Clark Kimberling, <a href="/A247522/b247522_1.txt">Table of n, a(n) for n = 1..1000</a>

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Numbers k such that d(r,k) = 1 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {1/2 + (golden ratio)/2}, and { } = fractional part. THIS ENTRY WILL BE REVISED SOON.

1, 4, 11, 5, 6, 7, 12, 15, 16, 18, 19, 20, 21, 24, 25, 28, 29, 32, 34, 35, 36, 37, 38, 39, 42, 47, 50, 40, 51, 62, 64, 52, 53, 54, 65, 66, 67, 68, 72, 73, 77, 78, 98, 82, 91, 101, 102, 106, 107, 109, 110, 113, 114, 123, 147, 124, 151, 152, 154, 157, 159, 155, 160, 161, 162, 163, 164, 168, 175, 169, 179, 180, 183, 192, 193, 194, 202, 195, 196, 197, 203, 210, 225, 246, 249, 256

Clark Kimberling, <a href="/A247522/b247522_1.txt">Table of n, a(n) for n = 1..1000</a>

{golden ratio} r has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...

{1/2 + golden ratio} s has binary digits 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...,

so that a(1) = 1 and a(2) = 5.

z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1 + 1/2];

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

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

Numbers k such that d(r,k) = 1 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {1/2 + golden ratio}, and { } = fractional part. THIS ENTRY WILL BE REVISED SOON.

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allocated for Clark KimberlingNumbers k such that d(r,k) = 1 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {1/2 + golden ratio}, and { } = fractional part.

1, 4, 11, 12, 15, 16, 18, 21, 24, 25, 29, 32, 34, 35, 36, 37, 39, 42, 47, 50, 51, 62, 64, 65, 68, 73, 78, 98, 102, 106, 107, 109, 110, 114, 123, 147, 151, 152, 154, 157, 159, 160, 161, 164, 168, 175, 180, 183, 192, 193, 194, 202, 203, 210, 225, 246, 249, 256

1,2

Every positive integer lies in exactly one of these: A247519, A247520, A247521.

Clark Kimberling, <a href="/A247522/b247522.txt">Table of n, a(n) for n = 1..1000</a>

{golden ratio} has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, ...

{1/2 + golden ratio} has binary digits 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...,

so that a(1) = 5.

z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1 + 1/2];

u = RealDigits[r, 2, z][[1]]

v = RealDigits[s, 2, z][[1]]

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

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

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

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

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

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

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

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

Cf. A247519, A247520, A247521.

allocated

nonn,easy,base

Clark Kimberling, Sep 19 2014

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editing

allocated for Clark Kimberling

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