Summary: I have a k-th order recurrence relation, and I need to find the terms a[n] in terms of a[0], a[1], ..., a[k-1]
Consider this recurrence relation:
2*(n+2)*(n+1)*a[n+2] + 3*(n+1)*a[n+1] + (n+3)*a[n] = 0
Since this is a second order homogeneous linear recurrence relation, each a[n] can be expressed as a linear combination of a[0] and a[1]:
a[n] = c0[n] * a[0] + c1[n] * a[1]
Where c0[n] and c1[n] are some constants. I am trying to find a[n] in the form above (having only a[0] and a[1] from the sequence).
This is how I defined the recurrence in Maxima:
rec: 2*(n+2)*(n+1)*a[n+2] + 3*(n+1)*a[n+1] + (n+3)*a[n] = 0;
eqns: makelist(''rec, n, 0, 6);
My current approach is setting a[0] to 0 and a[1] to 1 to find the coefficients of a[1] and vice versa:
c0 = solve(append(eqns, [a[0] = 1, a[1] = 0]));
c1 = solve(append(eqns, [a[0] = 0, a[1] = 1]));
This works fine, and I get the coefficients. But I wonder how I can tell Maxima that a[0] and a[1] are the unknowns, and it should find, e.g. a[2], in terms of a[0] and a[1].
