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

Author

Clark Kimberling, Aug 19 2011

Keywords

• nonn

Comments

The defining [sum] is equivalent to

...

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

...

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.

...

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...A118015

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

Examples

			a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]
    =[.718+.436+.154+.873+.591]
    =[2.77423]=2.
		

Links

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

Crossrefs

Cf. A194201.

Programs

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 *)