Monday, December 19, 2022
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Will the sequence ever change?


This sequence is generated by:

counting the variety of ‘I’s that seem within the Roman numeral illustration of the pure numbers 1, 2, 3, …

So, for instance, the numbers given within the puzzle relate to the phrases:

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

the place you possibly can clearly see the sample:

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

if you depend the variety of I’s.

As as to if this sequence would possibly go on ceaselessly…

Strictly talking, if making use of the development guidelines of Roman numerals utilizing the principle 7 characters ‘I’, ‘V’, ‘X’, ‘L’, ‘C’, ‘D’ and ‘M’ it is just potential to create numbers as much as a price of three,999 (MMMCMXCIX), since there isn’t any strategy to assemble a price for ‘4000’ utilizing solely these characters. (And so this sequence is terminated after 3999 phrases.)

If, nonetheless, we use the frequent conference {that a} numeral with a bar above it represents a a number of of 1000 (i.e. $bar{I}$ for 1000, $bar{V}$ for 5000, and many others.) then this would not be loads of assist both, as utilizing $overline{IV}$ to attempt to proceed the sequence to 4000 would produce an ‘I’ character in a spot the place there ought to be none if the sequence have been to proceed as initially…

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