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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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:

...

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

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Last modified November 29 04:38 EST 2021. Contains 349416 sequences. (Running on oeis4.)