Credit card interest is generally compounded daily. So if you want to compute this accurately, you need to figure out how many days you are being charged interest. There is usually a grace period for new purchases, where you aren't charged any interest until your first bill; however, the grace period only applies if you pay your statement in full. Since you were already carrying a balance on your card before this purchase, there was no grace period. The number of days would be from the date of purchase to the date of payment.
The formula for compound interest is:
where P is the principal ($3700), r is the daily interest rate (16.5%/365 = 0.0004521), and n is the number of days you are charging interest.
If the number of days is 330 (about 11 months), the interest would be $595.11.
This assumes that he didn't pay anything to you until the one payment after 11 months.
In the comments below, the OP said:
by having that amount on the card, the minimum payment was about $200 a month, and I made little headway in ever reducing the debt. Had that amount not been on my card, I would have been paying the same amount and making headway on the debt. Having that charge on my card cost me money, that is for certain. I'm trying to work out how much.
Your $200 monthly payment covered all of your interest and your friend's interest and helped pay down your debt. Yes, if you didn't have this purchase on your card and were still paying the $200 each month, you would have reduced your debt more. However, your friend did give you $500 in early payments, which helped you make your credit card payment.
Each of these monthly payments that you made includes a small amount that goes toward reducing your debt (principal). By charging your friend the $595 calculated above, you are crediting all of these principal payments for your own debt (as you should, since your friend wasn't really making any payments). However, you could have paid more toward the principal, and chose not to. It's not really your friend's fault that you didn't pay more.
However, I think I see where you are coming from. You are saying that because you had to pay his portion of the interest over these months, you missed out on the extra debt reduction and had to pay additional interest on your portion. For example, in the first month, $50.51 of your payment was for his interest. If that $50.51 had gone toward your principal instead, you would have saved $8.12 in interest over the next 11 months. Each of the following months you would have less savings, because there are fewer months left for you to save interest over. I put together a spreadsheet that shows the "lost savings" each month:
Based on this premise, having to pay your friend's interest each month means you lost out on $49.84 that you could have saved if you had instead been paying down your principal.
However, we need to look at the other side of this in Column B. You'll notice that your friend's interest charge goes up each month, because he hasn't been paying on it. However, you have been paying it each month, which means that the interest you have been paying on it has essentially been a flat $50.51 each month. By charging your friend the increasing amounts shown in column B, you are already essentially collecting the "lost savings" in column D. If you charge him both, you are really charging him twice, in my opinion.