But how can you, the intrepid math geek, work this out in your head? (It's a problem in Knuth, so it must be important...)

Easy! (Well, somewhat easily). Interpret tes' recurrence relations in binary (after all, they keep multiplying by 2).

What these recurrence relations are telling us to do is to take the binary representation of the number of people and reinterpret each 1 bit as "+1" and every 0 bit as "-1", keeping the same values for all positions. For example, for 9=10012 we have +8 and +1 for the 1's, and -4 and -2 for the 0's. Adding up, we get J(9)=3.

Can we do better? WATCH THIS SPACE! (Or add your easy-bit-operations below).