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 1-2 of 2 results.

A268339 Number of polyominoes with width and height equal to n that are invariant under all symmetries of the square.

Original entry on oeis.org

1, 1, 3, 3, 17, 17, 163, 163, 2753, 2753, 84731, 84731, 4879497, 4879497, 535376723, 535376723, 112921823249, 112921823249, 45931435159067, 45931435159067, 36048888105745113, 36048888105745113, 54568015172025197171, 54568015172025197171, 159197415409641803530753, 159197415409641803530753
Offset: 1

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Author

Craig Knecht, Feb 02 2016

Keywords

Comments

Percolation theory focuses on patterns that provide connectivity. Polyominoes that connect all boundaries of a square are in the percolation neighborhood. This subclass of symmetric polyominoes distinguishes itself for its beauty and its unusual enumeration pattern.

Examples

			The ones in this example provide the connective pattern that joins all boundaries of the square.
0 1 1 1 0
1 0 1 0 1
1 1 1 1 1
1 0 1 0 1
0 1 1 1 0

Links

Crossrefs

Cf. A054247 (all unique water retention patterns for an n X n square), A268311 (polyominoes that connect all boundaries on a square), A268758.

Formula

a(2*n) = a(2*n-1) = A268758(n). - Andrew Howroyd, May 03 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, May 03 2020

A331878 Number of (non-null) connected induced subgraphs in a right-triangular graph with n vertices on each side.

Original entry on oeis.org

1, 6, 32, 242, 3389, 94204, 5152184, 550554052, 114558123057, 46273463018266, 36187365370984792, 54676438562307219806, 159361336209506075254061, 894922393831489478287443136, 9673844907264825614989071514624, 201139020197000279384599867300494336
Offset: 1

Views

Author

Andrew Howroyd, May 03 2020

Keywords

Comments

a(n) is the number of lower triangular n X n binary arrays with all 1's connected.
a(n) is the number of ways to place a fixed polyomino on an n X n grid with all squares above the main diagonal unoccupied.

Examples

			The graphs for n = 2 and n = 3 are illustrated below:
                   o
        o          |
        |          o---o
        o---o      |   |
                   o---o---o
.
The a(2) = 6 connected induced subgraphs are:
     X     o     o     X     o     X
     o o   X o   o X   X o   X X   X X

Links

Crossrefs

Cf. A059525, A268758, A287689.
Showing 1-2 of 2 results.

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

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