cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Jonathan Vos Post, Apr 09 2005

Keywords

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

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.

A105448 Positive integers whose representation as Roman Fibonacci numbers has exactly three symbols.

Original entry on oeis.org

17, 25, 27, 28, 30, 38, 40, 41, 43, 44, 45, 46, 48, 49, 51, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 82, 83, 85, 93, 95, 96, 98, 99, 100, 101, 103, 104, 105, 107, 108, 109, 111, 112, 113, 115, 118, 124, 125, 126, 128, 132, 133, 134, 136, 140
Offset: 1

Views

Author

Jonathan Vos Post, Apr 09 2005

Keywords

Comments

A105446 gives the number of symbols in Roman Fibonacci numbers. A105447 gives positive integers whose representation as Roman Fibonacci numbers has exactly two symbols. A105449 gives positive integers whose representation as Roman Fibonacci numbers has exactly four symbols. A105450 gives positive integers whose representation as Roman Fibonacci numbers has exactly five symbols.

Examples

			In Roman Fibonacci number representation:
17 = "A22", 25 = "B22", 27 = "AA1", 28 = "AA2", 30 = "B81", 38 = "C22", 40 = "C33", 41 = "C52", 43 = "BB1", 44 = "BB2", 45 = "BB3", 46 = "1CA", 48 = "CA1", ..., 155 = "2FA", 156 = "1FA", 158 = "FA1", 159 = "FA2", 160 = "FA3".

Crossrefs

Cf. A000045, A006968, A105446, A105447, A105449-A105455.

Formula

a(n) is an element of A105447 iff A105446(n) = 3.

A105449 Positive integers whose representation as Roman Fibonacci numbers has exactly four symbols.

Original entry on oeis.org

80, 106, 114, 116, 117, 119, 120, 121, 122, 127, 129, 130, 135, 137, 138
Offset: 1

Views

Author

Jonathan Vos Post, Apr 09 2005

Keywords

Comments

A105446 gives the number of symbols in Roman Fibonacci numbers. A105447 gives positive integers whose representation as Roman Fibonacci numbers has exactly two symbols. A105448 gives positive integers whose representation as Roman Fibonacci numbers has exactly three symbols. A105450 gives positive integers whose representation as Roman Fibonacci numbers has exactly five symbols.

Examples

			In Roman Fibonacci number representation:
80 = "DB22", 106 = "EA22", 114 = "DD22", 116 = "DD33", 117 = "DD52", 119 = "DD81", 120 = "DD55", 121 = "DD2A", 122 = "DD1A", 127 = "BF22", 129 = "BF33", 130 = "BF52", 135 = "AF22", 137 = "AF33", 138 = "AF52".

Crossrefs

Cf. A000045, A006968, A105446-A105448, A105450-A105455.

Formula

a(n) is an element of A105449 iff A105446(n) = 4.

A051743 a(n) = (1/24)*n*(n + 5)*(n^2 + n + 6).

Original entry on oeis.org

2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, 1377, 1833, 2394, 3075, 3892, 4862, 6003, 7334, 8875, 10647, 12672, 14973, 17574, 20500, 23777, 27432, 31493, 35989, 40950, 46407, 52392, 58938, 66079, 73850, 82287, 91427, 101308, 111969, 123450
Offset: 1

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

Keywords

Comments

This is exactly the number of directed column-convex polyominoes. [Something is clearly missing from this sentence; as it stands, it makes no reference to the index n. - Jon E. Schoenfield, Dec 20 2016]
Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-3)=coeff(charpoly(A,x),x^(n-4)). [Milan Janjic, Jan 24 2010]

Links

Crossrefs

Cf. A005435, A006027, A105450.

Programs

  • Mathematica
    Table[(n (n + 5) (n^2 + n + 6))/24, {n, 50}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {2, 7, 18, 39, 75}, 50]
  • PARI
    Vec((x^3-3*x^2+3*x-2)/(x-1)^5 + O(x^50)) \\ G. C. Greubel, Dec 21 2016

Formula

a(n) = binomial(n+3, n-1) + binomial(n, n-1) = binomial(n+3, 4) + binomial(n, 1), n > 0.
From Harvey P. Dale, Nov 29 2011: (Start)
a(1)=2, a(2)=7, a(3)=18, a(4)=39, a(5)=75, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (x^3-3*x^2+3*x-2)/(x-1)^5. (End)
E.g.f.: (1/24)*(48*x + 36*x^2 + 12*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 21 2016
Showing 1-4 of 4 results.

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