A194161 -id:A194161 - cp's OEIS frontend

A194162 Partial sums of A194161.

0, 1, 2, 4, 6, 8, 11, 14, 18, 22, 27, 33, 39, 46, 53, 61, 69, 77, 86, 95, 105, 115, 126, 138, 150, 163, 176, 190, 204, 218, 233, 248, 264, 280, 296, 313, 331, 349, 368, 387, 407, 428, 449, 471, 493, 515, 538, 562, 586, 611, 636, 661, 687, 714, 741, 769

Offset: 1

Clark Kimberling, Aug 18 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A194161.

r = Sqrt[2];
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]
s[n_] := Sum[a[k], {k, 1, n}]   (* A194161 *)
Table[s[n], {n, 1, 100}]
b[n_] := Floor[n (n + 1) r/2] - Sum[Floor[k*r], {k, 1, n}]
Table[b[n], {n, 1, 90}]   (* A194162 *)

b(n) = floor(sum(k=1, n, frac(k*sqrt(2))));
a(n) = sum(k=1, n, b(k)); \\ Michel Marcus, Nov 06 2017

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

Original entry on oeis.org

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

Clark Kimberling, Aug 19 2011

nonn

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...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

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

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

Cf. A194201.

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 *)

cp's OEIS Frontend

A194162 Partial sums of A194161.

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