You don't want to sell me matchsticks

Question

Fix the image before completing the sequence. What do I want you to draw your attention to?

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103, 91, 79, 0, x, x, 28, x, x, x, 0, 78, x, x, x, x, x, x, 5

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Hints:

You need to place 11 new matches into the image to make it correct. The image represents something well-known that's needed to understand the sequence.

16x8, overlapped

Some weird patterns, huh?... The center matchstick of each 1x2 rectangle seems to always repeat in the pattern 01010101. Isn't it peculiar that halfway down the image, there's always a straight horizontal line of matchsticks across the entire width? What are the dimensions of the grid again? You can also look at the columns for potential hints, it's not like they're patternless. Take a close look at the raw patterns found in Lypyrhythm's partial answer. Anything weird you notice there?
Always keep in mind the 1x2 rectangles have overlapping edges...

Answer 1

Lukas wants to draw our attention to …

… the PSE Advent Calendar.

The completed sequence is:

103, 91, 79, 0, 119, 61, 28, 79, 21, 15, 0, 78, 119, 14, 79, 21, 61, 119, 5

The matches represent …

… the numbers from 0 to 127 in binary form. Each 1×2 “box” has seven possible positions for matches. These positions are bits in a binary representation of a seven-bit number. Bit 0 is the centre match, bits 1 through 6 are anticlockwise starting from the top left upright match.

These numbers are arranged in a straightforward fashion: Row-wise starting from 0 in the top left corner to 127 in the bottom right corner. The boxes overlap, so that a match on the border could be a bit that’s set on either of the neighbours or on both.

     -- 6 --
    |       |             (1 << 0) == 1
    1       5             (1 << 1) == 2
    |       |             (1 << 2) == 4
     -- 0 --              (1 << 3) == 8
    |       |             (1 << 4) == 16
    2       4             (1 << 5) == 32
    |       |             (1 << 6) == 64
     -- 3 --

And the sequence?

The seven-match boxes are, of course, seven-segment displays in disguise. You can assemble the bits of the numbers or look up the index in the match pattern. (But be careful with overlapping matches.) The numbers read:

 103  91  79   0   x   x  28   x   x   x   0  78   x   x   x   x   x   x   5
  _   _   _       _   _       _   _   _       _   _   _   _   _   _   _
 |_| |_  |_      |_| |_|     |_| |_| |_|     |   |_| |_| |_| |_| |_| |_|  _
 |    _| |_      |_| |_| |_| |_| |_| |_|     |_  |_| |_| |_| |_| |_| |_| |

The missing matches …

… had been identified by Lypyrhythm based on the regular pattern of the binary boxes. Now that we know what the boxes are, we can verify that finding.

Eleven machsticks are missing and eleven numbers in the sequence are missing. If we look at the missing matches, we can see that they all belong to letter shapes. (There are some minor ambiguities: The stem of the d could belong to a U and the stem of the A could be part of an N. These disambiguities are resolved by the fact that u/v and n are already represented unambiguously by lower-case seven-stick symbols.)

Below are the letters drawn out, i.e. no longer overlapped, and with the missing sticks highlighted with ugly DOS-blocks. (Sorry for that.)

    _ |  |_     _ |  |_     _ |  |_     _ |  ▐_
            |  |  |  |   _  _  _  _ |_ |_ ▐_ |_

    _ |  |_     ▄ |  |_     _ |  |_     _ |  |_
  |  |  |  || |▐ || || | _| _| _| _||_||_||_||_|

  | _|| ||_|  | _|| ||_|  | _|| ||_|  | _|| ||_|
            |  |  |  |   _  _  _  _ |_ |_ |_ |_

  | _|| ||_|  | _|| ||_|  | _|| ||_|  | ▄▌| ||_|
  |  |  |  || || || || | _| _| _| _||_||_||_||_|
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
    _ |  |_     _ |  |_     _ |  |_     _ |  ▐▄
            |  |  |  |   _  _  _  _ |_ |_ |_ |_
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
    _ |  |_     _ |  |_     _ |  |_     _ |  |_
  |  |  |  || || || || | _| _| _| _||_||_||_||_|
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
  | _|| ||_|  | _|| ||_|  | _|| ||_|  | _|| ||_|
            |  |  |  |   _  _  _  _ |_ |_ |_ |_
 _  _  _  _  _  _  _  _  _  _  _  _  _  _  _  _
  | _|| ||_|  | _|| |▐▄|  | _|| ||_|  | _|| ||_|
  |  |  |  || || || || ▌ _| _| _| _||_||_||_||_|

We can fill in the blanks:

The x's are:

    3 × A   2 × E   2 × d   2 × n   1 × L   1 × t

and we get:

 103  91  79   0 119  61  28  79  21  15   0  78 119  14  79  21  61 119   5
  _   _   _       _           _               _   _       _           _
 |_| |_  |_      |_|  _|     |_   _  |_      |   |_| |   |_   _   _| |_|  _
 |    _| |_      | | |_| |_| |_  | | |_      |_  | | |_  |_  | | |_| | | |

Answer 2

Partial answer

Look at the matchsticks one by one from left to right – you should realize they are placed in a particular order. As you can see in the image below, the pattern in the first row is:

• blank, blank, stick, stick, blank, blank, stick, stick, etc.

The second row consists of horizontal matchsticks placed in the following sequence:

• blank, stick, blank, stick, blank, stick, etc.

Do this for the remaining 30 rows of matchsticks. If you notice there is a gap where there is supposed to be a matchstick, insert one to correct the pattern. They should add up to exactly 11 new matchsticks (1 = matchstick, 0 = gap):

1. 001100110011001(1)0
2. 0101010101010101
3. 00001111000011(1)10
4. 0000000011111111
5. 00110011001100110
6. 01010(1)0101010101
7. 1111(1)11111111111
8. 0000000011111111
9. 1111111111111111
10. 0101010101010101
11. 00001111000011110
12. 0000000011111111
13. 1111111111111(1)11
14. 0101010101010(1)01
15. 1111111111111111
16. 1111111111111111
17. 001100110011001(1)0
18. 010101010101010(1)
19. 00001111000011110
20. 1111111111111111
21. 00110011001100110
22. 0101010101010101
23. 1111111111111111
24. 1111111111111111
25. 1111111111111111
26. 0101010101010101
27. 00001111000011110
28. 1111111111111111
29. 111111(1)111111111
30. 0101010(1)01010101
31. 1111111(1)11111111
32. 0000000011111111

The 11 matchsticks to add are highlighted green:

I am still trying to figure out what the corrected image is supposed to represent. It somehow reminds me of a rot13(sybbe cyna bs n (jryy-xabja?) ohvyqvat).