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

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

G. C. Greubel, Table of n, a(n) for n = 1..5000

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

Adjacent sequences: A194197 A194198 A194199 * A194201 A194202 A194203

Keyword

nonn

Author

Clark Kimberling, Aug 19 2011

Status

approved