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GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 17

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The generating function of the sequence $1,1,2,3,5,8, \dots$ is $\frac{1}{1-x-x^2}.$ The sequence generated by generating function $\frac{1}{1-3x-9x^2}$ is $\_\_\_\_\_$

  1. $1, 3, 18, 81, 405, \ldots$
  2. $1, 0, 3, 0,18, 0, 81, 0, 405 \ldots$
  3. $3,3,6,9,15,24,\ldots$
  4. None of the above
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The generating function of the sequence $1,1,2,3,5,8, \dots$ is $\frac{1}{1-x-x^2}.$

This means,

$1.x^0 + 1.x^1 + 2.x^2 + 3.x^3 + 4.x^4 + 8.x^5 + \ldots = \frac{1}{1-x-x^2} $

Replacing $x$ by $3x$ we get

$1.(3x)^0 + 1.(3x)^1 + 2.(3x)^2 + 3.(3x)^3 + 4.(3x)^4 + 8.(3x)^5 + \ldots = \frac{1}{1-3x-(3x)^2} $

$\implies 1 + 3x + 18x^2 + 81x^3 + \ldots = \frac{1}{1-3x-9x^2}$

So, the sequence generated by the function $\frac{1}{1-3x-9x^2}$ is $1,3,18,81,\ldots.$

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