Hi Netguru5, the solution by Dougy83 looks simpler, if you want you can try this, here you can design the time you want the LEDs to be on, and the 1 sec. delay is exactly designed with those components, it will give 0.99 sec, the transistor in the CLOCK will invert that signal, so when the time delay goes high, it will be a LOW in the CLOCK and when it goes LOW after that 0.99 sec. the transistor will send the rising edge that will turn on the other LED.
That should do what you need, the first two images are the explanation of how is that going to work, and the last is what you need, it will also remember the last LED on turn on, but by turnin' the opposite when you turn on again, you won't exactly remove the power supply but by moving on and off that switch, and by the way, when you turn it off the on LED won't turn off instantly, it will be on for the time you programmed it, and then the other LED won't light.
It is not complicated, and not as big as it looks, by doing 1.1*C*R you will have the time ON of the monostable multivibrator, that will light the LEDs.