Calvin’s comments on John’s prologue are all really good. Here he discusses the relationship between faith and regeneration, and it shows that he is a certainly aware of the difficulties in making any sort of an ordo. He opts for a both/and approach:
It may be thought that the Evangelist reverses the natural order by making regeneration to precede faith, whereas, on the contrary, it is an effect of faith, and therefore ought to be placed later. I reply, that both statements perfectly agree; because by faith we receive the incorruptible seed, (1 Peter 1:23,) by which we are born again to a new and divine life. And yet faith itself is a work of the Holy Spirit, who dwells in none but the children of God. So then, in various respects, faith is a part of our regeneration, and an entrance into the kingdom of God, that he may reckon us among his children. The illumination of our minds by the Holy Spirit belongs to our renewal, and thus faith flows from regeneration as from its source; but since it is by the same faith that we receive Christ, who sanctifies us by his Spirit, on that account it is said to be the beginning of our adoption.
On John 1: 13
We can certainly sympathize with Calvin here. On the one hand, we wish to say that faith is an effect of regeneration, for only those who are born of God can accept the gospel offer. But on the other hand, new life is located in the person of Christ, and thus we must lay hold of him in faith before we can be created anew.
Calvin’s answer is “Yes.”
Jesus Christ is the regeneration, and He must be apprehended by faith, yet we cannot believe until the Holy Spirit first enlightens us.
Wedgewords 
So, you’re saying Calvin was an Arminian… : )