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.
The rule for this sequence appears to be adding consecutive prime numbers. The sequence starts with 3, then adds the next prime number 2 to get 4. It then adds 3 (the next prime number) to 4 to get 7, then 5 to 7 to get 12, and so on. Each number in the sequence is the sum of the previous number and the next prime number in order.
3
19. Add 3, then add 4, then add 5, then add 6,etc.
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.
The rule for this sequence appears to be adding consecutive prime numbers. The sequence starts with 3, then adds the next prime number 2 to get 4. It then adds 3 (the next prime number) to 4 to get 7, then 5 to 7 to get 12, and so on. Each number in the sequence is the sum of the previous number and the next prime number in order.
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