A194236 - OEIS

%I #19 Dec 12 2024 11:24:41

%S 0,0,0,1,2,4,7,10,13,16,20,24,29,35,42,49,56,63,70,78,86,95,105,115,

%T 125,135,146,157,169,182,196,210,224,238,252,267,282,298,315,332,349,

%U 366,384,402,421,441,462,483,504,525,546,568,590,613,637,661,685

%N Partial sums of A194235.

%H G. C. Greubel, <a href="/A194236/b194236.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

%F From _Chai Wah Wu_, Jun 10 2020: (Start)

%F a(n) = 2*a(n-1) - a(n-2) + a(n-16) - 2*a(n-17) + a(n-18) for n > 18.

%F G.f.: x*(-x^14 - x^13 - x^12 - x^10 - x^6 - x^5 - x^3)/(x^18 - 2*x^17 + x^16 - x^2 + 2*x - 1). (End)

%t r = 1/8;

%t a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]

%t Table[a[n], {n, 1, 90}]    (* A194235 *)

%t s[n_] := Sum[a[k], {k, 1, n}]

%t Table[s[n], {n, 1, 100}]   (* A194236 *)

%o (PARI) f(n) = floor(sum(k=1, n, frac(k/8)));

%o a(n) = sum(k=1, n, f(k)); \\ _Michel Marcus_, Nov 03 2017

%Y Cf. A194235.

%K nonn

%O 1,5

%A _Clark Kimberling_, Aug 20 2011