Look at the ULN2003 datasheet, page 12.
This shows you that the darlington drivers sink current to ground when on.
The 1st requirement is to reverse the LED directions.
Now look at the Vce saturation voltage on page 5.
This is about 1V - 1.2V depending on LED current.
So VLED_string can be about Vcc-Vcesat = 5V - 1V_ = 4V or less.
Now look at LED datasheet Vf values on the 3rd page.
These are TYPICALLY 1.2V - 1.4V at 20-100 mA and 1.5V-1.8V worst case.
So at even 10 mA typical Vf x 3 = 3 x 1.4V = 4.2V = more than the driver will supply.
Try TWO series LEDs.
2 x 1.4V = 2.8V at 100 mA TYPICAL.
Vresistor = 4V or less - 2.8V = 1.2V.
With R = 7 Ohms I = V/R = 1.2/7 ~= 0.017A ~= 15 - 20 mA.
Rework for desired currents.