cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A247631 Numbers k such that d(r,k) = 0 and d(s,k) = 0, where d(x,k) = k-th binary digit of x, r = {sqrt(2)}, s = {sqrt(8)}, and { } = fractional part.

Original entry on oeis.org

8, 9, 10, 11, 14, 20, 24, 28, 37, 47, 51, 54, 57, 58, 59, 62, 63, 69, 81, 82, 85, 92, 106, 121, 128, 129, 147, 148, 149, 150, 161, 162, 165, 168, 181, 182, 183, 186, 190, 200, 201, 214, 217, 218, 219, 225, 226, 227, 228, 232, 236, 241, 245, 248, 249, 258
Offset: 1

Views

Author

Clark Kimberling, Sep 23 2014

Keywords

Comments

Every positive integer lies in exactly one of these: A247631, A247632, A247633, A247634. Deleting the initial 1 from the representation of r gives the representation of s.

Examples

			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) = 8 and a(2) = 9.

Links

Crossrefs

Cf. A247632, A247633, A247634, A247519.

Programs

  • Mathematica
    z = 400; 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]]
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]]  (* A247631 *)
Flatten[Position[t2, 1]]  (* A247632 *)
Flatten[Position[t3, 1]]  (* A247633 *)
Flatten[Position[t4, 1]]  (* A247634 *)

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Sep 19 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A247519, A247521, A247522.

Programs

  • 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 *)

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Sep 19 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A247519, A247520, A247522.

Programs

  • 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 *)

A247522 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 = {(golden ratio)/2}, and { } = fractional part.

Original entry on oeis.org

1, 5, 6, 7, 12, 15, 16, 19, 20, 21, 25, 28, 29, 35, 36, 37, 38, 39, 40, 51, 52, 53, 54, 65, 66, 67, 68, 72, 73, 77, 78, 82, 91, 101, 102, 106, 107, 110, 113, 114, 124, 151, 152, 155, 160, 161, 162, 163, 164, 168, 169, 179, 180, 193, 194, 195, 196, 197, 203
Offset: 1

Views

Author

Clark Kimberling, Sep 19 2014

Keywords

Comments

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

Examples

			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) = 1 and a(2) = 5.

Links

Crossrefs

Cf. A247519, A247520, A247521.

Programs

  • 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 *)

A247523 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.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 12, 15, 16, 19, 20, 21, 23, 25, 28, 29, 31, 35, 36, 37, 38, 39, 40, 44, 49, 51, 52, 53, 54, 56, 57, 58, 59, 65, 66, 67, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101, 102, 104, 106, 107, 110
Offset: 1

Views

Author

Clark Kimberling, Sep 19 2014

Keywords

Comments

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

Examples

			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) = 1 and a(2) = 3.

Links

Crossrefs

Cf. A247524, A247519.

Programs

  • 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]];
t = Table[If[u[[n]] == v[[n]], 1, 0], {n, 1, z}];
Flatten[Position[t, 1]]  (* A247523 *)
Flatten[Position[t, 0]]  (* A247524 *)
Showing 1-5 of 5 results.

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