A105446 Number of symbols in the Roman Fibonacci number representation of n.

1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 4, 3, 3, 3, 2, 3, 3, 3, 4

Offset: 1

Author

Jonathan Vos Post, Apr 09 2005

Keywords

• base
• easy
• nonn

Comments

The Roman Fibonacci numbers are composed from the values of the Fibonacci Numbers (A000045) with the grammar of the Roman Numerals (A006968) and a few rules to disambiguate.

The alphabet: {1, 2, 3, 5, 8, A=13, B=21, C=34, D=55, E=89, F=144, ...}.

Rule one: of the infinite set of representations of integers by this grammar, always restrict to the subset of those with shortest length.

Rule two: if there are two or more in the subset of shortest representations, restrict to the subset with fewest subtractions [A31 preferred to 188, B31 preferred to 1AA, CA preferred to 8D, DB preferred to AE].

Rule three: if there are two or more representations per Rules one and two, restrict to the subset with the most duplications of characters [22 preferred to 31, 33 preferred to 51, 55 preferred to 82, 88 preferred to A3, BBB preferred to D53, CC preferred to BE]. We do not need a Rule four for a while...

Lemma: no Roman Fibonacci number requires three consecutive instances of the same symbol. Proof: 3*F(i) = F(i+2) + F(i-2).

Question: what is the asymptotic length of the Roman Fibonacci numbers?

Examples

			a(1) = 1 because 1 is a Fibonacci number, equal to its own representation as a Roman Fibonacci number.
a(4) = 2 because 4 is not a Fibonacci number, but can be represented as the sum or difference of two Fibonacci numbers, with its Roman Fibonacci number representation being "22" (not "31" per rule three).
a(17) = 3 because the Roman Fibonacci number representation of 17 has three symbols, namely "A22" (not "188" per rule two).
a(80) = 4 because the Roman Fibonacci number representation of 80 has four symbols, namely "DB22".
		

References

• Cajori, F. A History of Mathematical Notations, 2 vols. Bound as One, Vol. 1: Notations in Elementary Mathematics. New York: Dover, pp. 30-37, 1993.
• Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44-45 and 281, 1992.
• Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4-5, 1969.

Links

• Eric Weisstein's World of Mathematics, Roman Numerals.
• Eric Weisstein's World of Mathematics, Fibonacci Numbers.

Crossrefs

A105447 = integers with A105446(n) = 2. A105448 = integers with A105446(n) = 3. A105449 = integers with A105446(n) = 4. A105450 = integers with A105446(n) = 5. A023150 = integers with A105446(n) = 6. A105452 = integers with A105446(n) = 7. A105453 = integers with A105446(n) = 8. A105454 = integers with A105446(n) = 9. A105455 = integers with A105446(n) = 10.

Cf. A000045, A006968.

Appears to be a duplicate of A058978.

Formula

a(n) = number of symbols in the Roman Fibonacci number representation of n, as defined in "Comments." a(n) = 1 iff n is an element of A000045. a(n) = 2 iff the shortest Roman Fibonacci number representation of n is as the sum or difference of two elements of A000045 and n is not an element of A000045.