Explicit formula for a sequence of alternating differences Imagine this sequence: $1, 11, 12, 22, 23, 33, 34, 44$, ...
Obviously the sequence is alternating in adding 10 and adding 1. You can find the formula for the differences easily with $5*(-1)^x+6$, but I can't find a way to find the exact explicit formula for the sequence, assuming there is one. 

If you try to find a formula for the terms in positions $2,4,6,\ldots$ which are   $11,22,33,\ldots$ you get $11 \frac{x}2 = \frac{11}{2}x$
If you try to find a formula for the terms in positions $1,3,5,\ldots$ which are   $1,12,23,\ldots$ you get $11 \frac{x-1}2 +1 = \frac{11}{2}x-\frac{9}{2}$ 
These expressions are of the form $\frac{11}{2}x-\frac{9}{4} \pm \frac{9}{4}$  and you can see that the $x$th term is  $$\frac{11}{2}x-\frac{9}{4} + \frac{9}{4}(-1)^x$$
