Recassim Succession¶

Let’s consider an infinite mathematical succession defined for a given $a_0 \geq 0$ by the following recursive rules:

\[\begin{split}a_i= \begin{cases} a_0, &\mbox{if } i = 0 \\ a_{i-1} - i, &\mbox{if } t = (a_{i-1}-i) \mbox{ is such that } t>0 \mbox{ and } t \mbox{ does no belong to the succession so far.} \\ a_{i-1} + i, &\mbox{otherwise} \end{cases}\end{split}\]

for all $i \geq 0$. For instance, given $a_0 = 0$ the initial terms of the succession are: $$ 0,1,3,6,2,7,13,20,12,21,11,⋯ $$

Observe that, on the one hand the 3rd term is 6 since the 2nd term is 3 and 3-3 is not >0 therefore $t_3 = 3+3 = 6$. On the other hand, $t_4 = 2$ since $t_3−4 = 2$.

The next image graphically shows the first 100 terms for $a_0 = 0$:

Please implement the following Python function in the recassim module (recassim.py file):

: recassim(a0, n):

such that

given

a0, an int such that a0 >= 0.

n, an int such that belongs to the succession.

returns the index of the first occurrence of n in the succession (notice that a value may appear more than once).

For example,

>>> recassim(0, 1)
1
>>> recassim(0, 7)
5
>>> recassim(0, 20)
7

Doctests are available at the recassim.test file.