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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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A186227 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and heptagonal numbers. Complement of A186228.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 109, 110, 111, 113, 114, 116, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 132, 133, 135, 136, 138, 139, 140, 142, 143, 145
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186219 for a general discussion of adjusted joint rank sequences.

Examples

			First, write
1..3..6..10..15..21..28..36..45... (triangular)
1.......7......18......34.......55... (heptagonal)
Then replace each number by its rank, where ties are settled by ranking the triangular number before the heptagonal:
a=(1,3,4,6,7,9,10,12,...), A186227.
b=(2,5,8,11,15,18,21,...), A186228.

Crossrefs

Cf. A186219, A186228, A186237, A186238,
Cf. A000217 (triangular numbers)
Cf. A000566 (heptagonal numbers)

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=1/2; u=1/2; v=1/2; w=0; x=5/2; y=-3/2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n],{n,1,100}]  (* A186227 *)
Table[b[n],{n,1,100}]  (* A186228 *)

A186274 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and octagonal numbers. Complement of A186159.

Original entry on oeis.org

2, 5, 9, 12, 15, 19, 22, 26, 29, 33, 36, 40, 43, 46, 50, 53, 57, 60, 64, 67, 71, 74, 78, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 153, 157, 160, 164, 167, 171, 174, 178, 181, 184, 188, 191, 195, 198, 202, 205, 209, 212, 216, 219, 222, 226, 229, 233, 236, 240, 243, 247, 250, 253, 257, 260, 264, 267, 271, 274, 278, 281, 284, 288, 291, 295, 298, 302, 305, 309, 312, 316, 319, 322, 326, 329, 333, 336, 340, 343
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186159.

Examples

			First, write the triangular and octagonal numbers:
1..3..6.....10..15..21..28
1........8..........21......
Then replace each by its rank, where ties are settled by ranking the triangular number before the octagonal:
a=(1,3,4,6,7,8,10,11,13,...)=A186159.
b=(2,5,9,12,15,19,22,26,...)=A186274.

Crossrefs

Cf. A186159, A186219.

Programs

A186290 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and pentagonal numbers. Complement of A186291.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 120, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 150, 152, 154, 156, 158, 160, 161, 163, 165, 167, 169, 170, 172, 174, 176, 178, 180, 181
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the pentagonal:
a=(2,3,5,7,9,11,12,14,....)=A186290.
b=(1,4,6,8,10,13,15,17,...)=A186291.

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=-1/2; u=1; v=0; w=0; x=3/2; y=-1/2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186290 *)
Table[b[n], {n, 1, 100}]  (* A186291 *)

A186291 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and pentagonal numbers. Complement of A186290.

Original entry on oeis.org

1, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 26, 28, 30, 33, 35, 37, 39, 42, 44, 46, 48, 50, 53, 55, 57, 59, 62, 64, 66, 68, 70, 73, 75, 77, 79, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 104, 106, 108, 111, 113, 115, 117, 119, 122, 124, 126, 128, 131, 133, 135, 137, 139, 142, 144, 146, 148, 151, 153, 155, 157, 159, 162, 164, 166, 168, 171, 173, 175, 177, 179, 182, 184, 186, 188, 191, 193, 195, 197, 200
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Examples

			First, write
1..4...9....16....25..36..49..... (squares)
1....5...12....22....35......51.. (pentagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the pentagonal:
a=(2,3,5,7,9,11,12,14,....)=A186290.
b=(1,4,6,8,10,13,15,17,...)=A186291.

Crossrefs

Cf. A186219, A186288, A186289, A186291.

Programs

A186315 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and hexagonal numbers. Complement of A186316.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 159, 160, 162, 164, 165, 167, 169, 170
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4...9...16..25....36....49. (squares)
1....6...15.......28....45.... (hexagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the hexagonal:
a=(1,3,5,7,8,10,12,13,...)=A186315.
b=(2,4,6,9,11,14,16,18,...)=A186316.

Crossrefs

Cf. A186219, A186316, A186317, A186318,
A000290 (squares), A000384 (hexagonal numbers).

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=2; y=-1; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186315 *)
Table[b[n], {n, 1, 100}]  (* A186316 *)

A186320 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and heptagonal numbers. Complement of A186321.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 90, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108, 109, 111, 112, 114, 116, 117, 119, 121, 122, 124, 125, 127, 129, 130, 132, 134, 135, 137, 139, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 158, 160, 161, 163
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Examples

			First, write
1..4...9..16....25...36...49...64.. (squares)
1....7.......18....34........55.... (heptagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the heptagonal:
a=(1,3,5,6,8,10,11,13,...)=A186320
b=(2,4,7,9,12,15,17,20,...)=A186321.

Crossrefs

Cf. A186219, A186321, A186322, A186323,
A000290 (squares), A000566 (heptagonal numbers).

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=5/2; y=-3/2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186320 *)
Table[b[n], {n, 1, 100}]  (* A186321 *)

A186322 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f and g are the squares and heptagonal numbers. Complement of A186323.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 90, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108, 109, 111, 112, 114, 116, 117, 119, 121, 122, 124, 126, 127, 129, 130, 132, 134, 135, 137, 139, 140, 142, 143, 145, 147, 148, 150, 152, 153, 155, 157, 158, 160, 161, 163
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Examples

			First, write
1..4...9..16....25...36...49...64.. (squares)
1....7.......18....34........55.... (heptagonal)
Replace each number by its rank, where ties are settled by ranking the square number after the heptagonal:
a=(2,3,5,6,8,10,11,13,...)=A186322
b=(1,4,7,9,12,15,17,20,...)=A186323.

Crossrefs

Cf. A186219, A186320, A186321, A186323.

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=-1/2; u=1; v=0; w=0; x=2; y=-1; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186322 *)
Table[b[n], {n, 1, 100}]  (* A186323 *)

A186324 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the squares and octagonal numbers. Complement of A186325.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 11, 12, 14, 16, 17, 19, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 42, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 113, 115, 117, 118, 120, 121, 123, 124, 126, 128, 129, 131, 132, 134, 135, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 154, 156, 158
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..4...9..16....25..36....49..64...  (squares)
1....8.......21........40........65. (octagonal)
Replace each number by its rank, where ties are settled by ranking the square number before the octagonal:
a=(1,3,5,6,8,9,11,12,14,...)=A186324
b=(2,4,7,10,13,15,18,21,...)=A186325.

Crossrefs

Cf. A186219, A186325, A186326, A186327,
A000290 (squares), A000567 (octagonal).

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=1/2; u=1; v=0; w=0; x=3; y=-2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186324 *)
Table[b[n], {n, 1, 100}]  (* A186325 *)

A186328 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the pentagonal numbers and the hexagonal numbers. Complement of A186329.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 57, 59, 61, 63, 65, 67, 69, 71, 72, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125, 126, 128, 130, 132, 134, 136, 138, 140, 141, 143, 145, 147, 149, 151, 153, 154, 156, 158, 160, 162, 164, 166, 168, 169, 171, 173, 175, 177, 179, 181, 182, 184, 186
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..5...12....22.....35......  (pentagonal)
1....6....15....28.......45.. (hexagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the hexagonal:
a=(1,3,5,7,9,11,13,15,16,....)=A186328
b=(2,4,6,8,10,12,14,17,19,...)=A186329.

Crossrefs

Cf. A186219, A186329, A186330, A186331,
A000384 (pentagonal), A000384 (hexagonal).

Programs

  • Mathematica
    (* adjusted joint ranking; general formula *)
d=1/2; u=3/2; v=-1/2; w=0; x=2; y=-1; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186328 *)
Table[b[n], {n, 1, 100}]  (* A186329 *)

A186342 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the pentagonal numbers and the octagonal numbers. Complement of A186343.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 65, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111, 112, 114, 116, 118, 119, 121, 123, 124, 126, 128, 129, 131, 133, 135, 136, 138, 140, 141, 143, 145, 147, 148, 150, 152, 153, 155, 157, 158, 160, 162, 164, 165, 167, 169, 170
Offset: 1

Views

Author

Clark Kimberling, Feb 18 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.

Examples

			First, write
1..5...12....22..35..... (pentagonal)
1....8....21........40.. (octagonal)
Then replace each number by its rank, where ties are settled by ranking the pentagonal number before the octagonal:
a=(1,3,5,7,8,10,12,13,15,...)=A186342
b=(2,4,6,9,11,14,16,19,21,...)=A186343.

Crossrefs

Cf. A186219, A186343, A186344, A186345,
A000326 (pentagonal), A000567 (octagonal).

Programs

  • Mathematica
     (* adjusted joint ranking; general formula *)
d=1/2; u=3/2; v=-1/2; w=0; x=3; y=-2; z=0;
h[n_]:=-y+(4x(u*n^2+v*n+w-z-d)+y^2)^(1/2);
a[n_]:=n+Floor[h[n]/(2x)];
k[n_]:=-v+(4u(x*n^2+y*n+z-w+d)+v^2)^(1/2);
b[n_]:=n+Floor[k[n]/(2u)];
Table[a[n], {n, 1, 100}]  (* A186342 *)
Table[b[n], {n, 1, 100}]  (* A186343 *)
Previous Showing 11-20 of 34 results. Next

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