Given any number it is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question and the next is the given number. There are also non-polynomial solutions.
The simplest polynomial solution, of order 4, is
Un = (-n4 + 10n3 - 11n2 - 142n + 456)/24 for n = 1, 2, 3, ...
and accordingly, the next in the sequence is U6 = 3.
The nest number in the sequence is 18. Note that the difference between each number and the next number in the sequence follows the simple sequence of 1,2,3,4. Obviously the next in the sequence of increases is 5, so 13+5=18.
The next number in the sequence 5-7-13-31 is 69.
3
19. Add 3, then add 4, then add 5, then add 6,etc.
18. 3+1=4, 4+2=6, 6+3=9, 9+4=13, 13+5=18.
The nest number in the sequence is 18. Note that the difference between each number and the next number in the sequence follows the simple sequence of 1,2,3,4. Obviously the next in the sequence of increases is 5, so 13+5=18.
The next number in the sequence 5-7-13-31 is 69.
13
The next number in the sequence is 27.
Every next number is increased by 5. The next number in the sequence is 23.
3
19. Add 3, then add 4, then add 5, then add 6,etc.
18. 3+1=4, 4+2=6, 6+3=9, 9+4=13, 13+5=18.
-17, -21 Pattern is subtract 4.
25 is the next number that appears in that sequence.
5, 11, 17
34