If there's one child that can be zero interactions if there are two children the three interactions if their three children they can be 12 interactions if there are four children there can be 39 interactions which recursivity equation represents the pattern

T1 = 0

Tn+1 = 3Tn+3

To find the recursive equation that represents this pattern, we need to analyze the given information and identify the relationship between the number of children and the number of interactions.

Let's break down the information given:

- If there is one child, there are zero interactions.
- If there are two children, there are three interactions.
- If there are three children, there are twelve interactions.
- If there are four children, there are thirty-nine interactions.

We can see that the number of interactions is increasing as the number of children increases. Let's try to find the relationship between the number of children (n) and the number of interactions (I).

From the given information, we can observe that the number of interactions is related to the number of children by the following pattern:

- Number of children: 1; Number of interactions: 0
- Number of children: 2; Number of interactions: 3 = (2 * 2) - 1
- Number of children: 3; Number of interactions: 12 = (3 * 3) + 3
- Number of children: 4; Number of interactions: 39 = (4 * 4) + (4 * 3) + 3

By analyzing the above pattern, we can observe that for each additional child, the number of interactions increases by the square of the child's number, plus the product of the child's number and (child's number - 1), plus 3.

Therefore, the recursive equation that represents this pattern is:

I(n) = (n * n) + (n * (n - 1)) + 3

Where:
I(n) represents the number of interactions when there are 'n' children.

Note: It is important to note that this analysis assumes each child will have interactions with every other child, resulting in the number of interactions being equal to the total possible interactions among the children.