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  1. or download this
        my $n = ...;
    my $b = ...;
...
        @possibles;

    print( "$n: ", ('1' x $n) =~ /^$pat\z/ ?1:0, "\n" );
  2. or download this
        my $n    = ...;
    my $b    = ...;
...
    my $pat = qr/(1|((?-2))\g{-1}{$b_m1})/;

    print( "$n: ", ('1' x $n) =~ /^$pat\z/ ?1:0, "\n" );
  3. or download this
        local our $pat;
    $pat = qr/
...
        \g{-1}{$b_m1}
        (?(?{ $^N =~ $pat })(?=)|(?!))
    /x;