%I #20 Aug 11 2022 03:19:32
%S 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,
%T 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,
%U 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
%N (Lower Wythoff sequence mod 2)+(Upper Wythoff sequence mod 2).
%C 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]}).
%C The sequence A191329 can also be obtained by placing 1 before each term of 2*A078588.
%H Antti Karttunen, <a href="/A191329/b191329.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = (A000201(n) mod 2) + (A001950(n) mod 2).
%F a(n) = A085002(n) + A171587(n). - _Michel Dekking_, Jan 28 2021
%e u = (1,3,4,6,8,9,...)... = (1,1,0,0,0,1,...) in mod 2
%e v = (2,5,7,10,13,15,...) = (0,1,1,0,1,1,...) in mod 2,
%e so that......... A191329 = (1,2,1,0,1,2,...).
%t r = GoldenRatio; s = r/(r - 1); h = 500;
%t u = Table[Floor[n*r], {n, 1, h}] (* A000201 *)
%t v = Table[Floor[n*s], {n, 1, h}] (* A001950 *)
%t w = Mod[u, 2] + Mod[v, 2] (* A191329 *)
%t b = Flatten[Position[w, 0]] (* A191330=2*A005653 *)
%t c = Flatten[Position[w, 1]] (* A005408, the odds *)
%t d = Flatten[Position[w, 2]] (* A191331=2*A005652 *)
%t e = b/2; (* A005653 *)
%t f = d/2; (* A005652 *)
%t x = (1/3)^b; z = (1/3)^d;
%t k[n_] := x[[n]]; x1 = Sum[k[n], {n, 1, 100}];
%t N[x1, 100]
%t RealDigits[x1, 10, 100] (* A191332 *)
%t k[n_] := z[[n]]; z1 = Sum[k[n], {n, 1, 100}];
%t N[z1, 100]
%t RealDigits[z1, 10, 100] (* A191333 *)
%t N[x1 + z1, 100] (* Checks that x1+z1=1/8 *)
%t x = (1/3)^e; z = (1/3)^f;
%t k[n_] := x[[n]]; x2 = Sum[k[n], {n, 1, 100}];
%t N[x2, 100]
%t RealDigits[x2, 10, 100] (* A191334 *)
%t k[n_] := z[[n]]; z2 = Sum[k[n], {n, 1, 100}];
%t N[z2, 100]
%t RealDigits[z2, 10, 100] (* A191335 *)
%t N[x2 + z2, 100] (* checks that x2+z2=1/2 *)
%o (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
%o (Python)
%o from math import isqrt
%o def A191329(n): return m if (m:=((n+isqrt(5*n**2))&2)+(n&1))<3 else 1 # _Chai Wah Wu_, Aug 10 2022
%Y Cf. A000201, A001950, A085002, A171587, A191330, A191331.
%K nonn
%O 1,2
%A _Clark Kimberling_, May 31 2011