Reverse Engineering Part 3

In my previous post, Reverse Engineering Part 2, I ended up being quite satisfied with the reverse engineering that created as output an homogenous linear recurrence relation of order 3 after input of any positive integer greater than 9. I used 28206 and 28207 as examples to illustrate the process.

This got me thinking about creating as output an homogenous linear recurrence relation of order 2 after input of any positive integer greater than 9. I explained in my post Reverse Engineering Part 1 that Gemini's algorithm had failed when attempting this. I was trying to get Gemini to create the following:$$ \begin{align} &\text{a}(n)=p \times \text{a}(n-1)+q \times \text{a}(n-2) \\ &\text{where} -9 \leq p,q \leq 9 \text{ with } p \neq 0 \text{ and } q \neq 0 \\ &\text{ and } 0 \leq \text{a}(n-1), \text{a}(n-2) \leq 9 \end{align}$$So I asked Gemini to relax the conditions by specifying that \( |p + q|\) should be as small as possible. When applied to 28207 this produced values \(p=39\) and \(q=-38\). Not quite what I wanted. So in the end I specified that \(|p^2+q^2+a(0)^2+a(1)^2|\) should be as small as possible. Applied to 28206, this was the result (permalink):

Target Number: 28206

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Constants found: p = 13, q = 1

Constraint check: Minimum p^2 + q^2 + a(0)^2 + a(1)^2 = 395

Seed numbers:    a(0) = 9, a(1) = 12

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Sequence progression:

  a(0) = 9

  a(1) = 12

  a(2) = 165

  a(3) = 2157

  a(4) = 28206

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Comma-separated sequence:

9, 12, 165, 2157, 28206

Applied to 28207, this was the result (permalink):

 Target Number: 28207

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Constants found: p = 6, q = 5

Constraint check: Minimum p^2 + q^2 + a(0)^2 + a(1)^2 = 231

Seed numbers:    a(0) = 11, a(1) = 7

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Sequence progression:

  a(0) = 11

  a(1) = 7

  a(2) = 97

  a(3) = 617

  a(4) = 4187

  a(5) = 28207

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Comma-separated sequence:

11, 7, 97, 617, 4187, 28207

Overall I'm quite happy with these sequences. All terms are positive, the coefficients and seed values are not large and the terms increase steadily toward their targets, avoiding any wild gyrations. I have incorporated this program into my daily number analysis. 

Reverse Engineering Part 2

In my previous post Reverse Engineering Part 1, I had specified to Gemini that I wanted \(p + q + r \) to be the minimum possible within the specifications that each of these coefficients were to be between -9 and -9 inclusive. I was thinking in terms of the sum getting as close to zero as possible forgetting that the minimum possible sum would be -27. That's why I was getting coefficients in the output that were all negative. The algorithm was doing what I'd asked of it! What I should have instructed Gemini to do was to take the absolute value of \(p+q+r\). So to summarise, our starting point is:$$ \begin{align} &\text{a}(n)=p \times \text{a}(n-1)+q \times \text{a}(n-2) + r \times \text{a}(n-2)\\ &\text{with } -9 \leq p,q,r \leq 9, 0 \leq \text{a}(2), \text{a}(1),\text{a}(0) \leq 9 \\ &\text{and } |p+q+r| \text{ as close to zero as possible} \end{align}$$Having gotten Gemini to modify the algorithm, the result for 28206 becomes :$$ \begin{align} &\text{a}(n)=5 \times \text{a}(n-1)-7 \times \text{a}(n-2) + 2 \times \text{a}(n-2)\\ &a(0) = 2, a(1) = 2, a(2) = 6\end{align}$$The full details are (permalink):

Target Number: 28206

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Constants found: p = 5, q = -7, r = 2

Constraint check: Minimum |p + q + r| = 0 (Actual Sum = 0)

Seed numbers:    a(0) = 2, a(1) = 2, a(2) = 6

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Sequence progression:

  a(0) = 2

  a(1) = 2

  a(2) = 6

  a(3) = 20

  a(4) = 62

  a(5) = 182

  a(6) = 516

  a(7) = 1430

  a(8) = 3902

  a(9) = 10532

  a(10) = 28206

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Comma-separated sequence:

2, 2, 6, 20, 62, 182, 516, 1430, 3902, 10532, 28206

This is a longer sequence than previously (2, 4, 2, -48, 408, -3390, 28206) but it has no negative members and is free of the wild gyrations that characterise the former. Similarly for 28207, we have (permalink):$$ \begin{align} &\text{a}(n)=5 \times \text{a}(n-1)+4 \times \text{a}(n-2) -8 \times \text{a}(n-2)\\ &a(0) = 1, a(1) = 1, a(2) = 7\end{align}$$The full results are (permalink):

Target Number: 28207

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Constants found: p = 5, q = 4, r = -8

Constraint check: Minimum |p + q + r| = 1 (Actual Sum = 1)

Seed numbers:    a(0) = 1, a(1) = 1, a(2) = 7

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Sequence progression:

  a(0) = 1

  a(1) = 1

  a(2) = 7

  a(3) = 31

  a(4) = 175

  a(5) = 943

  a(6) = 5167

  a(7) = 28207

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Comma-separated sequence:

1, 1, 7, 31, 175, 943, 5167, 28207

This is shorter than the previously calculated sequence (3, 5, 1, -73, 243, -323, -311, 2207, -3445, -3595, 27729, -51001, -16797, 304365, -658279, 28207) and again it has no negative members and is free of the wild gyrations that characterise the former. So, a lesson learned. I've modified my daily number analysis algorithm accordingly.