[Borek]

This one is completely different. I remember answering this question in the past, but I don't remember where I got it from.

Add 4 more substances to the list:

H₂, Li, He, Be, N₂, B, O₂, C, F₂, Na, Ne, Mg...

[UG]

P Al S Si 

[XGen]

P₄, Al, S₈, Si?

Basically UG but in standard states?

[Borek]

Neither is what I expected. Can you explain the logic behind your answer?

[XGen]

This is borderline guess, but the first, third, fifth, etc. elements progress along the periodic table and so do the second, forth, sixth, etc elements.

[Borek]

I see.

OK, it is not the right answer 

[Ann1234]

H2, Li, He, Be, N2, B, O2, C, F2, Na, Ne, Mg...

I took the "math sequence" path-I may be way out but - I'll give this a try:

first I put the elements next to its atomic number- Then I found what do I need to add or subtract (in atomic number units) to get the next element:for example H is 1, plus 2 gives Li; Li is 3, minus 1 gives He, and so on:

H2--1  (+2)
Li---3  (-1)
He--2  (+2)

Be--4  (+3)
N2--7  (-2)
B---5  (+3)

02--8  (-2)
C---6  (+3)
F2--9  (+2)

Na--11 (-1)
Ne--10 (+2)
Mg--12  (?)

Then I studied the sequence of signs:
(+ - +) ( + - +)   (- + +) (-  +  ?)   ( ? ? ?)  ( ? ? ?)  and I continued the sequence as follows:
(+ - +) ( + - +)   (- + +) ( - +  +)   (+ - +)  (+ - +).

Lastly the number sequence:
(2 1 2) (3 2 3) (2 3 2)  (1 2 ?) ( ? ? ?) (? ? ?) I continued the sequence as:
(2 1 2) (3 2 3) (2 3 2 ) (1 2 1) (2 1 2) (3 2 3)

so putting all together: this will the completed sequence:
( +2 -1 +2) (+ 3 -2 +3) ( -2 +3 +2) (- 1+ 2+ 1) (+2 -1 +2) ( +3 -2 +3)

so completing the sequence on the top:

H2--1  (+2)
Li---3  (-1)
He--2  (+2)

Be--4  (+3)
N2--7  (-2)
B---5  (+3)

02--8  (-2)
C---6  (+3)
F2--9  (+2)

Na--11 (-1)
Ne--10 (+2)
Mg--12 (+1)

Al ---13 (+2)
P4----15 (-1)
Si---14 (+2)

S8----16

and I wrote them in standard states as all are on the list:so Al, P₄, Si, S₈ is my answer  

[Fluorine]

I think I got it after trying out many strange approaches for 20min...Cl, K, Ar, Ca?

Highlight text for explanation (possible spoiler): They are following 'gas, solid, gas, etc.' pattern on the periodic table going 1st gas, 1st solid, 2nd gas, 2nd solid so on. Very cool!

[Ann1234]

you' re right =) I think you got it- real cool!

[UG]

This question is definitely a different one, keen to see the answer.
And that's a lot of typing Ann1234

[Borek]

I think I got it after trying out many strange approaches for 20min...Cl, K, Ar, Ca?

You are on the right track - but still not there.

[DrCMS]

Cl₂, Al, Ar, Si?

[Borek]

Cl₂, Al, Ar, Si?

Yep 

[Fluorine]

Oops, I see where I made the silly mistakes.