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 1-4 of 4 results.

A186237 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 (A000217) and heptagonal numbers (A000566). Complement of A186238.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 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

Examples

			See A186227.

Crossrefs

Cf. A000217, A000566, A186227, A186228, A186238.

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}] (* A186237 *)
Table[b[n],{n,1,100}] (* A186238 *)

Extensions

Definition corrected by Georg Fischer, Sep 24 2021

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 *)

A186228 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 A186227.

Original entry on oeis.org

2, 5, 8, 11, 15, 18, 21, 24, 27, 31, 34, 37, 40, 44, 47, 50, 53, 57, 60, 63, 66, 70, 73, 76, 79, 82, 86, 89, 92, 95, 99, 102, 105, 108, 112, 115, 118, 121, 125, 128, 131, 134, 137, 141, 144, 147, 150, 154, 157, 160, 163, 167, 170, 173, 176, 180, 183, 186, 189, 192, 196, 199, 202, 205, 209, 212, 215, 218, 222, 225, 228, 231, 235, 238, 241, 244, 248, 251, 254, 257, 260, 264, 267, 270, 273, 277, 280, 283, 286, 290, 293, 296, 299, 303, 306, 309
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2011

Keywords

Comments

See A186227.

Examples

			See A186227.

Crossrefs

Cf. A186219, A186227, A186237, A186238.

Programs

A352623 a(n) is the maximum number k of queens that can be placed on an n X n chessboard such that, for each number j in 0..k-1, there is a queen attacking exactly j unoccupied squares.

Original entry on oeis.org

0, 1, 0, 5, 8, 11, 14, 18, 22
Offset: 0

Views

Author

Rodolfo Kurchan, Mar 24 2022

Keywords

Comments

On the 6 X 6 board in the illustration (see Example), each square containing a number is occupied by a queen, and the number in that square is the number of unoccupied squares attacked by that queen. (Each square with no number is unoccupied.)
Solvers:
a(4) = 8 by Rodolfo Kurchan
a(5) = 11 by Gustavo Piñeiro
a(6) = 14 by Antonio Misericordia
a(7) = 18 by Antonio Misericordia
a(8) = 22 by Giorgio Vecchi

Examples

			Solution illustrating a(6) = 14 by Antonio Misericordia:
  +---+---+---+---+---+---+
  | 2 | 7 |   | 5 | 4 | 0 |
  +---+---+---+---+---+---+
  | 10|   |   |   | 6 | 1 |
  +---+---+---+---+---+---+
  |   |   |   | 11|   | 3 |
  +---+---+---+---+---+---+
  |   |   |   | 13|   | 8 |
  +---+---+---+---+---+---+
  |   |   |   |   |   |   |
  +---+---+---+---+---+---+
  |   |   | 12|   | 9 |   |
  +---+---+---+---+---+---+

Links

Crossrefs

Cf. A186238.
Showing 1-4 of 4 results.

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