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Computation Workshop Solution Checker

Recursively Defined Sequence

A sequence \((t(0), t(1), t(2), t(3), \ldots) = (2,1,5,13, \ldots)\) begins with \(t(0)=2\) and \(t(1)=1\). Then

  • \(t(2) = t(1) + 2t(0)\),
  • \(t(3) = t(2) + 2t(1) + 3t(0)\),
  • \(t(4) = t(3) + 2t(2) + 3t(1) + 4t(0)\),
  • \(\ldots\) and so on.
In general: \(t(n+1) = t(n) + 2t(n-1) + 3t(n-2) + ... + (n+1)t(0)\) for all positive integers \(n\).

Part A Determine the last \(9\) digits of \(t(2021)\).
Part B Determine the last \(9\) digits of \(t(2021^{2021})\).