A194198 - OEIS

%I #11 Jul 17 2019 18:02:09

%S 0,0,0,1,2,3,3,3,4,4,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,12,13,13,14,

%T 15,15,15,16,16,17,18,18,18,18,19,19,20,21,21,21,22,23,23,24,24,25,25,

%U 26,26,27,27,28,28,29,29,30,30,31,31,32,32,33,33,33,34,34,35,36

%N [sum{(k*r) : 1<=k<=n}], where [ ]=floor, ( )=fractional part, and r=sqrt(10).

%H Robert Israel, <a href="/A194198/b194198.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ n/2 by Weyl's equidistribution theorem. - _Robert Israel_, Jul 17 2019

%p map(floor,ListTools:-PartialSums([seq(frac(k*sqrt(10)),k=1..100)])); # _Robert Israel_, Jul 17 2019

%t r = Sqrt[10];

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

%t Table[a[n], {n, 1, 90}]  (* A194198 *)

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

%t Table[s[n], {n, 1, 100}] (* A194199 *)

%Y Cf. A194199.

%K nonn

%O 1,5

%A _Clark Kimberling_, Aug 19 2011