A191329 (Lower Wythoff sequence mod 2)+(Upper Wythoff sequence mod 2).

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

Offset

1,2

Comments

Let r=(golden ratio)=(1+sqrt(5))/2 and let [ ]=floor. Let u(n)=[nr] and v(n)=n+[nr], so that u=A000201, v=A001950, the Wythoff sequences, and A191329=(u mod 2)+(v mod 2)=(number of odd numbers in {[nr],[ns]}).

The sequence A191329 can also be obtained by placing 1 before each term of 2*A078588.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = (A000201(n) mod 2) + (A001950(n) mod 2).

a(n) = A085002(n) + A171587(n). - Michel Dekking, Jan 28 2021

Example

u = (1,3,4,6,8,9,...)... = (1,1,0,0,0,1,...) in mod 2
v = (2,5,7,10,13,15,...) = (0,1,1,0,1,1,...) in mod 2,
so that......... A191329 = (1,2,1,0,1,2,...).

Mathematica

r = GoldenRatio; s = r/(r - 1); h = 500;
u = Table[Floor[n*r], {n, 1, h}]  (* A000201 *)
v = Table[Floor[n*s], {n, 1, h}]  (* A001950 *)
w = Mod[u, 2] + Mod[v, 2]  (* A191329 *)
b = Flatten[Position[w, 0]]  (* A191330=2*A005653 *)
c = Flatten[Position[w, 1]]  (* A005408, the odds *)
d = Flatten[Position[w, 2]]  (* A191331=2*A005652 *)
e = b/2; (* A005653 *)
f = d/2; (* A005652 *)
x = (1/3)^b; z = (1/3)^d;
k[n_] := x[[n]]; x1 = Sum[k[n], {n, 1, 100}];
N[x1, 100]
RealDigits[x1, 10, 100]  (* A191332 *)
k[n_] := z[[n]]; z1 = Sum[k[n], {n, 1, 100}];
N[z1, 100]
RealDigits[z1, 10, 100]  (* A191333 *)
N[x1 + z1, 100] (* Checks that x1+z1=1/8 *)
x = (1/3)^e; z = (1/3)^f;
k[n_] := x[[n]]; x2 = Sum[k[n], {n, 1, 100}];
N[x2, 100]
RealDigits[x2, 10, 100]  (* A191334 *)
k[n_] := z[[n]]; z2 = Sum[k[n], {n, 1, 100}];
N[z2, 100]
RealDigits[z2, 10, 100]  (* A191335 *)
N[x2 + z2, 100] (* checks that x2+z2=1/2 *)

Prog

(PARI) A191329(n) = { my(y=n+sqrtint(n^2*5)); (((y+n+n)\2)%2) + ((y%4)>1); }; \\ (after programs in A001950 and A085002) - Antti Karttunen, May 19 2021
(Python)
from math import isqrt
def A191329(n): return m if (m:=((n+isqrt(5*n**2))&2)+(n&1))<3 else 1 # Chai Wah Wu, Aug 10 2022

Crossrefs

Cf. A000201, A001950, A085002, A171587, A191330, A191331.

Sequence in context: A229707 A262680 A366128 * A096661 A199339 A323202

Adjacent sequences: A191326 A191327 A191328 * A191330 A191331 A191332

Keyword

nonn

Author

Clark Kimberling, May 31 2011

Status

approved