Maths puzzle -- What is the missing digit?

[.Scott]

One last clue:

$2 + 3 \approx 6$
$3 + 6 \approx 8$
$6 + 8 \approx 15$

I stopped typing my clue in as soon as I saw yours pop up.
Clearly, we are on the same page!

 

[engnrshyckh]

We're not suppose to just give you the solution. So here's another (hopefully final) clue:

We know (2, 3, 6, 8, 15, ...) isn't Fibonacci because:
6 <> 2+3 (just misses)
8 <> 3+6 (just misses)
15 <> 6+8 (just misses)
22 <>

Thanks got it in some terms we add and aubtract 1 alternatively

 

[.Scott]

You can however get an exact formula for the n-th element of the series, and it does involve (-1)^n.
If you type [insert recurrence relation here], a(0)=2, a(1)=3, a(2) = 6 in wolfram alpha, you'll get the exact solution.

And now that a solution is finally revealed, @willem2 solved it with this:
$v_i = 2v_{i-2}+v_{i-3}$

 

[DrClaude]

And now that a solution is finally revealed, @willem2 solved it with this:
$v_i = 2v_{i-2}+v_{i-3}$

It is also easy to show that both approaches are the same. Take
$$
v_i = v_{i-1} + v_{i-2} + (-1)^i
$$
and replace $v_{i-1}$ on the rhs using the same recurrence relation.

 

[.Scott]

The "absolute value" method I referred to was this:
$v_i = v_{i-2} - v_{i-1} +1$
can generate this sequence:
2, -3, 6, -8, 15, -22, ...

The absolute value of each element in that sequence gives you the sequence we were working with.

Oh yes. Someone should say it: 59.

 

[engnrshyckh]

The "absolute value" method I referred to was this:
$v_i = v_{i-2} - v_{i-1} +1$
can generate this sequence:
2, -3, 6, -8, 15, -22, ...

The absolute value of each element in that sequence gives you the sequence we were working with.

There are infinite methods to solve it so we can say that series is divergent?

 

[.Scott]

There are infinite methods to solve it so we can say that series is divergent?

The reason it is divergent is because the sum of the terms does not approach any particular finite value.
See Divergent Series (wiki)

The reason that there are infinite methods is because there are always infinite methods. Everyone knows there is more than one way to skin a cat. But Mathematicians know there are an infinite number of ways to do that.

 

[zoki85]

What on Earth the pyramidal structure of the puzzle has to do with that sequence recursive relation?

 

[phinds]

What on Earth the pyramidal structure of the puzzle has to do with that sequence recursive relation?

Nothing. It's to throw off the scent.

 

[jbriggs444]

And now that a solution is finally revealed, @willem2 solved it with this:
$v_i = 2v_{i-2}+v_{i-3}$

Nice. So the characteristic equation is a cubic. Which means that the result will [likely] be the sum of three exponentials. Which is how the $-1^n$ fits in.