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A057353 a(n) = floor(3n/4).

%I A057353 #81 Mar 10 2025 11:01:45
%S A057353 0,0,1,2,3,3,4,5,6,6,7,8,9,9,10,11,12,12,13,14,15,15,16,17,18,18,19,
%T A057353 20,21,21,22,23,24,24,25,26,27,27,28,29,30,30,31,32,33,33,34,35,36,36,
%U A057353 37,38,39,39,40,41,42,42,43,44,45,45,46,47,48,48,49,50,51,51,52,53,54
%N A057353 a(n) = floor(3n/4).
%C A057353 The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.
%C A057353 For n >= 2, a(n) is the number of different integers that can be written as floor(k^2/n) for k = 1, 2, 3, ..., n-1. Generalization of the 1st problem proposed during the 15th Balkan Mathematical Olympiad in 1998 where the question was asked for n = 1998 with a(1998) = 1498. - _Bernard Schott_, Apr 22 2022
%C A057353 For n > 1, a(n) is also the Hadwiger number of the (n+1)-cycle complement graph (up to at least n = 16). - _Eric W. Weisstein_, Mar 10 2025
%D A057353 N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
%D A057353 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.
%H A057353 Vincenzo Librandi, <a href="/A057353/b057353.txt">Table of n, a(n) for n = 0..10000</a>
%H A057353 Balkan Mathematical Olympiad, <a href="http://schoolexercisebooks.blogspot.com/2010/09/15th-balkan-mathematical-olympiad.html">Problem 1</a>, 15th Balkan Mathematical Olympiad 1998.
%H A057353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleComplementGraph.html">Cycle Complement Graph</a>.
%H A057353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HadwigerNumber.html">Hadwiger Number</a>.
%H A057353 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%H A057353 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>.
%H A057353 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%F A057353 G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
%F A057353 For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3
%F A057353 a(n) = A002378(n) - A173562(n). - _Reinhard Zumkeller_, Feb 21 2010
%F A057353 a(n+1) = A140201(n) - A002265(n+1). - _Reinhard Zumkeller_, Jan 26 2011
%F A057353 a(n) = n-1 - A002265(n-1) = ( A007310(n) + A057077(n+1) )/4 for n>0. a(n) = a(n-1)+a(n-4)-a(n-5) for n>4. - _Bruno Berselli_, Jan 28 2011
%F A057353 a(n) = 1/8*(6*n + 2*cos((Pi*n)/2) + cos(Pi*n) - 2*sin((Pi*n)/2) - 3). - _Ilya Gutkovskiy_, Sep 18 2015
%F A057353 a(4n) = a(4n+1). - _Altug Alkan_, Sep 26 2015
%F A057353 Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) (A073010). - _Amiram Eldar_, Sep 29 2022
%t A057353 Table[Floor[3 n/4], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)
%t A057353 Floor[3 Range[0, 20]/4] (* _Eric W. Weisstein_, Mar 10 2025 *)
%t A057353 LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 3, 3}, {0, 20}] (* _Eric W. Weisstein_, Mar 10 2025 *)
%t A057353 CoefficientList[Series[x^2 (1 + x + x^2)/(1 - x - x^4 + x^5), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 10 2025 *)
%o A057353 (Magma) [Floor(3*n/4): n in [0..90]]; // _Vincenzo Librandi_, Feb 12 2012
%o A057353 (PARI) a(n)=3*n\4 \\ _Charles R Greathouse IV_, Sep 02 2015
%Y A057353 Floors of other ratios: A004526, A002264, A002265, A004523, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.
%Y A057353 Cf. A002378, A007310, A057077, A073010, A173562, A182210.
%K A057353 nonn,easy
%O A057353 0,4
%A A057353 _Mitch Harris_