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%I #12 Nov 12 2017 03:03:50
%S 0,0,1,2,2,2,3,4,4,4,5,6,6,6,7,8,8,8,9,9,10,11,11,12,12,12,13,13,14,
%T 15,15,16,16,16,17,17,18,18,19,19,20,21,21,21,22,23,23,23,24,24,25,25,
%U 26,26,27,28,28,29,29,30,30,31,31,32,32,33,33,33,34,34,35,36,36
%N a(n) = [Sum_{k=1..n} (k*r)], where [ ]=floor, ( )=fractional part, and r = (1+sqrt(5))/2, the golden ratio.
%H G. C. Greubel, <a href="/A194165/b194165.txt">Table of n, a(n) for n = 1..5000</a>
%t r = GoldenRatio;
%t a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t Table[a[n], {n, 1, 90}] (* A194165 *)
%t s[n_] := Sum[a[k], {k, 1, n}]
%t Table[s[n], {n, 1, 100}] (* A194166 *)
%o (PARI) a(n) = floor(sum(k=1, n, frac((1+sqrt(5))*k/2))); \\ _Michel Marcus_, Nov 12 2017
%Y Cf. A194166.
%K nonn
%O 1,4
%A _Clark Kimberling_, Aug 18 2011
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