Problem #7

This month's problem was created by Philippe Fondanaiche.

In order to manage my overdrafts, I have found an agreement with my banker who is as fond of mathematical puzzles as I am. My bank account is updated every month. The agreement is the following:

In January, the sum of every series of 3 consecutive debits/credits is negative, but the sum of every series of 5 consecutive debits/credits is positive.
In February, the sum of every series of 4 consecutive debits/credits is negative, but the sum of every series of 6 consecutive debits/credits is positive.

In month k, the sum of every series of k + 2 consecutive debits/credits is negative, but the sum of every series of k + 4 consecutive debits/credits is positive.

For each month of the year, determine the maximum number of bank transactions that I can make subject to these conditions.

Note: This is a generalization of an International Mathematical Olympiad problem.