A190248 a(n) = [nu+nv+nw]-[nu]-[nv]-[nw], where u=(1+sqrt(5))/2, v=u^2, w=u^3, []=floor.

1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 2, 1, 0, 1, 1, 2, 1, 0, 1

Offset: 1

Author

Clark Kimberling, May 06 2011

Keywords

• nonn

Comments

a(n) = A190440(n) - A078588(n). This follows from substituting w = 1+2u, v = 1+u, and taking 2n, n and n out of the floor functions. - Michel Dekking, Oct 21 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Burghard Herrmann, How integer sequences find their way into areas outside pure mathematics, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 67-71.

Crossrefs

Cf. A190249, A190250, A190251.

Programs

Magma

[Floor(2*n*(2+Sqrt(5))) - Floor(n*(1+Sqrt(5))/2) - Floor(n*(3 + Sqrt(5))/2): n in [1..30]]; // G. C. Greubel, Dec 26 2017

Mathematica

u = GoldenRatio; v = u^2; w=u^3;
f[n_] := Floor[n*u + n*v + n*w] - Floor[n*u] - Floor[n*v] - Floor[n*w]
t = Table[f[n], {n, 1, 120}] (* A190248 *)
Flatten[Position[t, 0]]      (* A190249 *)
Flatten[Position[t, 1]]      (* A190250 *)
Flatten[Position[t, 2]]      (* A190251 *)

PARI

for(n=1,30, print1(floor(2*n*(2+sqrt(5))) - floor(n*(1+sqrt(5))/2) - floor(n*(3 + sqrt(5))/2) - floor(n*(2 + sqrt(5))), ", ")) \\ G. C. Greubel, Dec 26 2017

Formula

a(n) = [2n+4nu]-[nu]-[n+nu]-[n+2nu], where u=(1+sqrt(5))/2. - Michel Dekking, Oct 21 2016

Extensions

Name corrected by Michel Dekking, Oct 21 2016