What and why is the next number in the following sequence of 2 6 3 9 6 18?

Answer 1

Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 6 such that the first six numbers are as listed in the question followed by the chosen next number. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

What number would you like? For example,

• if you want 1 then use the rule U(n) = (-31*n^6 + 727*n^5 - 6735*n^4 + 31265*n^3 - 75734*n^2 + 89268*n - 38620)/120;
• if you want 3 then use the rule U(n) = (-46*n^6 + 1080*n^5 - 10015*n^4 + 46530*n^3 - 112789*n^2 + 133020*n - 57420)/180;
• if you want 11 then use the rule U(n) = (-44*n^6 + 1038*n^5 - 9665*n^4 + 45060*n^3 - 109541*n^2 + 129492*n - 55980)/180;

and so on.