Given this sequence:
0, 4, 84, 212, 1940, 358036, 14113428, 127377044
What is the next item?
Hint:
The sequence is not based on any mathematical operations
Hint:
base
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$\begingroup$ Not in the OEIS, impressive. $\endgroup$– IronEagleCommented Oct 30, 2019 at 15:59
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11$\begingroup$ It's not difficult to come up with sequences that aren't in OEIS. What's impressive is coming up with ones that aren't there but deserve to be. (Which isn't quite the same thing as being suitable for a puzzle here, of course.) $\endgroup$– Gareth McCaughan ♦Commented Oct 30, 2019 at 17:22
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1 Answer
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The sequence is
The classical look-and-say sequence, but starting with a 0 and evaluated in base 4.
Proof:
$$\begin{align}0_4 &= 0 \\10_4 &= 4 \\1110_4 &= 84 \\3110_4 &= 212 \\132110_4 &= 1940 \\1113122110_4 &= 358036 \\311311222110_4 &= 14113428 \\13211321322110_4 &= 127377044\end{align}$$
Trivia:
Using base 4 is actually an interesting and meaningful idea because no digits can appear 4 times in a row in this sequence, so all "digits" are 3 or lower and thus valid in base 4.
Therefore, the next term in the sequence is
6006904289940, because $$1113122113121113222110_4 = 6006904289940$$
