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A194200
[sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.
2
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, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 34, 35, 36, 37
OFFSET
1,4
COMMENTS
The defining [sum] is equivalent to
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a(n)=[n(n+1)r/2]-sum{[k*r] : 1<=k<=n},
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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.
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Examples:
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r...........a........s....
1/2......A002265...A001972
1/3......A002264...A001840
2/3......A002264...A001840
1/4......A194220...A194221
1/5......A194222...A118015
2/5......A057354...A011858
3/5......A194222...A011815
4/5......A057354...A011858
1/6......A194223...A194224
3/7......A057357...A194229
1/8......A194235...A194236
3/8......A194237...A194238
sqrt(2)..A194161...A194162
sqrt(3)..A194163...A194164
sqrt(5)..A194169...A194170
sqrt(6)..A194173...A194174
tau......A194165...A194166; tau=(1+sqrt(5))/2
e........A194200...A194201
2e.......A194202...A194203
e/2......A194204...A194205
pi.......A194206...A194207
LINKS
EXAMPLE
a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]
=[.718+.436+.154+.873+.591]
=[2.77423]=2.
MATHEMATICA
r = E;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}] (* A194200 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A194201 *)
CROSSREFS
Cf. A194201.
Sequence in context: A029123 A025777 A269862 * A242736 A194237 A145707
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 19 2011
STATUS
approved