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

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