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a(n) = [Sum_{k=1..n} (r*k)], where [ ]=floor, ( )=fractional part, and r=(1+sqrt(2))/2.
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%I #12 Aug 20 2018 07:52:56

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

%T 14,15,16,16,16,16,17,18,18,18,19,20,20,21,21,21,22,23,23,24,24,25,26,

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

%N a(n) = [Sum_{k=1..n} (r*k)], where [ ]=floor, ( )=fractional part, and r=(1+sqrt(2))/2.

%H G. C. Greubel, <a href="/A194239/b194239.txt">Table of n, a(n) for n = 1..10000</a>

%t r = (1+Sqrt[2])/2;

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

%t Table[a[n], {n, 1, 90}] (* A194239 *)

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

%t Table[s[n], {n, 1, 100}] (* A194240 *)

%Y Cf. A194240.

%K nonn

%O 1,4

%A _Clark Kimberling_, Aug 20 2011