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[ghostofmars]

It's very easy to cheat with google docs based information sharing. The general unwritten rule in PBF games is don't .

Hmm...

I've added all the cells up to 100. You initially gave 11 numbers. Do you think that's enough to spot the 'complex' bit that's added completely?

Looks like the Quadratic without the extra bit, adding in those new terms, is most closely approximated by about 0.9x² - 1.2x + 0.85. Is this correct?

Further looking, the period of the extra term seems to be 50. Also correct?

I feel like I've deduced (almost) everything I need now and just need to piece it together correctly to solve this now...

With the initial numbers, you can figure it out if you recognize the pattern. Perhaps some later elements are easier
  n    a_n
 100   9061
1000 900601
This assumes starting that the first element is a_0.

[Tables]

Hmm... that's not a property I'd ever heard of before, but I see it. I presume what you're referring to is looking at the last two digits of nx²? They appear to exhibit the required behavior - cyclic with period 50 and inverses after 25 terms. There's an offset (this reverses at 0 and 25, not 9 and 34) but that's probably a secondary concern.

Now can I derive the extra term from this data? Let's have a look...

PPE: Okay correcting for assuming term 1 is generated from x=0 gives an actually slightly nicer base formula of 0.9x²+0.6+0.75. No major changes elsewhere. Pondering that new information through now.

[Tables]

Urgh. I tried using the above property, and it gives me graphs that look similar to the one generated by difference in terms from the original equation, but not quite the same. So I looked for patterns and noticed something important shared between them: The value every 10th term is the same along two sequences: 8-18-28... (=1.85) and 13-23-33... (=-0.65) for the differences and of course, 0-10-20... (=0) and 5-15-25... (=25) for the nx^2. So that's a clear similarity in the patterns.

But I can't make them match up. Trying any sort of linear mapping approach kind of works, but I get numbers out ranging over 152 possible values, including negatives - which if I'm taking the result

As I typed this up I noticed, while the 5-15-25... sequence always equals SOMETHING, it doesn't have to be 25. Let's see what I can do now...

Also if I'm barking up the wrong tree knowing that would be nice I suppose...

Edit: Nope, this still doesn't work. The only other results make things worse. Short version of the issue: I should be able to map into a range of 100 numbers. I can't. I can take abs, but then I also need to explain why and when negatives are used, which considering the nice 3rd derivatives, seems like a bad idea.

[ghostofmars]

There is no floating point number necessary. As usual, you shouldn't think to complicated. If you look at the last digit, you will notice that some numbers are missing. Why?

[scott_pilgrim]

I presume what you're referring to is looking at the last two digits of nx²? They appear to exhibit the required behavior - cyclic with period 50 and inverses after 25 terms. There's an offset (this reverses at 0 and 25, not 9 and 34) but that's probably a secondary concern.

Note that (50+x)^2=2,500+100x+x^2.  Since only the x^2 term affects the last two digits (the other terms are multiples of 100), then (50+x)^2 will have the same final two digits as x^2; hence the cyclic behavior of period 50.

Similarly, (50-x)^2=2,500-100x+x^2.  Again, only the x^2 term affects the last two digits, so (50-x)^2 will have the same final two digits as x^2; hence the reversal after 25 terms.

[ghostofmars]

I think, you have basically all you need to solve this:
a) the trend for large n is 0.9n^2 + 0.6n + 1
b) the row shows some periodic behavior similar to n^2
c) I gave you the hint that the last digit is important

One final tip:
Write down the n^2 series and compare to the one I gave you.

[sudgy]

Alright, necroing an age-old thread because I have an interesting series.  I use it all the time to pass the time when there's lots of numbers around me.

The series only applies to natural numbers.  I'll select certain parts to show you of the series:

1, 2, 3, ... (all numbers included up to) 94, 95, 96, 98, 100, 101, 102, ...

A random point later on: ... 28482, 28483, 28486, 28487, 28490, ...

The frequency of natural numbers in the series diminishes as the number gets larger, with the last number in the series being 13,876,543,210.

This isn't nearly enough information to go off of, so feel free to ask questions about other numbers.