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.

A357234 a(n) is the maximum length of a snake-like polyomino in an n X n square that starts and ends at opposite corners.

Original entry on oeis.org

1, 3, 5, 7, 17, 23, 31, 39, 51, 63, 75, 89, 105, 121, 139, 159
Offset: 1

Views

Author

Yi Yang, Sep 18 2022

Keywords

Comments

Snake-like polyominoes have all cells with at most two neighbor cells, and have at least one cell that has only one neighbor cell, where neighbors are horizontal or vertical (not diagonal).
Lower bounds for a(10)-a(22) are 63, 75, 89, 105, 121, 139, 159, 179, 201, 225, 249, 275, 303. Is it true that a(n) = round((2*n*n-4*n+28)/3) for n >= 9?

Examples

			Longest snakes for 5 <= n <= 8:
  X X X X X   X X X X X X   X X X . X X X   X . X X X X X X
  . . . . X   . . . . . X   . . X . X . X   X . X . . . . X
  X X X X X   X X X X X X   X X X . X . X   X . X X X X . X
  X . . . .   X . . . . .   X . . X X . X   X X . . . X . X
  X X X X X   X . X X X X   X . . X . X X   . X . X X X . X
              X X X . . X   X . . X . X .   X X . X . . X X
                            X X X X . X X   X . . X . . X .
                                            X X X X . . X X

Links

Crossrefs

Cf. A331968, A357516.

Formula

a(n) ~ 2*n^2/3. - Pontus von Brömssen, Sep 19 2022
a(n) <= A331968(n). - Pontus von Brömssen, Sep 21 2022

Extensions

a(1)-a(9) confirmed by Pontus von Brömssen, Sep 21 2022. - N. J. A. Sloane, Sep 30 2022
a(10)-a(13) confirmed by Elijah Beregovsky, Nov 27 2022
a(14)-a(16) from Andrew Howroyd, Feb 28 2023

A360200 Number of induced paths in the n X n grid graph.

Original entry on oeis.org

0, 8, 94, 1004, 14864, 334536, 11546874, 629381852, 56094263348, 8343512638896, 2074276200162230, 853966325494701152, 578432462293854136504, 646135466408339553958096, 1200595044818176185884236342
Offset: 1

Views

Author

Andrew Howroyd, Jan 29 2023

Keywords

Comments

Paths of length zero are not counted here.
Equivalently, a(n) is the number of snake-like polyominoes in an n X n square. Rotations, reflections and translations are counted separately.

Examples

			The a(2) = 8 induced paths are:
  O O   O .   . .   . O   O O   O .   . O   O O
  . .   O .   O O   . O   O .   O O   O O   . O

Links

Crossrefs

Main diagonal of A360199.
Cf. A059525, A297664 (induced cycles), A331968, A331986 (of maximum length), A357516.

A375299 Number of longest winning paths in n X n Hex.

Original entry on oeis.org

1, 3, 1, 4, 23, 51, 20, 115, 5568, 12, 3521, 40, 1058, 2104, 668, 7540, 1298, 83648, 16631833, 70630
Offset: 1

Views

Author

Peter Selinger, Aug 13 2024

Keywords

Comments

A winning path is a set of cells connecting the top edge to the bottom edge, minimal with respect to inclusion.

Examples

			For n = 4, the longest winning path has length 8, and the a(4) = 4 winning paths of length 8 are
   ========     ========     ========     ========
    X . . .      . X . .      X . . .      . X . .
     X . X X      X . X X      X . X X      X . X X
      X X . X      X X . X      X X . X      X X . X
       . . . X      . . . X      . . X .      . . X .
       ========     ========     ========     ========

Links

Crossrefs

Length of longest path is A375298.
Cf. A357516.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.