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 1-6 of 6 results.

A125709 Number of n-indecomposable polyominoes with at least n cells.

Original entry on oeis.org

1, 5, 32, 444, 13375, 684215, 52267513
Offset: 1

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Author

N. J. A. Sloane, Feb 01 2007

Keywords

Comments

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
MacKinnon incorrectly gives a(3) = 42.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.

Examples

			The five 2-indecomposable polyominoes:
...................X.
XX..XXX..XX..XXX..XXX
..........X...X....X.

Links

Crossrefs

Row sums of A125753. Cf. A125759, A125761, A126742, A126743.

Extensions

a(4) and a(5) from Peter Pleasants, Feb 13 2007
a(6) and a(7) from David Applegate, Feb 16 2007

A125753 Triangle read by rows: T(n,k) (n>=1) gives the number of n-indecomposable polyominoes with k cells (k >= n).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 0, 2, 5, 12, 6, 5, 1, 1, 0, 0, 0, 5, 12, 35, 108, 73, 76, 80, 25, 15, 15, 0, 0, 0, 0, 12, 35, 108, 369, 1285, 1044, 1475, 2205, 2643, 983, 1050, 1208, 958, 0, 0, 0, 0, 0, 35, 108, 369, 1285, 4655, 17073, 15980, 26548, 48766, 79579, 99860, 45898, 60433, 89890, 109424, 84312, 0, 0, 0, 0, 0, 0, 108, 369, 1285, 4655, 17073, 63600, 238591, 245955, 458397, 948201, 1857965, 3160371, 4153971, 2217787, 3402761, 5855953, 9067535, 11402651, 9170285, 0, 0, 0, 0, 0, 0, 0, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 3807508, 7710844, 17354771, 37983463
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Feb 04 2007, Feb 14 2007

Keywords

Comments

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
Row n has 4n-3 terms of which the first n-1 are zero.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.

Examples

			Triangle begins:
1
0,1,2,1,1
0,0,2,5,12,6,5,1,1
0,0,0,5,12,35,108,73,76,80,25,15,15
0,0,0,0,12,35,108,369,1285,1044,1475,2205,2643,983,1050,1208,958
0,0,0,0,0,35,108,369,1285,4655,17073,15980,26548,48766,79579,99860,45898,60433,89890,109424,84312
0,0,0,0,0,0,108,369,1285,4655,17073,63600,238591,245955,458397,948201,1857965,3160371,4153971,2217787,3402761,5855953,9067535,11402651,9170285
0,0,0,0,0,0,0,369,1285,4655,17073,63600,238591,901971,3426576,3807508,7710844,17354771,37983463,...

Links

Crossrefs

Row sums give A125709. Cf. A125759, A125761, A126742, A126743.

Extensions

Rows 5, 6, 7 and 8 from David Applegate, Feb 16 2007

A125761 Triangle read by rows: T(n,k) (n>=1) gives the number of n-indecomposable polyominoes with k cells (k >= 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 12, 6, 5, 1, 1, 1, 1, 2, 5, 12, 35, 108, 73, 76, 80, 25, 15, 15, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 1044, 1475, 2205, 2643, 983, 1050, 1208, 958, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 15980, 26548, 48766, 79579, 99860, 45898, 60433, 89890, 109424, 84312, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 245955, 458397, 948201, 1857965, 3160371, 4153971, 2217787, 3402761, 5855953, 9067535, 11402651, 9170285, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 3807508, 7710844, 17354771, 37983463
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Feb 05 2007

Keywords

Comments

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
Row n has 4n-3 nonzero terms.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.
Rows converge to A000105. - Andrey Zabolotskiy, Dec 26 2017

Examples

			Triangle begins:
1;
1,1,2,1,1;
1,1,2,5,12,6,5,1,1;
1,1,2,5,12,35,108,73,76,80,25,15,15;
1,1,2,5,12,35,108,369,1285,1044,1475,2205,2643,983,1050,1208,958;
1,1,2,5,12,35,108,369,1285,4655,17073,15980,26548,48766,79579,99860,45898,60433,89890,109424,84312;
1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,245955,458397,948201,1857965,3160371,4153971,2217787,3402761,5855953,9067535,11402651,9170285;
1,1,2,5,12,35,108,369,1285,4655,17073,63600,238591,901971,3426576,3807508,7710844,17354771,37983463,...

Links

Crossrefs

Row sums give A125759.
Cf. A125709, A125753, A126742, A126743; A000105, A195738, A195739, A049430.

Extensions

Rows 5, 6, 7 and 8 from David Applegate, Feb 16 2007

A126742 Number of n-indecomposable polyominoes with at least 2n cells.

Original entry on oeis.org

0, 2, 13, 284, 13375, 660690, 51941832
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Feb 01 2007

Keywords

Comments

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.

Examples

			The five 2-indecomposable polyominoes:
...................X.
XX..XXX..XX..XXX..XXX
..........X...X....X.
Only the last two have >= 4 cells, so a(2) = 2.

Links

Crossrefs

Row sums of A126743. Cf. A000105, A125759, A125761, A125709, A125753.

Extensions

a(4) and a(5) from Peter Pleasants, Feb 13 2007
a(6) and a(7) from David Applegate, Feb 16 2007

A126743 Triangle read by rows: T(n,k) (n>=1) gives the number of n-indecomposable polyominoes with k cells (k >= 2n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 6, 5, 1, 1, 0, 0, 0, 0, 0, 0, 0, 73, 76, 80, 25, 15, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1044, 1475, 2205, 2643, 983, 1050, 1208, 958, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15980, 26548, 48766, 79579, 99860, 45898, 60433, 89890, 109424, 84312, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 245955, 458397, 948201, 1857965, 3160371, 4153971, 2217787, 3402761, 5855953, 9067535, 11402651, 9170285, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3807508, 7710844, 17354771, 37983463
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Feb 04 2007

Keywords

Comments

A polyomino is called n-indecomposable if it cannot be partitioned (along cell boundaries) into two or more polyominoes each with at least n cells.
Row n has 4n-3 terms of which the first 2n-1 are zero.
For full lists of drawings of these polyominoes for n <= 6, see the links in A125759.

Examples

			Triangle begins:
0
0,0,0,1,1
0,0,0,0,0,6,5,1,1
0,0,0,0,0,0,0,73,76,80,25,15,15
0,0,0,0,0,0,0,0,0,1044,1475,2205,2643,983,1050,1208,958
0,0,0,0,0,0,0,0,0,0,0,15980,26548,48766,79579,99860,45898,60433,89890,109424,84312
0,0,0,0,0,0,0,0,0,0,0,0,0,245955,458397,948201,1857965,3160371,4153971,2217787,3402761,5855953,9067535,11402651,9170285
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3807508,7710844,17354771,37983463,...

Links

Crossrefs

Row sums give A126742. Cf. A000105, A125759, A125761, A125709, A125753.

Extensions

Rows 5, 6, 7 and 8 from David Applegate, Feb 16 2007

A342430 Number of prime polyominoes with n cells.

Original entry on oeis.org

0, 0, 1, 2, 1, 12, 5, 108, 145, 974, 2210, 17073, 31950, 238591, 587036, 3174686, 9236343, 50107909
Offset: 0

Views

Author

Drake Thomas, Mar 11 2021

Keywords

Comments

We say that a free polyomino is prime if it cannot be tiled by any other free polyomino besides the 1 X 1 square and itself.
The tiling of P must be with a single polyomino, and that single polyomino may not be the unique monomino or P itself. For example, decomposing the T-tetromino into a 3 X 1 and a 1 X 1 would use multiple tiles, and this is not permitted.
It can be shown that a(n) > 0 for all n >= 4, by considering the polyomino whose cells are at (0,1), (-1,1), (0,2), and (x,0) for all x = 0, 1, ..., n-4.

Examples

			For n = 4, the T-tetromino cannot be decomposed into smaller congruent polyominoes:
      +---+
      |   |
  +---+   +---+
  |           |
  +-----------+
The other four free tetrominoes can, however:
  +---+
  |   |
  |   |    +---+
  |   |    |   |
  +---+    |   |         +---+---+        +---+---+
  |   |    |   |         |   |   |        |       |
  |   |    +---+---+     |   |   |    +---+---+---+
  |   |    |       |     |   |   |    |       |
  +---+    +-------+     +---+---+    +---+---+
Thus a(4) = 1.

Links

Crossrefs

Cf. A000105, A125759, A213376.

Formula

a(n) = A000105(n) if n is prime.

Extensions

a(14)-a(17) from John Mason, Sep 16 2022
a(1) corrected by John Mason, Feb 27 2023
Showing 1-6 of 6 results.

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

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