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How might the Pell sequence model rabbit population growth?

Start off with 1 pair which after a month (or whatever) gives birth to 2 pairs. After another month, 1 of those pairs then gives birth to 2 pairs, but the other is more fecund and produces 3, making 5 altogether. Next month the 2 pairs each respectively produce a 2 and a 3, while the 3 produce a 2 and a 2 and a 3, resulting in 12 altogether. And so on. (It's relatively easy to visualise this process using say one dot linked by lines down the page to n other dots to represent one pair giving birth to n others.) Fast eh? Might be a better model for virus than rabbit reproduction.

How might the Pell sequence be Lucasated, that is what sequence would more faithfully reflect the powers of the Silver ratio just as the Lucas sequence does with the Golden?

1 1 3 7 17 41 99 239 577 . . . (OEIS A001333 "Numerators of continued fraction convergents to √2")
Starting with the second number as index 1, 41 for example is at index 5 and is much closer to ((1+√2)^5)/2 than the corresponding number in the "standard" Pell in the article, though admittedly you have to divide by 2.

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