A247522 -id:A247522

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A247519

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.

+10
6

3, 9, 10, 23, 31, 44, 49, 56, 57, 58, 59, 70, 75, 80, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96, 97, 104, 116, 119, 120, 121, 122, 128, 129, 130, 131, 134, 135, 136, 139, 140, 141, 142, 145, 146, 149, 166, 173, 174, 177, 182, 185, 190, 191, 199, 200, 201, 208

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

OFFSET

1,1

COMMENTS

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.

LINKS

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

EXAMPLE

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) = 3.

MATHEMATICA

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]]

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 *)

CROSSREFS

Cf. A247520, A247521, A247522.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 19 2014

STATUS

approved

A247520

Numbers k such that d(r,k) = 0 and d(s,k) = 1, where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

+10
4

2, 8, 13, 17, 22, 26, 30, 33, 41, 43, 46, 48, 55, 61, 63, 69, 74, 79, 83, 92, 99, 103, 108, 111, 115, 118, 125, 127, 133, 138, 144, 148, 153, 156, 158, 165, 170, 172, 176, 181, 184, 187, 189, 198, 204, 207, 212, 214, 216, 221, 227, 229, 235, 242, 248, 250

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

OFFSET

1,1

COMMENTS

Every positive integer lies in exactly one of these: A247519, A247520, A247521, A247522.

LINKS

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

EXAMPLE

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.

MATHEMATICA

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]]

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 *)

CROSSREFS

Cf. A247519, A247521, A247522.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 19 2014

STATUS

approved

A247521

Numbers k such that d(r,k) = 1 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.

+10
4

4, 11, 14, 18, 24, 27, 32, 34, 42, 45, 47, 50, 60, 62, 64, 71, 76, 81, 90, 98, 100, 105, 109, 112, 117, 123, 126, 132, 137, 143, 147, 150, 154, 157, 159, 167, 171, 175, 178, 183, 186, 188, 192, 202, 205, 210, 213, 215, 220, 224, 228, 233, 240, 245, 249, 252

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

OFFSET

1,1

COMMENTS

Every positive integer lies in exactly one of these: A247519, A247520, A247522.

LINKS

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

EXAMPLE

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) = 4.

MATHEMATICA

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]]

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 *)

CROSSREFS

Cf. A247519, A247520, A247522.

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Sep 19 2014

STATUS

approved

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