Pseudorandom increasing sequence
Given that you said it would be enough for the sequence to be unpredictable, here is a very simple solution.  The $i$th value in the sequence is given by

$$x_i = 2^{128} i + y_i$$

where $y_i$ is any pseudorandom sequence where you can efficiently find the $i$th value ($y_i$) and where each $y_i$ is 128 bits wide (i.e., $0 \le y_i &lt; 2^{128}$).  For instance, you could use

$$y_i = F(K,i)$$

where $F$ is any pseudorandom function (e.g., AES) with a 128-bit output, and $K$ is a key (the seed from which the entire sequence is generated).