Aug 24, 2012

### haruspex

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.

3. Aug 24, 2012

### 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:

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

4. Aug 24, 2012

### Staff: Mentor

5. Aug 24, 2012

### uart

Another series for the first one is,

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

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