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A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

%I A194200 #13 Dec 12 2024 11:14:21
%S A194200 0,1,1,2,2,3,3,3,4,4,5,6,6,6,7,7,7,8,9,9,9,10,11,11,12,13,13,13,14,15,
%T A194200 15,16,16,17,17,18,18,19,19,19,20,20,21,22,22,22,23,23,23,24,25,25,25,
%U A194200 26,27,27,28,28,29,29,30,30,31,32,32,33,33,34,34,34,35,36,37
%N A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.
%C A194200 The defining [sum] is equivalent to
%C A194200 ...
%C A194200 a(n)=[n(n+1)r/2]-sum{[k*r] : 1<=k<=n},
%C A194200 ...
%C A194200 where []=floor and r=sqrt(2).  Let s(n) denote the n-th partial sum of the sequence a; then the difference sequence d defined by d(n)=s(n+1)-s(n) gives the runlengths of a.
%C A194200 ...
%C A194200 Examples:
%C A194200 ...
%C A194200 r...........a........s....
%C A194200 1/2......A002265...A001972
%C A194200 1/3......A002264...A001840
%C A194200 2/3......A002264...A001840
%C A194200 1/4......A194220...A194221
%C A194200 1/5......A194222...A118015
%C A194200 2/5......A057354...A011858
%C A194200 3/5......A194222...A118015
%C A194200 4/5......A057354...A011858
%C A194200 1/6......A194223...A194224
%C A194200 3/7......A057357...A194229
%C A194200 1/8......A194235...A194236
%C A194200 3/8......A194237...A194238
%C A194200 sqrt(2)..A194161...A194162
%C A194200 sqrt(3)..A194163...A194164
%C A194200 sqrt(5)..A194169...A194170
%C A194200 sqrt(6)..A194173...A194174
%C A194200 tau......A194165...A194166; tau=(1+sqrt(5))/2
%C A194200 e........A194200...A194201
%C A194200 2e.......A194202...A194203
%C A194200 e/2......A194204...A194205
%C A194200 pi.......A194206...A194207
%H A194200 G. C. Greubel, <a href="/A194200/b194200.txt">Table of n, a(n) for n = 1..5000</a>
%e A194200 a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]
%e A194200     =[.718+.436+.154+.873+.591]
%e A194200     =[2.77423]=2.
%t A194200 r = E;
%t A194200 a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
%t A194200 Table[a[n], {n, 1, 90}]  (* A194200 *)
%t A194200 s[n_] := Sum[a[k], {k, 1, n}]
%t A194200 Table[s[n], {n, 1, 100}] (* A194201 *)
%Y A194200 Cf. A194201.
%K A194200 nonn
%O A194200 1,4
%A A194200 _Clark Kimberling_, Aug 19 2011