We are organising a trip in the car. Someone asks, “What about the dog?” I reply, “I counted the dog with the children.” That means that I put the dog in the same category [group] as the children.
Suppose that we are counting our stock, and someone asks me why I got 56 for a type of part, when they can see that there are about 25. I say, “I counted with the plastic ones included.” That means that the other person was only thinking of the metal ones.
You don’t say “Count with me.”; you say “Include me in your counting”. For your example about an equation, where you say, “Yes, I counted with that.”… you would say, “Yes, I did include that.”
( [Duplicating “mlecoz”.] “Count me in!” means, for instance, “I want to come too!”.)
“I counted on him being there.” means that it has turned out badly because he was not there. (It does not mean • that he did, nor • that he did not, know that I was counting on his being there [or “him being there”].)
“I am counting on you!” means that I need you to do what we decided you would, and otherwise everying will go awry [badly].
I can say, “I am banking on them not realising what is going on, until the door opens.”… to mean that I expect that they will not realise {whatever it is}, and that, if they do realise, my plan will fail.
“Remember that tomorrow we are leaving early.”
“Keep in mind that we are leaving early, tomorrow.”
“He is leaving in two weeks, so you need to keep that in mind.”
“He is leaving in two weeks, so we need to plan accordingly.”
“He is leaving in two weeks, so after that it will be too late.”
“He is leaving in two weeks, so you need to get it done by then.”
“Just be aware that he is leaving in two weeks.”
“He is leaving in two weeks, so make sure that you do not tell anyone before then.”
[I need more context, for this.]
“I included the velocity, so the result is correct.”
“I calculated the velocity using the [fancy maths term] equation, so the result is correct.”
[Actually, you would say, “… the result should be correct.”]
“I realised that the velocity is squared, in this case, so… .”
“I was aware that… .”
“I remembered that… .”
“I know that we are supposed to do this using the [fancy maths term] method, because of the [] being 0, so I used the linear velocity equation.”
“Don’t forget that I want to be there when you talk to him.”
“Don’t forget that I need to go with you to the shop.”
“I am depending on you to be there by 5pm.”
“I am depending on the train being a few minutes late.”
“It all depends on how the weather is over the next few days.”
“It [the plan] all depends on the English not making it through to this bridge until after sunset.”
Less common:
“His calculations [plot] depended on John talking to Mary before the concert.”
