Putnam Problem of the Day

[http://math.harvard.edu/putnam(...) !!84u21X0KjW+ No.10545260]

[math]
\text{Consider the power series expansion}
\\
\qquad \qquad \qquad \qquad \qquad \qquad \frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.
\\
\text{Prove that, for each integer }n\geq 0\text{, there is an integer }m\text{ such that}
\\
\qquad \quad \qquad \qquad \qquad \qquad \qquad a_n^2 + a_{n+1}^2 = a_m .
[/math]

[http://math.harvard.edu/putnam(...) !!84u21X0KjW+ No.10545262]

Previous Thread >>10543223

[Anonymous No.10545363]

>>10545260
[math] a_n^2 + a_{n+1}^2 = a_{2n + 2} [/math]
and [math] \displaystyle a_n = \frac{1}{2 \sqrt{2}} (\alpha^{n+1} +(-1)^n \beta^{n+1}) [/math] where [math] \alpha = \sqrt{2} + 1, \beta = \sqrt{2} - 1 [/math]

[Anonymous No.10545461]

it's just a generating function for a recursive relation lmao

[Anonymous No.10546690]

thinly veiled homework thread

[Anonymous No.10547508]

>>10545461
This.
You can do the problem without even solving the recurrence as >>10545363 unnecessarily did.

[Anonymous No.10548979]

>>10545260
>>10545363
1) my wife flan is so cute
2) how do you put you equations into 4chan(nel) like that?

[Anonymous No.10549021]

>>10548979
FUCKING LEAVE
KILL YOURSELF YOU SCHIZOPHRENIC FUCK
YOU SHIT UP /JP/ WITH YOUR SHITPOSTING YOU WILL NOT DO THE SAME TO THIS BOARD
FUCK OFF AND DIE
TAKE YOUR PILLS AND OVERDOSE MOTHERFUCKER
FUCK YOU

[Anonymous No.10549267]

>>10549021
dude chill

[Anonymous No.10550420]

>>10547508
>massage the equation
kek