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 1-8 of 8 results.

A228856 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,y+x), (y,2y+x), and (y,3y+x) are edges.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 39, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84
Offset: 1

Views

Author

Clark Kimberling, Sep 05 2013

Keywords

Examples

			Taking the first generation of edges of the tree to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5), (2,7)}, which grows G(3) = {(3,5), (3,8), (3,11), (5,7), (5,12), (5,17), (7,9), (7,16), (7,23)}, ... Expelling duplicate nodes and sorting leave {1,2,3,5,7,8,9,...}.

Links

Crossrefs

Cf. A141832, A228853, A228854, A228856.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228854 Nodes of tree generated as follows: (1,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 17, 18, 24, 25, 26, 27, 29, 37, 40, 41, 43, 44, 47, 56, 58, 61, 63, 64, 65, 67, 68, 69, 71, 76, 89, 91, 93, 97, 98, 99, 100, 101, 104, 105, 106, 108, 109, 111, 112, 115, 123, 137, 138, 140, 147, 149, 152, 153, 154, 155, 157, 159, 160
Offset: 1

Views

Author

Clark Kimberling, Sep 05 2013

Keywords

Examples

			Taking the first generation of edges of the tree to be G(1) = {(1,3)}, the edge (1,3) grows G(2) = {(3,4), (3,7)}, which grows G(3) = {(4,7), (4,11), (7,10),(7,17)}, ... Expelling duplicate nodes and sorting leave {1,3,4,7,10,11,...}.

Links

Crossrefs

Cf. A141832, A228853, A228855, A228856.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 1; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228855 Nodes of tree generated as follows: (2,3) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

2, 3, 5, 8, 11, 13, 18, 19, 21, 27, 29, 30, 31, 34, 41, 44, 46, 47, 49, 50, 55, 65, 67, 68, 71, 73, 75, 76, 79, 80, 81, 89, 100, 101, 106, 108, 109, 111, 112, 115, 116, 117, 119, 121, 123, 128, 129, 131, 144, 153, 157, 163, 165, 166, 171, 172, 173, 175, 176
Offset: 1

Views

Author

Clark Kimberling, Sep 05 2013

Keywords

Examples

			Taking the first generation of edges of the tree to be G(1) = {(2,3)}, the edge (2,3) grows G(2) = {(3,5), (3,8)}, which grows G(3) = {(5,8), (5,13), (8,11),(8,19)}, ... Expelling duplicate nodes and sorting leave {2,3,5,8,11,13,...}.

Links

Crossrefs

Cf. A141832, A228853, A228854, A228856.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 3; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228894 Nodes of tree generated as follows: (2,1) is an edge, and if (x,y) is an edge, then (y,y+x) and (y,2y+x) are edges.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 67, 68, 69, 71, 75, 76, 78, 79, 80, 83, 84, 85, 87, 88, 89, 91, 92, 93, 97, 98, 99
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2013

Keywords

Comments

The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). For example, the branch 2->1->3->4->7->11-> contributes the Lucas sequence, A000032. The other extreme branch, 1->4->9->22->53-> contributes A048654.

Examples

			Taking the first generation of edges to be G(1) = {(2,1)}, the edge (2,1) grows G(2) = {(1,3), (1,4)}, which grows G(3) = {(3,4), (3,7), (4,5), (4,9)}, ... Expelling duplicate nodes and sorting leave (1,2,3,4,5,7,9,...).

Links

Crossrefs

Cf. A228853.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x + 2 y}}; x = 2; y = 1; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228897 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x+y) and (y,x*y) are edges.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 24, 26, 30, 32, 34, 35, 39, 40, 42, 48, 52, 54, 55, 60, 63, 66, 68, 70, 72, 75, 84, 88, 89, 90, 96, 102, 104, 108, 110, 112, 117, 126, 130, 135, 136, 138, 144, 145, 150, 160, 165, 168, 174, 176, 178
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2013

Keywords

Comments

The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). The extreme branches are (1,2)->(2,3)->(3,5)->(5,8)->... and (1,2)->(2,4)->(4,8)->(8,32)->... These branches contribute to A228897, as subsequences, the Fibonacci numbers, A000045, and the sequence 2^(A000045) = A000302.

Examples

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,4)}, which grows G(3) = {(3,5), (3,6), (4,6), (4,8)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 4, 5, 6, 8, 9, 10, 12,...).

Crossrefs

Cf. A228853.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x* y}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {12}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228898 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x+y) and (y,x^2 + y^2) are edges.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 12, 13, 16, 19, 21, 29, 31, 34, 39, 45, 50, 55, 63, 73, 74, 81, 89, 97, 112, 119, 131, 144, 155, 160, 178, 185, 186, 191, 193, 205, 212, 233, 236, 246, 257, 283, 297, 312, 343, 369, 377, 391, 398, 417, 425, 441, 469, 479, 482, 505, 524, 555
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2013

Keywords

Comments

The tree has infinitely many branches which are essentially linear recurrence sequences (and infinitely many which are not). The extreme branches are (1,2)->(2,3)->(3,5)->(5,8)->... and (1,2)->(2,5)->(5,29)->(29,866)->... These branches contribute to A228898, as subsequences, the Fibonacci numbers, A000045, and A000283.

Examples

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,13), (5,7), (5,29)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 5, 7, 8, 12, 13, 16, 19,...).

Crossrefs

Cf. A228853, A000045.

Programs

  • Mathematica
    f[x_, y_] := {{y, x + y}, {y, x^2 + y^2}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {18}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228939 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y,x*y) and (y,x^2 + y^2) are edges.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 20, 29, 32, 50, 68, 80, 125, 128, 145, 256, 320, 416, 500, 544, 640, 866, 1088, 1250, 1600, 2048, 2600, 4205, 4688, 5120, 6464, 6800, 8192, 8320, 15725, 16640, 21866, 25000, 25114, 34816, 36992, 51200, 66560, 102656, 128000, 130000
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2013

Keywords

Examples

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,2), (2,5)}, which grows G(3) = {(2,4), (2,8), (5,10), (5,29)}, ... Expelling duplicate nodes and sorting leave (1, 2, 4, 5, 8, 10, 16, 20, 29, 32,...).

Crossrefs

Cf. A228853, A228939.

Programs

  • Mathematica
    f[x_, y_] := {{y, x* y}, {y, x^2 + y^2}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {18}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]

A228940 Nodes of tree generated as follows: (1,2) is an edge, and if (x,y) is an edge, then (y, y^2 - x^2) and (y, y^2 + x^2) are edges.

Original entry on oeis.org

1, 2, 3, 5, 13, 16, 21, 29, 34, 160, 178, 231, 281, 416, 466, 816, 866, 1131, 1181, 25431, 25769, 31515, 31853, 53105, 53617, 78705, 79217, 172615, 173497, 216715, 217597, 665015, 666697, 749115, 750797, 1278005, 1280317, 1393605, 1395917, 646710161
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2013

Keywords

Examples

			Taking the first generation of edges to be G(1) = {(1,2)}, the edge (1,2) grows G(2) = {(2,3), (2,5)}, which grows G(3) = {(3,5), (3,14), (5,21), (5,29)}, ... Expelling duplicate nodes and sorting leave (1, 2, 3, 5, 13, 16, 21, 29, 34, 160,...).

Links

Crossrefs

Cf. A228853, A228940.

Programs

  • Mathematica
    f[x_, y_] := {{y, y^2-x^2}, {y, y^2 + x^2}}; x = 1; y = 2; t = {{x, y}};
u = Table[t = Flatten[Map[Apply[f, #] &, t], 1], {15}]; v = Flatten[u];
w = Flatten[Prepend[Table[v[[2 k]], {k, 1, Length[v]/2}], {x, y}]];
Sort[Union[w]]
Showing 1-8 of 8 results.

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