alternating each Week

You'd need to know the number of weeks in each year, so you could work out whether your condition was enabled for odd weeks or even weeks. You can't just work out whether this year or last year has an even or odd number of weeks in the year, as it's basically cumulative.

If you only need this for a limited range of dates (e.g. 2000 to 2100) then you could just calculate a table. If you need it over a very wide range of dates, you could build up a table of 400 years and take advantage of everything cycling every 400 years.

Example of the simpler case, using the java.time API:

For example:

private static final int MIN_YEAR_INCLUSIVE = 2000;
private static final int MAX_YEAR_EXCLUSIVE = 2100;
private static final boolean[] ENABLED_FOR_ODD_WEEKS = 
    buildTable(MIN_YEAR_INCLUSIVE, MAX_YEAR_EXCLUSIVE);

private static final boolean[] buildTable(int minYear, int maxYear) {
    boolean[] table = new boolean[maxYear - minYear];        
    boolean current = true;
    for (int year = minYear; year < maxYear; year++) {
        table[year - minYear] = current;
        Year accessor = Year.of(year);
        int weeksInYear = IsoFields
            .WEEK_OF_WEEK_BASED_YEAR
            .rangeRefinedBy(accessor)
            .getMaximum();
        if (weeksInYear == 53) {
            current = !current;
        }
    }
}

public boolean isEnabled(LocalDate date) {
    int year = date.getYear();
    if (year < MIN_YEAR_INCLUSIVE || year >= MAX_YEAR_EXCLUSIVE) {
        throw new IllegalArgumentException("Invalid year: " + year);
    }
    int weekOfWeekYear = date.get(IsoFields.WEEK_OF_WEEK_BASED_YEAR);
    boolean oddWeek = weekOfWeekYear % 2 == 1;
    return oddWeek == ENABLED_FOR_ODD_WEEKS[year - MIN_YEAR_INCLUSIVE];
}