

A278313


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


1



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
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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



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
a(n) = a(n5) 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*(n3)*Pi/5) + 2*cos(4*(n3)*Pi/5) + cos(2*(n2)*Pi/5) + cos(4*(n2)*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



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

(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



KEYWORD

nonn,base


AUTHOR



STATUS

approved



