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
Examples
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.
Links
- 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).
Crossrefs
Cf. A006968.
Programs
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Magma
&cat [[1, 2, 3, 1, 0]^^30]; // Wesley Ivan Hurt, Dec 26 2016
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Maple
A278313:= n -> [1, 2, 3, 1, 0][(n mod 5)+1]: seq(A278313(n), n=0..100); # Wesley Ivan Hurt, Dec 26 2016
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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 *) -
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
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
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