Fun with integer sequences (A016103)

I needed a closed form of twice the sequence defined by A016103 (see if you can guess the connection with psychology :-)). When looking at its description, “Expansion of \frac{1}{(1-4x)(1-5x)(1-6x)}“, I wasn’t quite sure how to calculate a closed form. But the nearby sequences helped narrow down the search to something of the form a_1 \cdot  b_1^n + a_2 \cdot b_2^n + a_3 \cdot b_3^n + a_4 \cdot b_4^n… and this seems to do the trick:

(4^n + 6^n - 2 \cdot 5^n) /2

I found it by brute-force search, looking only at the first four terms of the sequence. This matches the sequence in the encyclopaedia entry at least for these values:

0, 1, 15, 151, 1275, 9751, 70035, 481951, 3216795

I can’t go further—I’m using R and haven’t yet found to time to get it to use long integers :-(

But aren’t sequences fun? :-)


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