A194236 Partial sums of A194235.

0, 0, 0, 1, 2, 4, 7, 10, 13, 16, 20, 24, 29, 35, 42, 49, 56, 63, 70, 78, 86, 95, 105, 115, 125, 135, 146, 157, 169, 182, 196, 210, 224, 238, 252, 267, 282, 298, 315, 332, 349, 366, 384, 402, 421, 441, 462, 483, 504, 525, 546, 568, 590, 613, 637, 661, 685

Offset: 1

Author

Clark Kimberling, Aug 20 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, 0, 0, 0, 0, 1, -2, 1).

Crossrefs

Cf. A194235.

Programs

Mathematica

r = 1/8;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]    (* A194235 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194236 *)

PARI

f(n) = floor(sum(k=1, n, frac(k/8)));
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Nov 03 2017

Formula

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

a(n) = 2*a(n-1) - a(n-2) + a(n-16) - 2*a(n-17) + a(n-18) for n > 18.

G.f.: x*(-x^14 - x^13 - x^12 - x^10 - x^6 - x^5 - x^3)/(x^18 - 2*x^17 + x^16 - x^2 + 2*x - 1). (End)