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.

A125759 Number of n-indecomposable polyominoes.

Original entry on oeis.org

1, 6, 34, 448, 13384, 684236, 52267569
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.
MacKinnon incorrectly implies that the sequence is 1,6,44.
MacKinnon only allows polyominoes with >= n cells, leading to A125709 and A125753.
The polyominoes with < 2n cells are uninteresting, leading to A126742 and A126743.
There is a sense in which n-decomposable polyominoes with >3n-2 cells are also uninteresting: they are precisely the "n-spiders", where an n-spider is a polyomino with a cell whose removal splits it into 4 components each with

Examples

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

Links

Crossrefs

Row sums of A125761. Cf. A125709, A125753, A126742, A126743, A000105.

Formula

a(n) = A125709(n) + Sum_{i=1..n-1} A000105(i).

Extensions

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

A124593 Number of 4-indecomposable trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188, 1260, 1341, 1413, 1494, 1584, 1665, 1755, 1855, 1945
Offset: 0

Views

Author

David Applegate and N. J. A. Sloane, Feb 14 2007, extended with generating function Feb 25 2007

Keywords

Comments

A connected graph is called k-decomposable if it is possible to remove some edges and leave a graph with at least two connected components in which every component has at least k nodes.
Every connected graph with < 2k nodes is automatically k-indecomposable.
Necessary conditions are that a 4-indecomposable tree may not contain a path with >= 8 nodes, nor two node-disjoint paths with >= 4 nodes each.
From Brendan McKay, Feb 15 2007: (Start)
A necessary and sufficient condition seems to be that there are no two node-disjoint subtrees each of which is P_4 or K_{1,3}.
Alternatively, a tree with n vertices is k-decomposable iff, for each edge, removing that edge leaves a component with at most k-1 vertices. Finding the maximal k such that a tree is k-decomposable is easy to do in linear time. (End)
The counts of 1-indecomposable (1,0,0,0,...), 2-indecomposable (1,1,1,1,1,1,...) or 3-indecomposable (1,1,1,2,3,3,4,4,5,5,6,6,7,7,...) trees with number of nodes = 1,2,3,4,... are all trivial.

Examples

			Rather than show some 4-indecomposable trees, instead we show all four 3-indecomposable trees with 7 nodes:
O-O-O-O-O....O..........O.O...O...O
....|........|..........|/.....\./.
....O....O-O-O-O-O..O-O-O-O...O-O-O
....|........|..........|....../.\.
....O........O..........O.....O...O
On the other hand, O-O-O-O-O-O-O is 3-decomposable, because removing the third edge gives O-O-O O-O-O-O, with 2 connected components each with >= 3 nodes.

Links

Crossrefs

Cf. A000055, A125709.

Programs

  • PARI
    Vec((1 -x^2 -2*x^3 +x^4 +3*x^5 +3*x^6 +2*x^7 -4*x^8 -5*x^9 -3*x^10 +3*x^11 +4*x^12 +x^13 -x^14 -x^15) / ((1 -x)^4*(1 +x)*(1 +x +x^2)^2) + O(x^50)) \\ Colin Barker, May 27 2016

Formula

G.f.: f(x) / ((1-x)*(1-x^2)*(1-x^3)^2) where f(x) = 1 - x^2 - 2*x^3 + x^4 + 3*x^5 + 3*x^6 + 2*x^7 - 4*x^8 - 5*x^9 - 3*x^10 + 3*x^11 + 4*x^12 + x^13 - x^14 - x^15.

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
Showing 1-6 of 6 results.

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

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