A194224 Partial sums of A194223.

0, 0, 1, 2, 4, 6, 8, 11, 14, 18, 23, 28, 33, 38, 44, 50, 57, 64, 71, 79, 87, 96, 106, 116, 126, 136, 147, 158, 170, 182, 194, 207, 220, 234, 249, 264, 279, 294, 310, 326, 343, 360, 377, 395, 413, 432, 452, 472, 492, 512, 533, 554, 576, 598, 620, 643, 666

Offset: 1

Author

Clark Kimberling, Aug 19 2011

Keywords

• nonn

Links

• 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, 0, 0, 0, 0, 1, -2, 1).

Crossrefs

Cf. A194223.

Programs

Mathematica

r = 1/6;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A194223 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194224 *)

PARI

for(n=1,50, print1(sum(j=1,n, floor(sum(k=1,j, frac(k/6)))), ", ")) \\ G. C. Greubel, Oct 29 2017

Formula

From Chai Wah Wu, Jun 10 2020: (Start)

a(n) = 2*a(n-1) - a(n-2) + a(n-12) - 2*a(n-13) + a(n-14) for n > 14.

G.f.: x*(-x^10 - x^9 - x^7 - x^4 - x^2)/(x^14 - 2*x^13 + x^12 - x^2 + 2*x - 1). (End)