In this case you have to use "an". The combination of "denken" + "an" is basicly a set phrase meaning "to think about/of sth." As a native speaker I can only think of two instances where you wold use "über" with the verb "denken".
I. The closely related verb "nachdenken" meaning "to think deeply" or "to contemplate" always uses "über" when used as a transitive verb.
Modified example: "Wir denken über euren Besuch nach."
II. Expressing your feelings about a person. But honestly, this usage is really rare when used affirmatively as in example b). Example a) is a relatively common phrase though.
a) "Wie denkt er über mich?" - "What does he think of me?" b) "Meine Schwiegereltern denken schlecht über mich." - "My parents-in-law think badly of me."
Could someone please tell me why it's "euren" in this sentence? That's accusative, right? I know "an" is one of the two-way prepositions that sometimes take dative & sometimes accusative. The "Tips and notes" says with two-way prepositions, that if there is movement from one place to another, then it's accusative, but if there's no movement, or movement within a certain place, then it's dative. So, I thought it would be dative.
Your comment was made two years ago, have you found out why already? I have the same doubt.
Try to find on the internet "verbs with prepositions/verben mit präpositionen", because it's a "denken+an+Akk." structure, and you'll find out why it takes the accusative case.
Probably because that is not the translation the course creators had in mind, and it's not the first one that comes to mind in my opinion. However, "guest(s)" should be a possible translation for "Besuch (m)", although "visitor(s)" might be a bit more literal. In practice I would prefer using "Gast" instead in this context, to avoid confusion.
Perhaps not "we are thinking about your guest" as I would expect a guest to be a Besucher, but I was curious why not "we are thinking about/considering your company." (Your company as in "your guest(s)", "your presence/companionship", or as in "your company/flock/troupe" [a group of people whos society one shares], not a company as in business or military structure). Besuch was used as a polite way to refer to one's company in a previous Duo lesson. At least something to the effect of "sie/er hat Besuch" was explained to me to be a polite way to say someone has company without specifying number or gender. Is Besuch in that sense just a gender/quantity neutral abbreviation of Besucher((in)nen) or can it mean company in the broader sense?
I had understood that in cases of the split verbs, the preposition goes to the end of the phrase. Why is the word order different in this case, with the an right after the verb? Why is it not " Wir denken euren Besuch an"?