Electronic – Ramp generator using 555 timer

You can't use the main timing resistor and capacitor to generate your trigger signal. As you discovered, this is exactly equivalent to the astable configuration.

Instead, you need to provide a separate resistor (to Vcc) and capacitor (to ground) for the Trigger pin. The time constant for this pair should be short relative to the main timing period, but long relative to the risetime of the power supply.

These components will hold the trigger low while the chip powers up, but then allow it to go high a short time later. Once the timer has timed out, the main timing capacitor will be discharged, but the trigger capacitor will not.

You have two equations in two unknowns.. you need to solve them for R1 and R2. From the duty cycle equation:-

\$ R_1 + R_2 = 0.9R_1 + 1.8 R_2\$

Solving--I'll let you do your own algebra, you'll get unique values for R1 and R2 (given C = 1uF). I'll note that vvy's answer is slightly off (s/he does say "around").