Question

Find the formula for the given two series:
{1,2,4,5,7,8,10,11,13,14,16,17,19,20,22...}
{1,2,4,5,8,9,13,14,19,20,26,27,34,35,43...}

1st series = +1, +2, +1, +2, +1, +2...
2nd series = +1, +2, +1, +3, +1, +4...

Answer 1

formula as in.... like arithmetic sequence formula?

Answer 2

No, the difference between the terms aren't constant

Answer 3

right, but we can still write a formula

Answer 4

Yes, but heavens if I can find it

Answer 5

wait i almost got it

Answer 6

nope nevermind sorry

Answer 7

You can define the associated sequences recursively. Not so sure about a closed formula for the \(n\)-th term though...

For the first sequence, you could say
\[\begin{cases}a_n=\begin{cases}a_{n-1}+2&\text{for odd }n\\a_{n-1}+1&\text{for even }n\end{cases}\\a_1=1\end{cases}\]
For the second,
\[\begin{cases}a_n=\begin{cases}a_{n-1}+\left\lceil\dfrac{\color{red}{n+1}}{2}\right\rceil&\text{for odd }n\\a_{n-1}+1&\text{for even }n\end{cases}\\a_1=1\end{cases}\]
(\(n+1\) can also be replaced with \(n+2\)).