Question

Find a generating function in closed form for the sequence:
{1,2,3,4,1,2,3,4,1,2,3,4,...}

Answer 1

Answer on Question #76325 – Math – Discrete Mathematics

Question

Find a generating function in closed form for the sequence: $\{1,2,3,4,1,2,3,4,1,2,3,4,\ldots\}$.

Solution

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

Answer: $\frac{1 + 2x + 3x^2 + 4x^3}{1 - x^4}$.

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