all I have been able to find was the ripple counter, which adds binary according to clocks signals.

Is there a chip or a circuit design that does this?

And what would be the name for it...?

 

The IC is a 4017; it is called a Decade counter (you'd just not use the extra counts) Look at this project page http://www.doctronics.co.uk/4017.htm

 

If you need it to count 0, 1, 2, 3, tie output 4 back to the reset pin on the 4017

 

Doesn't the animation represent a shift register with a single '1' on or am I missing something?

 

Either would work. A 4022 would also work.

LEDs, 555s, Flashers, and Light Chasers
Chapter 9, Light Chasers
Chapter 11, Making Patterns

 

The image name in the OP is "ripple counter 01.gif"

Here is a circuit that will do what you want; as well as the details:


http://www.allaboutcircuits.com/vol_4/chpt_11/2.html

 

I think ring counter is the correct name

 

Whatever the name the pattern of the animation shows the sequence:
1-2-4-8
and the easiest implementation is a shift register with the MSB fed back to the LSB. There should also be "RESET" logic to decode illegal states and make a transition to a legal state on the next clock, or a mechanism to load a value with a single bit on at power up.

BTW -- a ring counter is also known as a cyclic shift register
http://en.wikipedia.org/wiki/Ring_counter
and the twisted-ring counter or Johnson counter feeds back the complement of the MSB

 

Whatever the name the pattern of the animation shows the sequence:
1-2-4-8
and the easiest implementation is a shift register with the MSB fed back to the LSB. There should also be "RESET" logic to decode illegal states and make a transition to a legal state on the next clock, or a mechanism to load a value with a single bit on at power up.

BTW -- a ring counter is also known as a cyclic shift register
http://en.wikipedia.org/wiki/Ring_counter
and the twisted-ring counter or Johnson counter feeds back the complement of the MSB

The 4017 is a 5 stage Johnson counter with the 10 unique states decoded to 10 individual outputs. As others have pointed out, it is probably the easiest way to make the counter illustrated by the OP.