Assuming that your input file contains only FASTA header records and sequence records (i.e., no blank lines, comment lines, etc.), you might do something like (untested):

my $rx_fasta_header = qr{ \A > }xms;  # header pattern - very naive
my $rx_sequence     = qr{ \A [ACDEFGHIKLMNPQRSTVWY_]+ \z }xms;  # plea
+se check

my $sequence;  # sequence accumulator - initially undefined

RECORD:
while (my $record = <$read_filehandle>) {

    chomp $record;  # get rid of newline, if any

    # fasta header - process any sequence accumulated so far
    if ($record =~ $rx_fasta_header) {
        
        # do not process if no sequence accumulated so far (i.e., firs
+t header)
        next RECORD if not defined $sequence;
        
        my $seq_len = length $sequence;
        my $seq_n_A = $sequence =~ tr/A//;;
        do_something_with($sequence, $seq_len, $seq_n_A);
        
        undef $sequence;  # prepare to process next sequence
        next RECORD;
        
        }

    # part of sequence - accumulate
    if ($record =~ $rx_sequence) {
        $sequence .= $record;  # accumulate sub-sequences (records)
        next RECORD;
        }

    die "bad record: '$record'";  # don't know what this is

    }
[download]

Following is the script that I tried with open:

#!/usr/bin/perl
use strict;
use warnings;

print 'Please enter protein sequence filename: ';
chomp( my $prot_filename = <STDIN> );

open my $PROTFILE, '<', $prot_filename 
  or die "Cannot open '$prot_filename' because: $!";

my $report_name = $prot_filename.'_alanine_report';
open my $out_file, '>', $report_name
     or die "Cannot open '$report_name' because: $!";

my $rx_fasta_header = qr{ \A > }xms;  # header pattern - very naive
my $rx_sequence     = qr{ \A [ACDEFGHIKLMNPQRSTVWY_]+ \z }xms;  # plea
+se check

my $sequence;  # sequence accumulator - initially undefined

RECORD:
while (my $record = <$PROTFILE>) {

    chomp $record;  # get rid of newline, if any

    # fasta header - process any sequence accumulated so far
    if ($record =~ $rx_fasta_header) {
        
        # do not process if no sequence accumulated so far (i.e., firs
+t header)
        next RECORD if not defined $sequence;
        
        my $seq_len = length $sequence;
        my $seq_n_A = $sequence =~ tr/A//;;
        do_something_with($sequence, $seq_len, $seq_n_A);
        
        undef $sequence;  # prepare to process next sequence
        next RECORD;
        
        }

    # part of sequence - accumulate
    if ($record =~ $rx_sequence) {
        $sequence .= $record;  # accumulate sub-sequences (records)
        next RECORD;
        }

    die "bad record: '$record'";  # don't know what this is

    }

# show all results
print $out_file;
close $out_file;
[download]

But, it gave me the error: bad record: '' at ../alanine_vs_sequence_length_2.pl line 46, <$PROTFILE> line 42. And the outfile is empty. What am I doing wrong here?