I'm looking for a sequence that includes every number from $1$-$9$ and then ends with $0$. It should look random on first glance (or as random as possible given the mental arithmetic constraint) but have some pattern I can explain, which I can also exploit to help me remember it.
For example, this almost works:
I only need to memorize $192$ and then I can crank out the rest in my head. Unfortunately it doesn't end in $0$.
http://www.worldofnumbers.com/ninedig1.htm helped me find this pattern but squeezing out something with zeroes from the sites it links to hasn't been easy for me.
These worked for a while then stopped:
$7418529630 = 7n$ $mod$ $10$ iterated
$3692581470 = 3n$ $mod$ $10$ iterated
Obviously I don't want to turn this question subjective. I don't know what the standard is around here for mental arithmetic but I'll take it, and any suggestions are welcome, even if they might fail the terribly subjective "look random" test. But to tell the story, my child loves "the ants go marching one by one", and I love singing it to her, but she's insatiable and I'm sick of singing it to her in order. Now I'm just trying to stay awake and singing, and maybe set her up for an interesting preschool experience, because at heart I'm a horrible person. It has to end with $0$ because the last line of the song is "The ants were marching ten by ten, the little one stopped to shout 'The End'" and I'm terrible at remembering which numbers I've already sung.
Sequences that start with $1$ are choice and a set of sequences I could rotate through would make me so very happy.