A194220 a(n) = floor(Sum_{k=1..n} frac(k/4)).

0, 0, 1, 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 11, 12, 12, 12, 12, 13, 13, 13, 14, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 25, 26, 27, 27, 27, 27, 28

Offset: 1

Author

Clark Kimberling, Aug 19 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A194221.

Programs

Mathematica

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 *)

PARI

a(n) = floor(sum(k=1, n, frac(k/4))); \\ Michel Marcus, Nov 24 2024

Formula

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

a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.

G.f.: x*(x^6 + x^5 + x^2)/(x^9 - x^8 - x + 1). (End)

a(n) = floor((n+1)/4) + floor((n+2)/4) - floor((n+6)/8). - Ridouane Oudra, Nov 24 2024

Extensions

Name edited by Michel Marcus, Nov 24 2024