Three methods to generate the sequence b(n):
(Codes in Mathematica format)

1) b(n) = b(n-1) + b(n-2) - b(n-3) + b(n-5), with b(0) = 0, b(1) = 1, b(2) = 1, b(3) = 2 and b(4) = 2
LinearRecurrence[{1, 1, -1, 0, 1}, {0, 1, 1, 2, 2}, 50]

2) vector matrix (Markov): 
m = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 0, -1, 1, 1}};
CharacteristicPolynomial[m, x];
v[0] := {0, 1, 1, 2, 2};
v[n_] := v[n] = m.v[n - 1];
Table[v[n][[1]], {n, 0, 50}]

3) polynomial expansion:
p[x_] := -(1 - x^2 + x^3 + x^4 - x^5);
q[x_] := ExpandAll[x^4*p[1/x]];
Table[SeriesCoefficient[Series[1/q[x], {x, 0, 50}], m], {m, 0, 50}]