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 1-3 of 3 results.

A186225 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 triangular numbers and pentagonal numbers. Complement of A186226.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158
Offset: 1

Views

Author

Clark Kimberling, Feb 15 2011

Keywords

Examples

			See A186223.

Crossrefs

A186219, A186223, A186224, A186226.

Programs

  • Mathematica
    (* Program for adjusted rank sequences as described at A186219 *)
d=-1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0; (* triangular & pentagonal *)
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}]  (* A186225 *)
Table[b[n],{n,1,100}]  (* A186226 *)

A186223 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 pentagonal numbers. Complement of A186224.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104, 106, 107, 109, 110, 112, 114, 115, 117, 118, 120, 121, 123, 125, 126, 128, 129, 131, 132, 134, 136, 137, 139, 140, 142, 143, 145, 147, 148, 150, 151, 153, 155, 156, 158
Offset: 1

Views

Author

Clark Kimberling, Feb 15 2011

Keywords

Comments

See A186219 for a general description.

Examples

			First, write
1..3...6..10....15...21.....28......36...45...  (triangular)
1....5.........12...........22......35........... (pentagonal)
Replace each number by its rank, where ties are settled by ranking the triangular number before the pentagonal:
a=(1,3,5,6,8,9,11,13,...)
b=(2,4,7,10,12,15,18,...).

Crossrefs

Cf. A186219, A186224, A186225, A186226,
A000217 (triangular), A000326 (pentagonal).

Programs

  • Mathematica
    d=1/2; u=1/2; v=1/2; w=0; x=3/2; y=-1/2; z=0;
(* triangular & pentagonal *)
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}]  (* A186223 *)
Table[b[n],{n,1,100}]  (* A186224 *)

A186224 Adjusted joint rank sequence of (g(i)) and (f(j)) with f(i) before g(j) when f(i)=g(j), where f and g are the triangular numbers and pentagonal numbers. Complement of A186223.

Original entry on oeis.org

2, 4, 7, 10, 12, 15, 18, 21, 23, 26, 29, 32, 34, 37, 40, 42, 45, 48, 51, 53, 56, 59, 62, 64, 67, 70, 72, 75, 78, 81, 83, 86, 89, 92, 94, 97, 100, 103, 105, 108, 111, 113, 116, 119, 122, 124, 127, 130, 133, 135, 138, 141, 144, 146, 149, 152, 154, 157, 160, 163, 165, 168, 171, 174, 176, 179, 182, 184, 187, 190, 193, 195, 198, 201, 204, 206, 209, 212, 215, 217, 220, 223, 225, 228, 231, 234, 236, 239, 242, 245, 247, 250, 253, 256, 258, 261, 264, 266, 269, 272
Offset: 1

Views

Author

Clark Kimberling, Feb 15 2011

Keywords

Comments

See A186223.

Examples

			See A186223.

Crossrefs

Cf. A186223, A186225, A186226.

Programs

Showing 1-3 of 3 results.

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