How quickly can you do this?

Let \(a\) be the number of tuples \((a_1,a_2,a_3,\ldots)\) that satisfy the equation given below : \[\sum_{i=1}^{\infty} ia_{i}=45\]

Let \(b\) be the number of tuples \((b_1,b_2,b_3,\ldots)\) that satisfy the inequality given below : \[\sum_{i=1}^{\infty} (i+1)b_{i}\leq 45\]

Enter your answer by concatenating \(a\) and \(b\). For example, if \(a=23\) and \(b=65\), then enter \(2365\) as your answer.


Details and Assumptions

  • \(a_{i}\geq 0\ \forall\ i\in \Bbb Z^{+}\)
  • \(b_{i}\geq 0\ \forall\ i\in \Bbb Z^{+}\)
  • \(\Bbb Z^{+}\) denotes the set of all positive integers.
  • By title, I mean how quickly your program can do this.

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