A194165 a(n) = [Sum_{k=1..n} (k*r)], where [ ]=floor, ( )=fractional part, and r = (1+sqrt(5))/2, the golden ratio.

0, 0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 12, 12, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 33, 34, 34, 35, 36, 36

Offset: 1

Author

Clark Kimberling, Aug 18 2011

Keywords

• nonn

Links

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

Crossrefs

Cf. A194166.

Programs

Mathematica

r = GoldenRatio;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]  (* A194165 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A194166 *)

PARI

a(n) = floor(sum(k=1, n, frac((1+sqrt(5))*k/2))); \\ Michel Marcus, Nov 12 2017