The number is
416
As @MikeQ and @boboquack noted in the comments,
The italicised letters are ioi iio ioi, which, when taking "basic" aka hint to binary", leads to 565, the first number in the "sequence". In addition, each line in the blockquote has three italicised letters, confirming this interpretation.
If we then do this for each number, we get
101 110 101
111 010 001
101 111 101
100 010 101
110 110 111
formatting according to the "looking down upon this sequence" hint.
Now, we want three digits, and our grid is neatly split into three blocks, but none look particularly like a digit. However, with some nudging from the OP,
we can re-group the bits so that all the fours bits are together, all the twos bits are together, and all the ones bits are together. For example, looking at the first row, the fours bits from left to right are 1__ 1__ 1__
, the twos bits are _0_ _1_ _0_
, and the ones bits are __1 __0 __1
. We can put them together to form 111 010 101
.
Doing so for each row yields
111 010 101
100 110 101
111 010 111
101 010 001
111 111 001
If you can't see it yet:
*** * * *
* ** * *
*** * ***
* * * *
*** *** *
That doesn't start with a 4 though.
Let's flip it. (Or, more technically, take the bits in little endian order).
* * * ***
* * ** *
*** * ***
..* * * *
..* *** ***
101
,110
,101
. $\endgroup$