Hi BigJay,
first off there is a math error in your description.
An assembly of 36 parallel branches with six LEDs each in series totals 216 LEDs.
Connecting 36 parallel branches with four each makes the 144 you mentioned.
Anyway, the minimum supply voltage for six LEDs in series is 7.8V (number of LEDs multiplied with LED forward voltage assuming 1.3V per LED)
1.3V refers to the LD274-3, there are some with a forward voltage of 1.5V as well on the market.
To calculate for the appropriate current limiting resistor use this formula:
R(limit)=US-(NLED*Uf)/If, where R(limit) is the current limiting resistor (Ω), US is the supply voltage, NFLED is the number of LEDs connected in series, Uf is the LED forward voltage (V) and If is the LED forward current (A)
Example: US=12V, number of serial LEDs=4, Uf=1.3V, If=0.02A
R(limit)=12-(4*1.3)/0.02; R(limit)=340Ω. The closest standard resistor value is 330Ω, which you might use since most IRLEDs can be operated up to 100mA If.
The power dissipation of the current limiting resistor is calculated by the voltage drop it causes multiplied with the current flow.
P(W)=6.8V*0.02A; P=0.136W. Using a standard resistor rated 250mW suffices.
Don't use carbon film resistors and use metal film resistors instead, since carbon film resistors have a high positive temperature drift and increasing the resistance means decreased LED power.
Boncuk