Consider the squares alternating with the triangular numbers $$a_n=0,0,1,1,4,3,9,6,16,10,25,15,36,21,49,28,64,36,81,\ldots$$
Is it trivial that $4a_n+1$ is the largest odd divisor of $n^2+1$. In particular it appears that $$4a_n+1 = 1,1,5,5,17,13,37,25,65,41,101,61,145,\ldots$$ In OEIS language:$\text{ }$$\text{ }$$4*$A123596$+1=$A228564.
This is just based on pure speculation and checking out the first few terms. A counter example would be nice.