Finds all solutions of the linear recurrence equation with constant coefficients.

Optionally, one can specify output=basis  or output=gensol, which specifies respectively that one wants a basis of the solutions or a generic solution.  The initial conditions that may be associated to the problem are only used in the generic solution case.  Arbitrary constants for the basis case are represented as _C[1], _C[2], ..., _C[n], where the _C is an escaped local variable.

See the help page for LREtools[REcreate] for the definition of the format of a problem.

$\mathrm{with}\left(\mathrm{LREtools}\right)\:$

$\mathrm{rec}≔a\left(n\right)-9a\left(n+1\right)+26a\left(n+2\right)-34a\left(n+3\right)+21a\left(n+4\right)-5a\left(n+5\right)=0\:$

$\mathrm{constcoeffsol}\left(\mathrm{rec}\,a\left(n\right)\,\varnothing \right)$

$-\frac{\left(-\frac{3125a\left(4\right)}{256}-\frac{3125a\left(0\right)}{256}+\frac{3125a\left(1\right)}{64}-\frac{9375a\left(2\right)}{128}+\frac{3125a\left(3\right)}{64}\right){\left(\frac{1}{5}\right)}^n}{5}+\frac{\left(\frac{5a\left(4\right)}{4}+\frac{a\left(0\right)}{4}-2a\left(1\right)+\frac{9a\left(2\right)}{2}-4a\left(3\right)\right)\left(n+1\right)\left(n+2\right)\left(n+3\right)}{6}-\frac{\left(\frac{85a\left(4\right)}{16}+\frac{21a\left(0\right)}{16}-\frac{41a\left(1\right)}{4}+\frac{175a\left(2\right)}{8}-\frac{73a\left(3\right)}{4}\right)\left(n+1\right)\left(n+2\right)}{2}+\left(\frac{565a\left(4\right)}{64}+\frac{181a\left(0\right)}{64}-\frac{341a\left(1\right)}{16}+\frac{1343a\left(2\right)}{32}-\frac{517a\left(3\right)}{16}\right)\left(n+1\right)-\frac{1845a\left(4\right)}{256}-\frac{821a\left(0\right)}{256}+\frac{1461a\left(1\right)}{64}-\frac{5023a\left(2\right)}{128}+\frac{1781a\left(3\right)}{64}$

$\mathrm{rec}≔a\left(n+4\right)=-a\left(n+3\right)+3a\left(n+2\right)+5a\left(n+1\right)+2a\left(n\right)\:$

$\mathrm{constcoeffsol}\left(\mathrm{rec}\,a\left(n\right)\,\varnothing \right)$

$-2\left(-\frac{a\left(3\right)}{54}-\frac{a\left(2\right)}{18}-\frac{a\left(1\right)}{18}-\frac{a\left(0\right)}{54}\right)2^n+\left(-\frac{13a\left(3\right)}{27}-\frac{4a\left(2\right)}{9}+\frac{14a\left(1\right)}{9}+\frac{68a\left(0\right)}{27}\right){\left(\mathrm{-1}\right)}^n+\frac{\left(-\frac{a\left(3\right)}{3}+a\left(1\right)+\frac{2a\left(0\right)}{3}\right){\left(\mathrm{-1}\right)}^n\left(n+1\right)\left(n+2\right)}{2}+\left(\frac{7a\left(3\right)}{9}+\frac{a\left(2\right)}{3}-\frac{8a\left(1\right)}{3}-\frac{20a\left(0\right)}{9}\right){\left(\mathrm{-1}\right)}^n\left(n+1\right)$

$\mathrm{rec}≔a\left(n+2\right)-2a\left(n+1\right)-a\left(n\right)=0\:$

$\mathrm{constcoeffsol}\left(\mathrm{rec}\,a\left(n\right)\,\varnothing \right)$

$-\frac{\left(\left(-\frac{3a\left(0\right)}{4}+\frac{a\left(1\right)}{4}\right)\left(\sqrt{2}-1\right)+\frac{a\left(0\right)}{4}-\frac{a\left(1\right)}{4}\right){\left(\sqrt{2}-1\right)}^{-n}}{\sqrt{2}-1}-\frac{\left(\left(-\frac{3a\left(0\right)}{4}+\frac{a\left(1\right)}{4}\right)\left(-1-\sqrt{2}\right)+\frac{a\left(0\right)}{4}-\frac{a\left(1\right)}{4}\right){\left(-1-\sqrt{2}\right)}^{-n}}{-1-\sqrt{2}}$