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A194220 a(n) = floor(Sum_{k=1..n} frac(k/4)).

%I A194220 #21 Dec 11 2024 22:42:29
%S A194220 0,0,1,1,1,2,3,3,3,3,4,4,4,5,6,6,6,6,7,7,7,8,9,9,9,9,10,10,10,11,12,
%T A194220 12,12,12,13,13,13,14,15,15,15,15,16,16,16,17,18,18,18,18,19,19,19,20,
%U A194220 21,21,21,21,22,22,22,23,24,24,24,24,25,25,25,26,27,27,27,27,28
%N A194220 a(n) = floor(Sum_{k=1..n} frac(k/4)).
%H A194220 G. C. Greubel, <a href="/A194220/b194220.txt">Table of n, a(n) for n = 1..5000</a>
%H A194220 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,1,-1).
%F A194220 From _Chai Wah Wu_, Jun 10 2020: (Start)
%F A194220 a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
%F A194220 G.f.: x*(x^6 + x^5 + x^2)/(x^9 - x^8 - x + 1). (End)
%F A194220 a(n) = floor((n+1)/4) + floor((n+2)/4) - floor((n+6)/8). - _Ridouane Oudra_, Nov 24 2024
%t A194220 r = 1/4;
%t A194220 a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t A194220 Table[a[n], {n, 1, 90}]    (* A194220 *)
%t A194220 s[n_] := Sum[a[k], {k, 1, n}]
%t A194220 Table[s[n], {n, 1, 100}]   (* A194221 *)
%o A194220 (PARI) a(n) = floor(sum(k=1, n, frac(k/4))); \\ _Michel Marcus_, Nov 24 2024
%Y A194220 Cf. A194221.
%K A194220 nonn,easy
%O A194220 1,6
%A A194220 _Clark Kimberling_, Aug 19 2011
%E A194220 Name edited by _Michel Marcus_, Nov 24 2024