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Solving Recurrence equation for $n$

asked 2015-02-11 19:52:16 -0500

student gravatar image

updated 2015-02-11 19:54:19 -0500

I have a recurrence equation $t_n$ which is defined as

$$ t_n=2.5t_{n-1}-1.5t_{n-4},\qquad \text{for }n\geq5 $$

Where $t_1=5$, $t_2=10.5$, $t_3=26.25$, and $t_4=62.625$.

I need to solve my recurrence equation for $n$ when $t_n=13\times10^{10}$. How can I achieve this in Sage?

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answered 2015-02-11 20:36:18 -0500

kcrisman gravatar image

Nothing native, but sympy has rsolve and Maxima has solve_rec (see e.g. this sage-support thread). Unless this ticket has something that helped?

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answered 2015-02-16 02:45:16 -0500

updated 2015-02-16 02:47:41 -0500

Is the exact answer needed? I guess not, since the coefficients are given in floating point. Then an estimate with five significant digits should do. This is usually done symbolically by constructing the (rational) generating function of the recurrence, computing the roots of the denominator polynomial, and from the largest root r the asymptotics which is of form c*r^n for linear recurrences. This can be done manually, with Sage performing these step by step, or (when you have the g.f.) using code from ticket #10519. For examples see my paper.

However, even more simple, but without getting any asymptotics as byproduct, would be to use matrix exponentiation. See for example This looks like a Project Euler question anyway.

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Asked: 2015-02-11 19:52:16 -0500

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Last updated: Feb 16 '15