\(\displaystyle 0,1,1,1,1,1,\dots\)

\(\displaystyle 1,0,0,1,0,0,1,0,0,1,0,0,1,\dots\)

\(\displaystyle 1,2,4,8,16,32,\dots\)

\(\displaystyle 0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,\dots\)

\(\displaystyle 1,-1,1,-1,1,-1,1,-1,1,-1,\dots\)

\(\displaystyle \displaystyle 2^8,2^7\binom{8}{1}, 2^6\binom{8}{2},\dots,\binom{8}{8},0,0,0,\dots\)

\(\displaystyle 1,1,1,0,0,1,1,1,1,1,1,1,1,1,\dots\)

\(\displaystyle 0,0,0,1,2,3,4,5,6,\dots\)

\(\displaystyle 3,2,4,1,1,1,1,1,1,\dots\)

\(\displaystyle 0,2,0,0,2,0,0,2,0,0,2,0,0,2,\dots\)

\(\displaystyle 6,0,-6,0,6,0,-6,0,6,\dots\)

\(\displaystyle \displaystyle 1,3,6,10,15,\dots,\binom{n+2}{2},\dots\)