Subscript[c, 2 + n] == -(Subscript[c, n]/((n + 1) (n + 2)))

f[x_] := f[x] = f[x - 1] + f[x - 2]

f[0] = f[1] = 1

sol = c[n] /. RSolve[ {c[n + 2] ==
   -c[n]/((n + 1) (n + 2)),
   c[0] == c0, c[1] == c1}, c[n], n][[1, 1]];
sol /. n -> # & /@ Range[0, 5] // InputForm

{c0, c1, -c0/2, -c1/6, c0/24, c1/120}

ClearAll[c];
c[0] = c0; c[1] = c1;
c[m_] := c[m] = With[{n = m - 2},
   -c[n]/((n + 1) (n + 2))];
c /@ Range[0, 5] // InputForm

ClearAll[y,x];
ode = y''[x] + y[x] == 0;
sol = AsymptoticDSolveValue[ode, y[x], {x, 0, 5}];
sol /. {C[1] -> y[0], C[2] -> y'[0]}