As I understand your task (which seems to be different from BrowserUk), this should do it:

#!/usr/bin/perl -w
use strict;

my %hash = (a=>2, b=>1, c=>5, d=>3, e=>4);

my $input = "abcdaeec";

for (2, 3) {
    print "n=$_: ";
    my $n = $_ - 1;

    my @sums;
    while ( $input =~ /(.)(?=.{$n}(.))/g ) {
        push @sums, $hash{$1} + $hash{$2};
    }
    print "@sums\n";
}

__END__
n=2: 7 4 7 7 6 9
n=3: 5 3 9 7 7
[download]

Update: let YAPE::Regex::Explain explain what the regex does:

use YAPE::Regex::Explain;

print YAPE::Regex::Explain->new( qr/(.)(?=.{1}(.))/ )->explain;

__END__
The regular expression:

(?-imsx:(.)(?=.{1}(.)))

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    .                        any character except \n
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  (?=                      look ahead to see if there is:
----------------------------------------------------------------------
    .{1}                     any character except \n (1 times)
----------------------------------------------------------------------
    (                        group and capture to \2:
----------------------------------------------------------------------
      .                        any character except \n
----------------------------------------------------------------------
    )                        end of \2
----------------------------------------------------------------------
  )                        end of look-ahead
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

And see perlop for what m//g does in scalar context (like the while (...) here).