A194220 -id:A194220 - cp's OEIS frontend

A194221 Partial sums of A194220.

0, 0, 1, 2, 3, 5, 8, 11, 14, 17, 21, 25, 29, 34, 40, 46, 52, 58, 65, 72, 79, 87, 96, 105, 114, 123, 133, 143, 153, 164, 176, 188, 200, 212, 225, 238, 251, 265, 280, 295, 310, 325, 341, 357, 373, 390, 408, 426, 444, 462, 481, 500, 519, 539, 560, 581, 602

Offset: 1

Clark Kimberling, Aug 19 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A194220.

seq(coeff(series( x^3*(1+x^3+x^4)/((1-x)^2*(1-x^8)) , x, n+1), x, n), n = 1..60); # G. C. Greubel, Jun 10 2020

r = 1/4;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A194220 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194221 *)

From Chai Wah Wu, Jun 10 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10) for n > 10.
G.f.: x^3*(1 + x^3 + x^4)/((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x^4)). (End)

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

A194221 Partial sums of A194220.

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