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C#

By using 7 that is allowed by question, start everything of wordcount from "0000"

int four ="0000".Length;
int eight = four + four;
int fifteen = eight + 7;
int sixteen = eight + eight;
int twentythree = sixteen + 7;
int fourtytwo = four + fifteen + twentythree;
List<int> list = new List<int> { four, eight, fifteen, sixteen, twentythree, fourtytwo };

//while (true) //for (; ;)
//{
//    foreach (int number in list)
//    {
//        Console.WriteLine(number);
//    }
//}

//infinite loop via number 0,7,9 that are allowed
//for (int i=0; i<7; i++)
//{
//    Console.WriteLine(list[i]);
//    if (i == (7-9+7))
//        i = 0;
//}

for (int i=0;;i++)
{
    Console.WriteLine(list[i]);
    i = i % (7-9+7);
}

C#

By using 7 that is allowed by question, start everything of wordcount from "0000"

int four ="0000".Length;
int eight = four + four;
int fifteen = eight + 7;
int sixteen = eight + eight;
int twentythree = sixteen + 7;
int fourtytwo = four + fifteen + twentythree;
List<int> list = new List<int> { four, eight, fifteen, sixteen, twentythree, fourtytwo };

//while (true)
//{
//    foreach (int number in list)
//    {
//        Console.WriteLine(number);
//    }
//}

//infinite loop via number 0,7,9 that are allowed
for (int i=0; i<7; i++)
{
    Console.WriteLine(list[i]);
    if (i == (7-9+7))
        i = 0;
}

C#

By using 7 that is allowed by question, start everything of wordcount from "0000"

        int four ="0000".Length;
        int eight = four + four;
        int fifteen = eight + 7;
        int sixteen = eight + eight;
        int twentythree = sixteen + 7;
        int fourtytwo = four + fifteen + twentythree;
        List<int> list = new List<int> { four, eight, fifteen, sixteen, twentythree, fourtytwo };

        while (true)
        {
            foreach (int number in list)
            {
                Console.WriteLine(number);
            }
        }

int four ="0000".Length;
int eight = four + four;
int fifteen = eight + 7;
int sixteen = eight + eight;
int twentythree = sixteen + 7;
int fourtytwo = four + fifteen + twentythree;
List<int> list = new List<int> { four, eight, fifteen, sixteen, twentythree, fourtytwo };

while (true)
{
    foreach (int number in list)
    {
        Console.WriteLine(number);
    }
}