How to solve this number-sequence?

Question

My friend gave me this fun puzzle to solve but I still cannot figure out the logic between them...

1,3,4,7,8,11,14,15,19,22,23,25,26,?,?,?,?

Could anyone help me solve it?

Answer 1

I think the answer is:

29, 30, 33, 34

I found that the sequence has the following property:

A number is in the sequence IF AND ONLY IF it is the sum of a consecutive sequence of other numbers in the pattern.

Here is a demonstration (not quite a proof, but convincing enough) that the property holds, and how I got the next numbers in the sequence:

Start with {1, 3}. (You have to start somewhere, right?) Add them:
1 + 3 = 4.

Add 4 to the sequence. Now we have {1, 3, 4}. Take each consecutive sub-sequence we haven't already considered:
3 + 4 = 7
1 + 3 + 4 = 8
(Note that we do not consider 1 + 4 = 5 because 1 and 4 are not consecutive in the sequence.)

Now the sequence is {1, 3, 4, 7, 8}.

Keep adding consecutive subsequences, then adding the results to the sequence.
*NOTE: Be careful of the order you do this in! Notice if we add 3 + 4 = 7, then we add 7 to the sequence and calculate 4 + 7 = 11, we have skipped over 8 (oh no!) and 7 + 11 = 18 is, in fact, NOT in the sequence. I would advise going step by step, in increasing order, calculating ALL possible sub-sequence sums for one number before moving on to the next. (i.e. Finish subsequences whose biggest number is 4 before adding 7 into the mix.)
Let's repeat for 7 and 8, just to check that this works...

4 + 7 = 11
3 + 4 + 7 = 14
1 + 3 + 4 + 7 = 15

7 + 8 = 15
4 + 7 + 8 = 19
3 + 4 + 7 + 8 = 22
1 + 3 + 4 + 7 + 8 = 23

Now we've added all subsequences of {1, 3, 4, 7, 8} and got {1, 3, 4, 7, 8, 11, 14, 15, 19, 22, 23}.
Note that at this point, we can't be sure that we aren't missing some numbers near the end of our partially-built sequence. But, we can "Lock in" {1, 3, 4, 7, 8} as the first 5 numbers, and we can guarantee that 11 is the next number in the sequence. We can also guarantee that 14, 15, 19, 22, and 23 are in the sequence (but we can't guarantee that there are no others which we haven't found yet.)
I won't show all my work for the rest, but I'll tell you the first time I saw each of the next numbers appear:

11 + 14 = 25
7 + 8 + 11 = 26
14 + 15 = 29 (Notice we have to work all the way up to 15 to find this one!)
4 + 7 + 8 + 11 = 30
3 + 4 + 7 + 8 + 11 = 33 (and 8 + 11 + 14 = 33)
1 + 3 + 4 + 7 + 8 + 11 = 34

After adding all consecutive subsequences with 15 as the greatest number, we have:
{1, 3, 4, 7, 8, 11, 14, 15, 19, 22, 23, 25, 26, 29, 30, 33, 34, 40, 44, 47, 48, 55, 59, 62, 63}
Of course, we can only verify that we aren't missing numbers up to 19. To fill in the rest, you keep following the steps above.

Answer 2

Possible answer:

29, 30, 33, 36

Reason:

1(+2)=3(+1)=4(+3)=7(+1)=8(+3)=11(+3)=14(+1)=15(+4)=19(+3)=22(+1)=23(+2)=25(+1)=26 The pattern resets at 23 with a +2.

Answer 3

The answer could be:

27, 30, 33, 37

because:

I honestly don't know how to describe the lengthy and unmathematical reasoning I came up with but if I'm correct at all I'll try and explain. Although I doubt I am. It basically came down to adding up previous numbers in the series in a pattern, so 4 = 1 + 3, 7 = 3 + 4, 11 = 1 + 3 + 7, 14 = 3 + 4 + 7, 15 = 1 + 3 + 4 + 7, 19 = 1 + 3 + 4 + 11, 22 = 1 + 3 + 7 + 11, 25 = 1 + 3 + 7 + 14, 26 = 1 + 3 + 7 + 15. At this point I assumed that one and three were mandatory numbers, and that the 4 and the 7 in the equations played on a pattern; at first there were no four or sevens, then one four, then one seven, then two fours, then two sevens. I added this pattern and then just continued adding on numbers in the sequence. Is this correct? No. Probably not. Did it make sense? No. Had I slept? No.