Question: what is the rule for working out the formula for a sequence with three lines of difference? eg.
1, 3, 7, 15, 29 1st line differences: 2, 4, 8, 14 2nd line differences: 2, 4, 6 3rd line differences: 2, 2
There is no actual rule. You are looking for a pattern that will allow you to continue the sequence. In this example the third differences are both 2. If this pattern is continued, and the next entry in the third difference row is also 2, then the 4th entry in the 2nd line of differences must be 8.
The 4th entry in the 2nd line of differences being 8 forces the 5th entry in the 1st line of differences to be 22. Finally this forces the next entry in the sequence to be 51.
The pattern that you see in some line of differences doesn't have to be a constant as in the example above. Here is another example
2 6 13 29 72 4 7 16 43 3 9 27In this example the pattern I see in the final row is that each term is three times the previous row. Thus, if this pattern repeats, then the next entry in the final row is 27x3=81 and you can then work up the rows to find the next term in the sequence.