A278313 Number of letters "I" in Roman numeral representation of n.

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

Offset

1,2

Comments

Period 5: repeat [1, 2, 3, 1, 0]. - Omar E. Pol, Nov 19 2016

For large numbers we have the examples: 8000 -> VMMM (underline the V); 80000 -> LXXX (underline the LXXX); 800000 -> DCCC (underline the DCCC); ... see The Rules of Roman Numerals under Links. - José de Jesús Camacho Medina, Nov 21 2016

Links

Table of n, a(n) for n=1..100.

K6Math.com, The Rules of Roman Numerals

Eric Weisstein's World of Mathematics, Roman numerals

Wikipedia, Roman numerals

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Formula

a(n) = (n mod 5) - ((3n + 2n^2 + 3n^3 + 2n^4) mod 5).

G.f.: x*(1 + 2*x + 3*x^2 + x^3)/((1 - x)*(1 + x + x^2 + x^3 + x^4)). - Ilya Gutkovskiy, Nov 20 2016

From Wesley Ivan Hurt, Dec 26 2016: (Start)

a(n) = a(n-5) for n > 5.

a(n) = (7 + (n mod 5) + 2*((n+1) mod 5) - ((n+2) mod 5) - ((n+3) mod 5) - ((n+4) mod 5))/5. (End)

a(n) = 1 + (2/5)*(1 + 2*cos(2*(n-3)*Pi/5) + 2*cos(4*(n-3)*Pi/5) + cos(2*(n-2)*Pi/5) + cos(4*(n-2)*Pi/5) - cos(2*n*Pi/5) - cos(4*n*Pi/5)). - Wesley Ivan Hurt, Oct 04 2018

Example

a(1) = 1 because 1 in Roman numerals is I, which contains only one I.
a(2) = 2 because 2 in Roman numerals is II, which contains two I's.
a(3) = 3 because 3 in Roman numerals is III, which contains three I's.
a(4) = 1 because 4 in Roman numerals is IV, which contains only one I.
a(5) = 0 because 5 in Roman numerals is V, which does not contain I's.
a(6) = 1 because 6 in Roman numerals is VI, which contains only one I.
a(7) = 2 because 7 in Roman numerals is VII, which contains two I's.
a(8) = 3 because 8 in Roman numerals is VIII, which contains three I's.
a(9) = 1 because 9 in Roman numerals is IX, which contains only one I.
a(10) = 0 because 10 in Roman numerals is X, which does not contain I's.
a(50) = 0 because 50 in Roman numerals is L, which does not contain I's.
a(100) = 0 because 100 in Roman numerals is C, which does not contain I's.
a(500) = 0 because 500 in Roman numerals is D, which does not contain I's.
a(551) = 1 because 551 in Roman numerals is DLI, which contains only one I.
a(1000) = 0 because 1000 in Roman numerals is M, which does not contain I's.
a(1001) = 1 because 1001 in Roman numerals is MI, which contains only one I.

Maple

A278313:= n -> [1, 2, 3, 1, 0][(n mod 5)+1]: seq(A278313(n), n=0..100); # Wesley Ivan Hurt, Dec 26 2016

Mathematica

Table[Mod[n, 5] - Mod[3n + 2n^2 + 3n^3 + 2n^4, 5], {n, 100}]
Table[StringCount[RomanNumeral@ n, "I"], {n, 105}] (* Michael De Vlieger, Nov 24 2016, Version 10.2 *)

Prog

(Magma) &cat [[1, 2, 3, 1, 0]^^30]; // Wesley Ivan Hurt, Dec 26 2016
(PARI) Vec(x*(1 + 2*x + 3*x^2 + x^3)/((1 - x)*(1 + x + x^2 + x^3 + x^4)) + O(x^50)) \\ G. C. Greubel, Dec 26 2016

Crossrefs

Cf. A006968.

Sequence in context: A321897 A050074 A346688 * A006705 A031269 A006703

Adjacent sequences: A278310 A278311 A278312 * A278314 A278315 A278316

Keyword

nonn,base

Author

José de Jesús Camacho Medina, Nov 17 2016

Status

approved