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A194224 Partial sums of A194223.

%I A194224 #18 Dec 12 2024 11:18:30
%S A194224 0,0,1,2,4,6,8,11,14,18,23,28,33,38,44,50,57,64,71,79,87,96,106,116,
%T A194224 126,136,147,158,170,182,194,207,220,234,249,264,279,294,310,326,343,
%U A194224 360,377,395,413,432,452,472,492,512,533,554,576,598,620,643,666
%N A194224 Partial sums of A194223.
%H A194224 G. C. Greubel, <a href="/A194224/b194224.txt">Table of n, a(n) for n = 1..1000</a>
%H A194224 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).
%F A194224 From _Chai Wah Wu_, Jun 10 2020: (Start)
%F A194224 a(n) = 2*a(n-1) - a(n-2) + a(n-12) - 2*a(n-13) + a(n-14) for n > 14.
%F A194224 G.f.: x*(-x^10 - x^9 - x^7 - x^4 - x^2)/(x^14 - 2*x^13 + x^12 - x^2 + 2*x - 1). (End)
%t A194224 r = 1/6;
%t A194224 a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t A194224 Table[a[n], {n, 1, 90}]    (* A194223 *)
%t A194224 s[n_] := Sum[a[k], {k, 1, n}]
%t A194224 Table[s[n], {n, 1, 100}]   (* A194224 *)
%o A194224 (PARI) for(n=1,50, print1(sum(j=1,n, floor(sum(k=1,j, frac(k/6)))), ", ")) \\ _G. C. Greubel_, Oct 29 2017
%Y A194224 Cf. A194223.
%K A194224 nonn
%O A194224 1,4
%A A194224 _Clark Kimberling_, Aug 19 2011