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Numbers k such that d(r,k) = 0 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.
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%I #14 Dec 03 2014 00:57:02

%S 3,9,10,23,31,44,49,56,57,58,59,70,75,80,84,85,86,87,88,89,93,94,95,

%T 96,97,104,116,119,120,121,122,128,129,130,131,134,135,136,139,140,

%U 141,142,145,146,149,166,173,174,177,182,185,190,191,199,200,201,208

%N Numbers k such that d(r,k) = 0 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

%C Every positive integer lies in exactly one of these: A247519, A247520, A247521, A247522. Prefixing the binary digits for r by 1 gives the binary digits for s.

%H Clark Kimberling, <a href="/A247519/b247519.txt">Table of n, a(n) for n = 1..1000</a>

%e r has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...

%e s has binary digits 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, ...

%e so that a(1) = 3.

%t z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1/2];

%t u = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[r, 2, z]]

%t v = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[s, 2, z]]

%t t1 = Table[If[u[[n]] == 0 && v[[n]] == 0, 1, 0], {n, 1, z}];

%t t2 = Table[If[u[[n]] == 0 && v[[n]] == 1, 1, 0], {n, 1, z}];

%t t3 = Table[If[u[[n]] == 1 && v[[n]] == 0, 1, 0], {n, 1, z}];

%t t4 = Table[If[u[[n]] == 1 && v[[n]] == 1, 1, 0], {n, 1, z}];

%t Flatten[Position[t1, 1]] (* A247519 *)

%t Flatten[Position[t2, 1]] (* A247520 *)

%t Flatten[Position[t3, 1]] (* A247521 *)

%t Flatten[Position[t4, 1]] (* A247522 *)

%Y Cf. A247520, A247521, A247522.

%K nonn,easy,base

%O 1,1

%A _Clark Kimberling_, Sep 19 2014