A194200 - OEIS

%I #13 Dec 12 2024 11:14:21

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

%U 26,27,27,28,28,29,29,30,30,31,32,32,33,33,34,34,34,35,36,37

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

%C The defining [sum] is equivalent to

%C ...

%C a(n)=[n(n+1)r/2]-sum{[k*r] : 1<=k<=n},

%C ...

%C 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 ...

%C Examples:

%C ...

%C r...........a........s....

%C 1/2......A002265...A001972

%C 1/3......A002264...A001840

%C 2/3......A002264...A001840

%C 1/4......A194220...A194221

%C 1/5......A194222...A118015

%C 2/5......A057354...A011858

%C 3/5......A194222...A118015

%C 4/5......A057354...A011858

%C 1/6......A194223...A194224

%C 3/7......A057357...A194229

%C 1/8......A194235...A194236

%C 3/8......A194237...A194238

%C sqrt(2)..A194161...A194162

%C sqrt(3)..A194163...A194164

%C sqrt(5)..A194169...A194170

%C sqrt(6)..A194173...A194174

%C tau......A194165...A194166; tau=(1+sqrt(5))/2

%C e........A194200...A194201

%C 2e.......A194202...A194203

%C e/2......A194204...A194205

%C pi.......A194206...A194207

%H G. C. Greubel, <a href="/A194200/b194200.txt">Table of n, a(n) for n = 1..5000</a>

%e a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]

%e     =[.718+.436+.154+.873+.591]

%e     =[2.77423]=2.

%t r = E;

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

%t Table[a[n], {n, 1, 90}]  (* A194200 *)

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

%t Table[s[n], {n, 1, 100}] (* A194201 *)

%Y Cf. A194201.

%K nonn

%O 1,4

%A _Clark Kimberling_, Aug 19 2011