A191329 - OEIS

%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