1 = 1
4 = 1 + 3
9 = 1 + 3 + 5
... 

##</code><code>##

use strict;
use warnings;
use 5.010;
for (1..100) {
    my $str = '1' x $_;
    my $prev;
    say $_, ' ', $str =~ m/^(1)(?{$prev=$^N})(?>((??{$prev})11)(?{$prev=$^N}))*$/ ? 'yes' : 'no';
}