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.

A030223 Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 30, 36, 80, 97, 213, 266, 578, 737, 1589, 2051, 4408, 5747, 12333, 16213, 34737, 45979, 98367, 131007, 279902, 374781, 799732, 1075793, 2293193, 3097415, 6596787, 8942350, 19031088, 25880367, 55043561, 75068945, 159570624, 218189681
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

These are the achiral polyominoes of the regular tiling with Schläfli symbol {3,6}. An achiral polyomino is identical to its reflection. This sequence can most readily be calculated by enumerating achiral fixed polyominoes for three situations with a given axis of symmetry: 1) fixed polyominoes with an axis of symmetry composed of cell edges, A364485; 2) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and a vertex as the highest polyomino point on this axis, A364486; and 3) fixed polyominoes with a vertical axis of symmetry composed of cell altitudes and an edge center as the highest polyomino point on this axis, A364487. Those three sequences include each achiral polyomino exactly twice. - Robert A. Russell, Jul 26 2023

Links

Crossrefs

Cf. A006534 (oriented), A000577 (unoriented), A030224 (chiral), A001420 (fixed).
Calculation components: A364485, A364486, A364487.
Other tilings: A030227 {4,4}, A030225 {6,3}.

Formula

From Robert A. Russell, Jul 27 2023: (Start)
a(n) = (A364486(n) + A364487(n)) / 2, n odd.
a(n) = (A364485(n/2) + A364486(n) + A364487(n)) / 2, n even.
a(n) = 2*A000577(n) - A006534(n) = A006534(n) - 2*A030224(n) = A000577(n) - A030224(n). (End)

Extensions

a(19) to a(28) from Joseph Myers, Sep 24 2002
Additional terms from Robert A. Russell, Jul 26 2023
Name edited by Robert A. Russell, Jul 27 2023

A364487 Number of fixed triangular n-ominoes of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell altitudes and the point of the polyomino farthest along that axis in a specified direction is a cell edge center.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 5, 2, 13, 5, 36, 16, 96, 45, 262, 128, 720, 368, 1991, 1047, 5549, 2995, 15583, 8607, 44027, 24788, 125043, 71620, 356706, 207412, 1021318, 601719, 2933861, 1748874, 8452723, 5091776, 24417793, 14848210, 70706750, 43364962, 205193316, 126828277
Offset: 1

Views

Author

Robert A. Russell, Jul 26 2023

Keywords

Comments

This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

Examples

			These are the n-ominoes for n<7. The highest point of the polyomino on the vertical axis of symmetry must be an edge center.
  ____   ____   ____________   ____      ____
  \  /  /\  /\  \  /\  /\  /  /\  /\    /\  /\
   \/  /__\/__\  \/__\/__\/  /__\/__\  /__\/__\
                             \  /\  /  \  /\  /
                              \/  \/    \/__\/

Links

Crossrefs

Cf. A030223, A364485, A364486.

Formula

a(n) = 2*A030223(n) - A364486(n), n odd.
a(n) = 2*A030223(n) - A364485(n/2) - A364486(n), n even.

A364485 Number of fixed triangular polyominoes with 2n cells of the regular tiling with Schläfli symbol {3,6} that have a common axis of symmetry coincident with cell edges.

Original entry on oeis.org

1, 2, 4, 9, 23, 59, 155, 418, 1136, 3122, 8663, 24201, 68059, 192471, 546899, 1560511, 4469000, 12839642, 36995629, 106875531, 309477998, 898075778, 2611239508, 7606064348, 22191694916, 64845964156, 189752911736, 555985221037, 1631053277370, 4790356866561
Offset: 1

Views

Author

Robert A. Russell, Jul 26 2023

Keywords

Comments

Half of the cells lie on each side of the axis of symmetry, so there must be an even number. This is one of three sequences used to calculate A030223, the number of achiral polyominoes for this tiling. Two fixed polyominoes are identical only if one is a translation of the other.

Examples

			These are halves above the axis of symmetry of the polyominoes for n<4. The bottom edge of each is the axis of symmetry.
                                             /\    /\
          ____  ____   ________   ____      /__\  /__\
   /\    /\  /  \  /\  \  /\  /  /\  /\    /\  /  \  /\
  /__\  /__\/    \/__\  \/__\/  /__\/__\  /__\/    \/__\

Crossrefs

Cf. A030223, A364486, A364487.

Formula

a(n) = 2*A030223(2n) - A364486(2n) - A364487(2n).
Showing 1-3 of 3 results.

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