find the pattern in this number sequence of 1's and 0's

Jelly, 11 10 bytes

Saved 1 byte thanks to @Dennis

ȷḶ×⁽q£:ȷ5Ḃ

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How?

I first noticed that the pattern alternates between runs of length 4 and length 3, skipping the length-4 step every few runs. This led me to look for a number which could be divided into the current index, then taken mod 2 and floored—i.e. retrieving the least significant bit—to give the bit at that index in the series. After much trial and error, I found that 3.41845 does exactly that, but multiplying by its approximate reciprocal (.29253) is a byte shorter.

ȷḶ×⁽q£:ȷ5Ḃ    Main link. Arguments: none
ȷ             Yield 1e3, i.e. 1000.
 Ḷ            Lowered range; yield [0, 1, 2, ..., 999].
  ×⁽q£        Multiply each item by 29253.
      :ȷ5     Floor-divide each item by 1e5, i.e. 100000.
         Ḃ    Take each item mod 2.

Dyalog APL, 99 83 82 bytes

a←{⍵/0 1}¨(↓3 2⍴4 3 3)
{a⊢←↓⍉↑a{⍺∘{⍵/⊂⍺}¨⍵}¨↓3 3⍴⍵}¨(9/5)∘⊤¨1386531 496098
1000⍴∊a

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Definitely not the intended solution as this still has a lot of hardcoded data, but it's a start.

Ruby, 34 29 26 22 bytes

$.+=184while p$./629%2

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Quick explanation: this works because of the magic number 629. I noticed that the sequence starts repeating after the 629th element, and I tried to "improve" some existing answer, using only integer math. I found that the other "magic number" (0.29253) is actually 184/629.