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Clear Speech Students .
Clear Speech Students 2.32 ACTUAL DRIVER.Q:

Solutions to recurrence relations with multiple different indices

I’m having problems with a problem of recursion relations involving multiple different indices. I was wondering if anyone can point me towards a source containing a step-by-step guide on how to solve this, or if this is just not possible.
Specifically I have: $a_{n+2}-a_n=b_n-c_n-d_n$ with $a_0=0,b_0=0,c_0=0,d_0=1$ and $a_1=0,b_1=1,c_1=1,d_1=0$
I’m having trouble finding the general solution and am wondering if I should just consider different cases.

A:

If you have such recurrence relations with $m$ different indices (in your case $m=3$), then you can try to see which subsequences are growing faster, which are growing with the same speed and which are shrinking.
In your case, it looks like the indices are $a,b,c,d,e$. You can think about the subsequence $a,b,c,d$ as sequences $a_n, b_n, c_n$ and as such they grow at the same speed. What then happens for the subsequence $d,e$? Consider the sequence $d_n:=d_n$ and $e_n:=b_n-c_n$. This sequence doesn’t change after time $n$, since $c_n=0$ and so $e_n=b_n$ is a constant. Therefore, the sequence $d_n,e_n$ will be shrinking with a different speed (see this question and the linked discussion for more).
Based on this, you can solve the problem in this way.

#!/usr/bin/env python3
# Copyright (c) Facebook, Inc. and its affiliates.
#
# Licensed under the Apache License, Version 2.0 (the “License”);
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#

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