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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 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

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

Sequence in context: A090787 A229707 A262680 * A096661 A199339 A323202

Adjacent sequences:  A191326 A191327 A191328 * A191330 A191331 A191332

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 31 2011

STATUS

approved

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Last modified January 22 02:46 EST 2022. Contains 350481 sequences. (Running on oeis4.)