A105446 - OEIS

%I #11 Dec 06 2022 09:50:04

%S 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,

%T 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,

%U 3,3,3,4,3,3,3,2,3,3,3,4

%N Number of symbols in the Roman Fibonacci number representation of n.

%C 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.

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

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

%C 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].

%C 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...

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

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

%D 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.

%D Menninger, K. Number Words and Number Symbols: A Cultural History of Numbers. New York: Dover, pp. 44-45 and 281, 1992.

%D Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 4-5, 1969.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RomanNumerals.html">Roman Numerals</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Numbers</a>.

%F 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.

%e a(1) = 1 because 1 is a Fibonacci number, equal to its own representation as a Roman Fibonacci number.

%e 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).

%e a(17) = 3 because the Roman Fibonacci number representation of 17 has three symbols, namely "A22" (not "188" per rule two).

%e a(80) = 4 because the Roman Fibonacci number representation of 80 has four symbols, namely "DB22".

%Y 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.

%Y Cf. A000045, A006968.

%Y Appears to be a duplicate of A058978.

%K base,easy,nonn

%O 1,4

%A _Jonathan Vos Post_, Apr 09 2005