Problem 6

A sequence

(u_n)

is defined by

u_0=2

u_1=\frac52

, and

u_{n+1}=u_n(u_{n-1}^2-2)-u_1

for

n=1,2,\ldots

. Prove that for every positive integer

n

\lfloor u_n\rfloor=2^{\frac{2^n-(-1)^n}{3}}

, where

\lfloor x\rfloor

denotes the greatest integer less than or equal to

x

Step 2 of 6: Introduce an auxiliary integer sequence

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In plain words

The exponent sequence grows by the same two-step rule that appears when multiplying the exponential expressions in the recurrence.

x_0=0,\quad x_1=1,\quad x_n=x_{n-1}+2x_{n-2}

Detailed analysis

Define $x_0=0$ , $x_1=1$ , and $x_n=x_{n-1}+2x_{n-2}$ for $n\ge2$ . Its characteristic equation is $r^2-r-2=0$ , with roots $2$ and $-1$ , so $x_n=\dfrac{2^n-(-1)^n}{3}$ .