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Here is a sequence that seems to go on forever. Will it? Will this sequence ever change?

1,2,3,1,0,1,2,3,1,0,1,2,3,1,0 and so on

What does this sequence relate to?

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1 Answer 1

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This sequence is generated by:

counting the number of 'I's that appear in the Roman numeral representation of the natural numbers 1, 2, 3, ...

So, for example, the numbers given in the puzzle relate to the terms:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

which in Roman numerals are represented as:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV

where you can clearly see the pattern:

1, 2, 3, 1, 0, 1, 2, 3, 1, 0, 1, 2, 3, 1, 0

when you count the number of I's.

As to whether this sequence might go on forever...

Strictly speaking, if applying the construction rules of Roman numerals using the main 7 characters 'I', 'V', 'X', 'L', 'C', 'D' and 'M' it is only possible to create numbers up to a value of 3,999 (MMMCMXCIX), since there is no way to construct a value for '4000' using only these characters. (And so this sequence is terminated after 3999 terms, with the same 1-2-3-1-0 pattern appearing 799 times in a row, culminating in a final 1-2-3-1.)

If, however, we use the common convention that a numeral with a bar above it represents a multiple of 1000 (i.e. $\bar{I}$ for 1000, $\bar{V}$ for 5000, etc.) then this wouldn't be a lot of help either, as using $\overline{IV}$ to try and continue the sequence to 4000 would produce an 'I' character in a place where there should be none if the sequence were to continue as originally...

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