You can always just make a Table of values:
Table[p[n], {n, 4, 14}]
But I'm guessing this isn't what you want... You can of course always define it to be a recurse function on lists:
p[1] := {5};
p[2] := {1};
p[3] := {2};
p[n_] := {p[n - 1], Last@p[n - 1] + Last@p[n - 3]} /; n > 3;
p[14]
{{{{{{{{{{{{2}, 7}, 8}, 10}, 17}, 25}, 35}, 52}, 77}, 112}, 164}, 241}
