4 … 15 … 10 … 9 … 16 … 12 … 13 … Next?
18?
Sorry, it does not appear in the Online database of integer sequences.
Chances are, just about any number can be argued as a good “next”.
There’s probably some non-mathematical solution, like “words in each sentence of the Declaration of Independence” or “verses in each chapter of the Bible” or “number of times each team in the Major Leagues, starting from the northeast, has won the world series”.
Looks like a Mensa sequence. Any hints? …
The hint, I wager, is that this is yet another SwingWing thread. I propose “YASWT” as a new standard acronym.
I got 17.
17
I got 193.
It’s 87
I’ve seen this before.
It’s a sequence of random numbers, so the next number is 46.
4 … 15 … 10 … 9 … 16 … 12 … 13 … … 275 !
Simple.
The first two pairs of entries sum to 19. The next pair sums to 18. The final number is 13, and assuming we want another pair summing to 18 my answer for the next entry is…
5.
The differences between the numbers are as follows:
+11 -5 -1 +7 -4 -1
The first pair of differences is equal to 6. The second pair of differences is equal to 6. The next pair is equal to -5.
Bollocks.
Hmm, OK then, add another one to that -5 to equal 6 and we have +11, so the next number is:
24
Or something.
My lame take on it:
4+15=19 15+10=25 10+9=19 9+16=25 16+12=25
Unfortunately the patterm seems to alter slightly and my brain is overheating trying to figure out any more…
DOH!!! (mental note: proof post before pressing submit!!)
4+15=19 15+10=25 10+9=19 9+16=25 16+12=28 12+13=25
So close to a pattern!
And now I review the OP I’ve just noticed I lost a 10 somewhere… must have hit backspace during cut’n’pasting or something…
(That’s my excuse and I’m sticking to it… 
I’m guessing 17, 14
I’m grouping them in threes myself:
4,15,10
9,16,12
13
It seems to better fit a pattern. The first column goes +5,+4. The second goes +1,+1. The third goes +2,+2. The difference between the last two columns is what’s added to the number in the first column. So (15-10)+4 =9
(16-12)+9 = 13. Where the 15 and 10 comes from, I don’t know, but I think it’s a valid pattern. You can extend it thusly:
4,15,10
9,16,12
13,17,14
16,18,16
18,19,18
19,20,20
But there’s probably a slightly better mathematical relationship that I’m missing on first glance. Still, I think 17 followed by 14 is a good answer.
Don’t know about “YASWT?” The only hint was that a portion of the sequence becomes repetitive.
The 5th differences are not constant, so if this is generated by a polynomial, its degree is at least 6.
This is intriguing. Could you fill me in on this bit of mathematical knowledge with which I am obviously unaware?
[symbol]D[/symbol][sub]h[/sub] is the finite difference operator, with [symbol]D[/symbol][sub]h[/sub]f = f(x + h) - f(x). [symbol]D[/symbol][sub]h[/sub][sup]n[/sup] is defined by [symbol]D[/symbol][sub]h[/sub][sup]1[/sup]f = [symbol]D[/symbol][sub]h[/sub]f, and [symbol]D[/symbol][sub]h[/sub][sup]n+1[/sup]f = [symbol]D[/symbol][sub]h[/sub][symbol]D[/symbol][sub]h[/sub][sup]n[/sup]f. We also define [symbol]D[/symbol][sub]h[/sub][sup]0[/sup]f = f.
It’s relatively straightforward to show that [symbol]D[/symbol][sub]h[/sub][sup]n[/sup]x[sup]n[/sup] is constant, and [symbol]D[/symbol][sub]h[/sub][sup]k[/sup]x[sup]n[/sup] = 0 whenever k > n, although the details can get messy. It’s also easy to show that [symbol]D[/symbol][sub]h[/sub][sup]n[/sup]([symbol]a[/symbol]f + [symbol]b[/symbol]g) = [symbol]aD[/symbol][sub]h[/sub][sup]n[/sup]f + [symbol]bD[/symbol][sub]h[/sub][sup]n[/sup]g. From this, it’s easy to show that [symbol]D[/symbol][sub]h[/sub][sup]degree§[/sup]p is constant for any polynomial function p.