OEIS sequence A277306

[Ganado]

Would somebody mind explaining what this sequence means?
https://oeis.org/A277306

I've been staring at it for a few minutes, and can't figure out what it's saying.
"G.f. [general formula?] satisfies: A(x + A(x)^2) = x + 2*A(x)^2."

The offset is 1, meaning A(1) = 1, A(2) = 1, A(3) = 0, ... (based on my understanding of the wiki, http://oeis.org/wiki/OEIS_FAQ)

Let's say x = 1, plugging in you get:

  A(1 + A(1)^2) = 1 + 2*A(1)^2
  A(1 + 1) = 1 + 2*1
  A(2) = 3
  1 = 3

So clearly I am interpreting this sequence wrong. I have found nothing on the OEIS FAQ on sequences like these, so I really don't know what to make of it.
Any help? It's probably something simple/obvious.

[lastchance]

Like you, I assumed A(1)=1 and got, by formula, A(2)=3 ... which is not what is given.

I thought "G.f" was generating function, but it doesn't really matter.

The graph of the function grows incredibly quickly.

[Ganado]

G.f. being generating function makes a lot more sense than general formula. I'll revisit the sequence with that in mind.

[lastchance]

It goes with "generating function" I think because if you look at the power series under "Example" then the so-called "series" are the coefficients of x, x^2, x^3, x^4 etc.

Thus, you have to insert a general power series into the formula and equate coefficients of each power of x.

Looks a nightmare to code up (not least because some of those integer coefficients are huge).

[Ganado]

Thanks for the help, I crunched some of the lower powers to confirm it works once it's understood to be the generating function coefficients.

A(x) = x + x^2 - 4*x^4 + 2*x^5 + 52*x^6 + ...

A(x + A(x)^2) = A(x + (x + x^2 - 4*x^4 + 2*x^5 + 52*x^6+ ...)^2)
              = A(x + x^2 + 2x^3 + x^4 - 8x^5 - 4x^6 + 108x^7 + 120x^8 - 16x^9 - 412x^10  + ...)
			  
              = (x + x^2 + 2x^3 + x^4 - 8x^5 - 4x^6 + 108x^7 + 120x^8 - 16x^9 - 412x^10  + ...)
              + (x + x^2 + 2x^3 + x^4 - 8x^5 - 4x^6 + 108x^7 + 120x^8 - 16x^9 - 412x^10  + ...)^2
              - 4(x + x^2 + ...)^4
			  
              = (x +  x^2 + 2x^3 +  x^4 - 8x^5 - 4x^6 + 108x^7 + 120x^8 - 16x^9 - 412x^10 )
              +       x^2 + 2x^3 + 5x^4 + 6x^5 + ...
              +                  - 4x^4 + ...
              =  x + 2x^2 + 4x^3 + 2x^4 + ...
			  
x + 2*A(x)^2  =  x + 2*(x + x^2 - 4*x^4 + 2*x^5 + 52*x^6+ ...)^2
              =  x + 2x^2 + 4x^3 + 2x^4 + ...