Electronic – the value of this 5-band resistor

Color code standards are defined in IEC 60062. See also wikipedia: Electronic Color Code

Generally speaking, the first, second [,third] band represent a digit each, followed by a multiplier, tolerance [, temperature coefficient]. Third digit and temperature coefficient are both optional. Given that gold cannot represent a digit (but only multiplier or tolerance), you have:

digit, digit, multiplier, tolerance, temp.coefficient.

Black has no tolerance assigned, which makes it the second digit in your case.

Applying this to your resistor leads to: 5 | 0 | 0.1 | 0.5% | 15ppm/K

Thus, you want a 5 Ohm resistor.

EDIT: It's 0.5 Ohm (multiplier 0.01), given that the middle band is silver (as stated in the question). It looked golden to me, but it's difficult to tell from the picture. However, the above applies to 'silver', where I've referred to 'gold'.

That resistor is toast for sure, but the spiral 'bump' indicates it is a wire-wound resistor of at least 1 watt. The first band is either red or orange, so if it is 0.33 ohms or 0.22 ohms that would make sense as a series current limiting or current sensing resistor. It may be so damaged that the nickle-chrome wire is no good, or its bond to the end caps has gone bad.

If you look at the top of the burnt resistor the nickle-chrome wire appears to be burnt away, creating a gap, so this resistor should measure as being 'open' or infinite resistance. It could also be in the category of "Fusible" resistor or at least "Flame proof".

If installing it as it is now causes problems I would try both a new 0.22 ohm 1 watt and a 0.33 ohm 1 watt to see which works best. The higher value is best as long as the motor runs at expected speed and torque. Note that this was a 1% tolerance resistor. Such a tight tolerance is normally used for accurate current sensing.