Puzzles/Easy Sequence 2/Solution

Add 1 2 3 4... OR x(x-1)/2+1 :

1 2 4 7 11 16 22 29 ...

$a_{n}=a_{n-2}+a_{n-1}+1$:

1 2 4 7 12 20 33 ...

Add the three previous numbers:

1 2 4 7 13 24 44 ...

The numbers with an odd number of ones in their binary representation:

1 2 4 7 8 11 13 ... or 1 10 100 111 1000 1011 1101 ...

A formula for the $n$^th number is: $a_{0}=1$, $a_{2n}=a_{n}+2n$, $a_{2n+1}=6n+3-a_{n}$.