Rebuild partition table on external HDD - not writable
OS X uses three different types of valid partition tables. Only one of those is written to a particular disk:


GUID partition table
Apple Partition Table
MBR


The default partition table on OS X is the GUID partition table.

Depending on the previously used partition table on your disk you have to use different tools to create/change/write it:


gpt for GUID partition table
pdisk for Apple Partition Table
fdisk for MBR


After additional informations given by the OP (testdisk results) and checking it in a virtual machine it's highly probable that the disk was formatted with an Apple Partition Map. Consequently use pdisk to rebuild the partition map.

pdisk:

A typical disk formatted with an Apple Partition Table looks like this:

Driver Descriptor Map: Block 0              Size:      1 Block
Apple Partition Table: Block 1 - 63         Size:     63 Blocks
Apple_Free:            Block 64 - 262207    Size: 262144 Blocks
1st partition:         Block 262208 - 
Apple_Free:                                 Size: 262144 Blocks
...
Apple_Free:                                 Size: 262144 Blocks
last partition
Apple_Free:                                 Size:     16 Blocks


To recreate the partition map do the following:


attach the external drive
detach any other external drive
open Terminal and enter diskutil list to get the DiskIdentifier (in the step below I assume your DiskIdentifier is disk1
enter diskutil unmountDisk /dev/disk1
enter sudo pdisk /dev/rdisk1  You will get the following output:


  Edit /dev/rdisk1 -
  Command (? for help):

Now enter c then enter the start block, the size of the first partition and the name. You have to hit the enter key after every input. Repeat this for the next two partitions. At the end it should look like this:


  Command (? for help): c
  First block: 262208
  Length in blocks: 419430400
  Name of partition: part1
  Command (? for help): c
  First block: 419954752
  Length in blocks: 838860800
  Name of partition: part2
  Command (? for help): c
  First block: 1259077696
  Length in blocks: 694447456
  Name of partition: part3
  Command (? for help):  

At the last prompt enter w then y to write the changes to disk and finally qto quit pdisk:


  Command (? for help): w
  Writing the map destroys what was there before. Is that okay? [n/y]: y
  The partition table has been altered!
  
  Command (? for help): 

The missing volumes should mount automatically. Otherwise enter diskutil mountDisk /dev/disk1. Enter q at the last prompt to quit pdisk, then quit Terminal, open Disk Utility and check the recovered volumes for errors.




gpt:

In the example below I assume the DiskIdentifier of your external disk is disk1 (check this with diskutil list)

First you have to unmount the external disk:

diskutil umountDisk disk1


Remove the current MBR and create a GPT with gpt:

sudo gpt create -f /dev/disk1


First rebuild the EFI entry with:

sudo gpt add -b 40 -i 1 -s 409600 -t C12A7328-F81F-11D2-BA4B-00A0C93EC93B disk1


The EFI partition is a fixed sized partition near the beginning of every GUID partitioned disk (using OS X partitioning tools)

Then add the JHFS+ partition entries with the TestDisk findings:

sudo gpt add -b StartBlock -i IndexNumber -s SizeOfVolume -t 48465300-0000-11AA-AA11-00306543ECAC disk1


Example for the first OS X partition:

sudo gpt add -b 409640 -i 2 -s SizeOfHFSVolume1 -t 48465300-0000-11AA-AA11-00306543ECAC disk1


After adding a partition with gpt you may have to unmount disk1 with diskutil umountDisk disk1 again if your get a "resource is busy" error adding additional partitions.

Increase the index number by 1 for every new partition.

Listed below are answers to similar questions (partly covering other file systems like ExFAT)


How to fix GUID hard drive corrupted to MBR?
HFS+ invalid number of allocation blocks 
Hard drive no longer accessible


If you run into problems, leave a comment to the answer with @klanomath.