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A191329
(Lower Wythoff sequence mod 2)+(Upper Wythoff sequence mod 2).
12
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 31 2011
STATUS
approved