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given recurrence, find a_(6n)

Source: VTRMC 2004 P2

August 2, 2021
Sequencesrecurrence relationfunctionalgebra

Problem Statement

A sequence of integers {f(n)}\{f(n)\} for n=0,1,2,…n=0,1,2,\ldots is defined as follows: f(0)=0f(0)=0 and for n>0n>0, f(n)=f(n−1)+3,if n=0 or 1(mod6),f(n−1)+1,if n=2 or 5(mod6),f(n−1)+2,if n=3 or 4(mod6).\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}Derive an explicit formula for f(n)f(n) when n≡0(mod6)n\equiv0\pmod6, showing all necessary details in your derivation.
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