I have 3.1V LED 20mA and wire 20 or more
You are correct that the combined VF for the LEDs adds when connected in series to give you 62V for 20 LEDs. This is of course only approximate as the LEDs will vary slightly.

HOWEVER, you need to provide current limiting for the LEDs by either using a constant current power supply, or some current limiting device if you use a battery.

Since you seem to be using a battery solution I'd suggest you consider a simple linear constant current device like the AL5809-20 (less than $0.50c each 10 off). This is a simple two terminal device that sets a constant current through a string of LEDs and you can use a simple power supply or battery.

The device comes in various current levels designated by the last two digits of the part number:

  

To use them you simply connect as you would a resistor, and it sets the current for the string. You need to supply a slightly higher voltage because of the CC device. The overall efficiency is quite high because of minimal voltage drop across the device. 

The recommended minimum voltage is 2.5V but the device will work down to a minimum overhead voltage of 1.75V.



You would then be able to use a higher voltage such as 2 - 6 LiPo cells in series to provide the power (and they are rechargeable). 
For example if you used 5S than you have a minimum of 5*3.6=18V and a maximum of 5*4.2=21V. With your LED forward voltages in the in the 3.1V range you could get 5 LEDs in each string with an ALS5809-20.

If you want to be really creative you could use a single LiPo cell (3.6 - 4.2V) and a boost DC-DC convertor (there are many on Ebay etc) to get say 20-50V and connect up to 15 LEDs in each string. The ALS5809 is limited to 60V, so you could easily support strings of 20 LEDs and it has to standoff only a small portion of the voltage. If you were to go above 60V I'd put a small resistor in parallel with the ALS5809 just in case (say 10kOhm).

An example of a very cheap Ebay converter is here, and you could go from a single LiPo cell (3.6-4.2V) to 28VDC allowing strings of 8 LEDs to be run.