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.

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A194412 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

3, 9, 12, 15, 21, 24, 27, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 75, 78, 81, 87, 90, 93, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 183, 195, 207, 210, 213, 219, 222, 225, 231
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is a multiple of 3; see A194368.

Links

Crossrefs

Cf. A002193, A194368, A194411, A194413, A194414.

Programs

  • Mathematica
    r = Sqrt[2]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]         (* A194411 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]         (* A194412 *)
%/3                              (* A194413 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t3, 1]]         (* A194414 *)

A194413 (A194412)/3.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 61, 65, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Links

Crossrefs

Cf. A194368, A194412.

Programs

  • Mathematica
    r = Sqrt[2]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]         (* A194411 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]         (* A194412 *)
%/3                              (* A194413 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t3, 1]]         (* A194414 *)

A194414 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 18, 19, 20, 22, 23, 25, 28, 29, 30, 31, 32, 34, 35, 37, 40, 46, 47, 49, 52, 58, 59, 61, 64, 76, 88, 100, 103, 104, 105, 106, 107, 109, 110, 112, 115, 116, 117, 118, 119, 121, 122, 124, 127, 128, 129, 130, 131, 133, 134
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Links

Crossrefs

Cf. A002193, A194368, A194411, A194412, A194413.

Programs

  • Mathematica
    r = Sqrt[2]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 200}];
Flatten[Position[t1, 1]]         (* A194411 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t2, 1]]         (* A194412 *)
%/3                              (* A194413 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t3, 1]]         (* A194414 *)

A194415 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) < 0, where r=sqrt(3) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 4, 5, 8, 16, 17, 19, 20, 23, 31, 32, 34, 35, 38, 46, 47, 49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 64, 65, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 83, 86, 87, 88, 89, 90, 91, 92, 94, 95, 98, 101, 102, 103, 104, 105, 106, 107, 109, 110, 112, 113, 114, 115, 116
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A002194, A194368, A194416, A194417, A194418.

Programs

  • Mathematica
    r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

A194416 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) = 0, where r=sqrt(3) and < > denotes fractional part.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 63, 66, 69, 78, 81, 84, 93, 96, 99, 108, 111, 123, 126, 138, 141, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 216, 219, 222, 231, 234
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is divisible by 3; see A194368.

Crossrefs

Cf. A002194, A194368, A194415, A194417, A194418.

Programs

  • Mathematica
    r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

A194417 (A194416)/3.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 26, 27, 28, 31, 32, 33, 36, 37, 41, 42, 46, 47, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 77, 78, 79, 82, 83, 84, 87, 88, 92, 93, 97, 98
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368, A194416.

Programs

  • Mathematica
    r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

A194418 Numbers m such that Sum_{k=1..m} (<1/3 + k*r> - ) > 0, where r=sqrt(3) and < > denotes fractional part.

Original entry on oeis.org

7, 10, 11, 13, 14, 22, 25, 26, 28, 29, 37, 40, 41, 43, 44, 52, 55, 67, 70, 82, 85, 97, 100, 160, 163, 164, 166, 167, 175, 178, 179, 181, 182, 190, 193, 194, 196, 197, 205, 208, 220, 223, 235, 238, 250, 253, 373, 376, 388, 391, 403, 406
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[3]; c = 1/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 150}];
Flatten[Position[t1, 1]]           (* A194415 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]           (* A194416 *)
%/3                                (* A194417 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 500}];
Flatten[Position[t3, 1]]           (* A194418 *)

A194422 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) < 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 4, 7, 13, 14, 16, 19, 25, 26, 28, 31, 43, 55, 67, 70, 71, 72, 73, 74, 76, 77, 79, 82, 83, 84, 85, 86, 88, 89, 91, 94, 95, 96, 97, 98, 100, 101, 103, 106, 112, 113, 115, 118, 124, 125, 127, 130, 142, 154, 166, 241, 253, 265, 310, 311, 313, 316, 322, 323
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Links

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[2]; c = 2/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t1, 1]]         (* A194422 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]         (* A194423 *)
%/3                              (* A194424 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]         (* A194425 *)

A194423 Numbers m such that Sum_{k=1..m} (<2/3 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 42, 45, 48, 54, 57, 60, 66, 69, 75, 78, 81, 87, 90, 93, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 141, 144, 147, 153, 156, 159, 165, 168, 171, 183, 195, 240, 243, 246, 252, 255, 258, 264
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

Every term is divisible by 3; see A194368.

Links

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[2]; c = 2/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t1, 1]]         (* A194422 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]         (* A194423 *)
%/3         (* A194424 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]         (* A194425 *)

A194424 (A194423)/3.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 61, 65, 80, 81, 82, 84, 85, 86, 88, 89, 90, 94, 98
Offset: 1

Views

Author

Clark Kimberling, Aug 24 2011

Keywords

Comments

See A194368.

Links

Crossrefs

Cf. A194368.

Programs

  • Mathematica
    r = Sqrt[2]; c = 2/3;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t1, 1]]         (* A194422 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]         (* A194423 *)
%/3         (* A194424 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t3, 1]]         (* A194425 *)
Previous Showing 11-20 of 67 results. Next

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