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A194223 a(n) = [sum{(k/6) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

%I A194223 #19 Dec 12 2024 11:17:35
%S A194223 0,0,1,1,2,2,2,3,3,4,5,5,5,5,6,6,7,7,7,8,8,9,10,10,10,10,11,11,12,12,
%T A194223 12,13,13,14,15,15,15,15,16,16,17,17,17,18,18,19,20,20,20,20,21,21,22,
%U A194223 22,22,23,23,24,25,25,25,25,26,26,27,27,27,28,28,29,30,30,30,30
%N A194223 a(n) = [sum{(k/6) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.
%H A194223 G. C. Greubel, <a href="/A194223/b194223.txt">Table of n, a(n) for n = 1..5000</a>
%H A194223 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
%F A194223 From _Chai Wah Wu_, Jun 10 2020: (Start)
%F A194223 a(n) = a(n-1) + a(n-12) - a(n-13) for n > 13.
%F A194223 G.f.: x*(x^10 + x^9 + x^7 + x^4 + x^2)/(x^13 - x^12 - x + 1). (End)
%t A194223 r = 1/6;
%t A194223 a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t A194223 Table[a[n], {n, 1, 90}]    (* A194223 *)
%t A194223 s[n_] := Sum[a[k], {k, 1, n}]
%t A194223 Table[s[n], {n, 1, 100}]   (* A194224 *)
%o A194223 (PARI) for(n=1,50, print1(floor(sum(k=1,n, frac(k/6))), ", ")) \\ _G. C. Greubel_, Oct 29 2017
%Y A194223 Cf. A194224.
%K A194223 nonn
%O A194223 1,5
%A A194223 _Clark Kimberling_, Aug 19 2011