What is the sequence 768 747 785 761 802 775?

Answer 1

There are infinitely many sequences that start with those 6 values.

However, if you look at the differences between each term and the next you will find:

768 - 21 = 747 + 38 = 785

785 - 24 = 761 + 41 = 802

802 - 27 = 775

So it looks like a sequence of alternately subtracting and adding something, the value of which increases by 3 each time: -21, +38, -24, +41, -27, ...

Alternatively it can be seen as two sequences interlaced, one term from each in sequence:

768, 785, 802, ... where each term is 17 more than the last: U{n} = 751 + 17n for n = 1, 2, 3, ...

747, 761, 775, ... where each term is 14 more than the last: U{n} = 733 + 14n for n = 1, 2, 3, ...

Combining the two gives:

U{n} = 751 + 17(n+1)/2 for n = odd (1, 3, 5, ...)

U{n} = 733 + 7n for n = even (2, 4, 6, ...)

Answer 2

According to Wittgenstein's Finite Rule Paradox every finite sequence of numbers can be a described in infinitely many ways and so can be continued any of these ways - some simple, some complicated but all equally valid.

The simplest polynomial solution is:

Un = (-127*n5 + 2215*n4 - 14500*n3 + 43940*n2 - 60238*n + 51750)/30, for n = 1, 2, 3, ...