Surely someone must remember this.
Played on a piano, all eighth notes. Every C is middle C except for the last three, which are an octave below.
C|ACGCECDC|CCC
Moving over to CS for you.
A above middle C or below? 
Thanks Idle Thoughts.
The whole thing is a descending line below middle C.
Gah, it’s the ending bit to a children’s song, like a nursery rhyme song! Cannot dig it out…
Ok, the one I am thinking of starts at Middle C, goes above then below for A and G, then back to Middle C and above and down again.
ECDClowAClowG|CECDClowAClowGC|lowAClowGClowElowD|(low)CCC
Try here? Thinking back I think that might have been just a beginner’s piano exercise. For some reason though I picture cartoon characters whiling away time bored… I think it’s one they made you play on one hand while the other played Swanee River, at that! (To teach syncopation, I think?)
I’m thinking it’s an exercise too. I kind of associate it with my brother taking lessons as a kid.
Eventually he played it as fast as he could?
That I don’t remember.
Go try that site? They have a key assist feature, press ctrl to make the notes linger some. You can mouseclick to get the notes, Middle C is the third C from the left, it corresponds to “T” on the keyboard. 
Ok, using the keyboard cheat thing here’s the song I remember that had those notes for an ending.
(t?)utyt|tetw|tutyt|etwt|utyt|etwt|t0t9t888
Try that and see if it is what you mean? The one I remember was supposed to be played getting a little faster with emphasis on the last three notes. Sometimes it started with a Middle C, but not always
Ok, I misremembered, this song was in the same section of the song they combined with Swanee River to teach syncopation. But it was Humoresque + Swannee River for that! Gah! Remembering more and more about that book! Wish I still had it…
That’s one of those things that lots of kids play when they can get their mitts on a piano. You play it on the black keys, starting with F# (or the Bflat above that, since you are starting with what I consider to be the second note).
Also I am probably throwing in ornamental notes, I remember it can be simpler. I think it could end t89000 Silvorange, but it does have some kind of name or I remember at least that the lesson did! 
Gah! Ok, correction!
tutyt|etwt|utyt|etw|tutyt|etwt|0t9t888
And the lyrics are the nursery rhyme “Peter, Peter, pumpkin eater”
It is, in fact, an excerpt from Peter, Peter, Pumpkin Eater. Transposed into the key of C (the all black keys version is F#), and sung with the lyrics “pumpkin shell and there he kept her very well.”
The entire song would go:
ECDC | ACGC
ECDC | ACGC
ECDC | ACGC
ECDC CCC
Where the bolded letters are in the octave above Middle C, and the italicized ones are the octave below.