A247523 -id:A247523 - cp's OEIS frontend

A247524 Numbers k such that d(r,k) != d(s,k), where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

2, 4, 8, 11, 13, 14, 17, 18, 22, 24, 26, 27, 30, 32, 33, 34, 41, 42, 43, 45, 46, 47, 48, 50, 55, 60, 61, 62, 63, 64, 69, 71, 74, 76, 79, 81, 83, 90, 92, 98, 99, 100, 103, 105, 108, 109, 111, 112, 115, 117, 118, 123, 125, 126, 127, 132, 133, 137, 138, 143

Offset: 1

Clark Kimberling, Sep 19 2014

Every positive integer lies in exactly one of the sequences A247423 and A247524.

			r has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...
s has binary digits 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, ...
so that a(1) = 2 and a(2) = 4.

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A247523.

z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1/2];
u = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[r, 2, z]];
v = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[s, 2, z]];
t = Table[If[u[[n]] == v[[n]], 1, 0], {n, 1, z}];
Flatten[Position[t, 1]]  (* A247523 *)
Flatten[Position[t, 0]]  (* A247524 *)
Module[{nn=150,gr,g2},gr=Rest[RealDigits[GoldenRatio,2,nn+1][[1]]];g2 = RealDigits[ GoldenRatio/2,2,nn][[1]];Position[Thread[{gr,g2}],?(#[[1]] != #[[2]]&),1,Heads->False]]//Flatten (* _Harvey P. Dale, Jun 28 2021 *)

A247635 Numbers k such that d(r,k) = d(s,k), where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(8)}, and { } = fractional part.

Original entry on oeis.org

2, 8, 9, 10, 11, 14, 16, 17, 18, 20, 22, 24, 26, 28, 30, 31, 32, 33, 34, 35, 37, 39, 40, 43, 44, 45, 47, 49, 51, 54, 57, 58, 59, 62, 63, 67, 69, 73, 74, 75, 76, 79, 81, 82, 85, 87, 90, 92, 94, 97, 98, 106, 114, 115, 116, 117, 121, 123, 124, 125, 126, 128

Offset: 1

Clark Kimberling, Sep 23 2014

Every positive integer lies in exactly one of the sequences A247635 and A247636.

			r has binary digits 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, ...
s has binary digits 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, ...
so that a(1) = 2 and a(2) = 8.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A247636, A247631, A247523.

z = 200; r = FractionalPart[Sqrt[2]]; s = FractionalPart[Sqrt[8]];
u = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[r, 2, z]];
v = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[s, 2, z]];
t = Table[If[u[[n]] == v[[n]], 1, 0], {n, 1, z}];
Flatten[Position[t, 1]]  (* A247635 *)
Flatten[Position[t, 0]]  (* A247636 *)

cp's OEIS Frontend

A247524 Numbers k such that d(r,k) != d(s,k), where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

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