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The function rsolve (from sympy) can deal with linear recurrence relations.

To get the documentation:

sage: from sympy import rsolve
sage: rsolve?

The recurrence relation you are interested in has squares, but you could set b(n) = a(n)^2.

This way you need to solve a linear recurrence relation to find b(n).

sage: from sympy import Function, rsolve
sage: from sympy.abc import n
sage: y = Function('y')
sage: f = y(n+2) - 5*y(n+1) + 6*y(n) - 7*n
sage: rsolve(f, y(n))
2**n*C0 + 3**n*C1 + 7*C0*n
sage: b = rsolve(f, y(n), {y(0): 1, y(1): 1})
sage: b
-2**n/3 + 4*3**n/3 - 7*n/3

From there you need to check that this b(n) is always non-negative.

Finding a(n) then just amounts to picking, for each n, one of the two possible square roots of b(n).