Clear[a, n, m, even];

even[1] = n/4;

even[m_] := even[m] = (n - even[m - 1])/2;

Take[Table[even[m], {m, 1, 100}], -3] // N

Clear[even]

even[m_] =
 a[m] /. RSolve[{a[m] == (n - a[m - 1])/2, a[1] == n/4}, a[m], m][[1]] //
  ExpandAll

even[m] == n/3 (1 + (1/2)*(-1/2)^m) //
 Simplify[#, {Element[m, Integers], m > 0}] &

Limit[n/3 (1 + (1/2)*(-1/2)^m), m -> Infinity]