|Necessary in sentences, not always necessary as a greeting message|
Basically, if you want to use it as a greeting message, i.e., for a postcard of a greeting card, you may say "A Happy New Year" or "Happy New Year". This is the same phenomenon as articles, "a", "an", and "the", are often omitted when they are used as book titles, not in sentences. In fact, if you look at cards sold in U.S. stores, some have "A Happy New Year" and others have "Happy New Year".
So, 2 can be said with an article, "a". In the Google research, we can find 15,200 examples of "happy new year to you", and, 3,630 examples out of 15,200 are "a happy new year to you".
As for 3, it has to have an "a", since it is used in a sentence. In Google, as well, we can find 1,650 examples of "I wish you a happy new year", but there are only 123 examples of "I wish you happy new year". However, "I wish you happy new year" could be possible, if you have a comma after "you", if you quote "happy new year", or if you distinguish them by using upper-case letters like "Happy New Year", as you see in the examples.
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