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  T_n == 1 (mod 3) when n == 1 (mod 3)
  T_n == 0 (mod 3) otherwise

  if n == 0 (mod 3), require A A A or B B B
  if n == 1 (mod 3), require A A B
  if n == 2 (mod 3), require A B B

  sub p_tri {
    my $num = shift;
...
    my $t = int((sqrt($x) + 1)/2);
    return +((2 * $t - 1) * (2 * $t - 1) == $x) ? 0 : --$t;
  }