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 1-10 of 18 results. Next

A052451 Number of n-node simple graphs having clique number 3.

Original entry on oeis.org

0, 0, 1, 3, 15, 82, 578, 6021, 101267, 2882460, 138787233, 11117715525, 1459391330953
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Also, number of n-node simple graphs having independence number 3. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 3 of A263341.
Cf. A052450, A052452, A052453, A077392, A077393, A077394.

Extensions

2 more terms from Eric W. Weisstein, Nov 03 2002
a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A052452 Number of n-node simple graphs having clique number 4.

Original entry on oeis.org

0, 0, 0, 1, 4, 30, 301, 4985, 142276, 7269487, 655015612, 103031645470, 28250197044039
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Also, number of n-node simple graphs having independence number 4. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 4 of A263341.
Cf. A052450, A052451, A052453, A077392, A077393, A077394.

Extensions

2 more terms from Eric W. Weisstein, Nov 03 2002
a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A077392 Number of n-node simple graphs having clique number 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 51, 842, 27107, 1724440, 210799447, 47337500562, 19053225506745
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 5. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 5 of A263341.
Cf. A052450, A052451, A052452, A052453, A077393, A077394.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A077393 Number of n-node simple graphs having clique number 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 80, 1995, 112225, 13893557, 3514580130, 1696127391214
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 6. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 6 of A263341.
Cf. A052450, A052451, A052452, A052453, A077392, A077394.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A263284 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n vertices with domination number k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 11, 16, 5, 1, 1, 34, 94, 21, 5, 1, 1, 156, 708, 152, 21, 5, 1, 1, 1044, 9384, 1724, 166, 21, 5, 1, 1, 12346, 221135, 38996, 1997, 166, 21, 5, 1, 1, 274668, 9877969, 1800340, 49961, 2036, 166, 21, 5, 1, 1
Offset: 1

Views

Author

Christian Stump, Oct 13 2015

Keywords

Comments

The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
For any graph the domination number is greater than or equal to the irredundance number (A332404) and less than or equal to the independent domination number (A332402). - Andrew Howroyd, Feb 13 2020

Examples

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       4,       5,       1,     1;
      11,      16,       5,     1,    1;
      34,      94,      21,     5,    1,   1;
     156,     708,     152,    21,    5,   1,  1;
    1044,    9384,    1724,   166,   21,   5,  1, 1;
   12346,  221135,   38996,  1997,  166,  21,  5, 1, 1;
  274668, 9877969, 1800340, 49961, 2036, 166, 21, 5, 1, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Columns k=1..2 are A000088(n-1), A332625.
Cf. A263341, A332400, A332401, A332402, A332403, A332404, A332405.

Formula

T(n,k) = T(n-1,k-1) for 2*(k-1) >= n. - Andrew Howroyd, Feb 17 2020

Extensions

Extended to 10 rows by Eric W. Weisstein, May 18 2017

A077394 Number of n-node simple graphs having clique number 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 117, 4210, 388547, 87269430, 42603563082
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2002

Keywords

Comments

Also, number of n-node simple graphs having independence number 7. - Andrew Howroyd, Oct 31 2017

Links

Crossrefs

Column 7 of A263341.
Cf. A052450, A052451, A052452, A052453, A077392, A077393.

Extensions

a(10) from Keith Briggs, Mar 15 2006
a(11) from Michael Sollami, Jan 29 2012
a(12) from Michael Sollami, Mar 26 2012
a(13) from Brendan McKay, May 07 2018

A126744 Triangle read by rows: T(n,k) gives number of connected graphs on n nodes with clique number n-k, (n>=2, k=0..n-2).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 11, 6, 1, 4, 25, 63, 19, 1, 5, 45, 266, 477, 59, 1, 6, 73, 785, 4646, 5339, 267, 1, 7, 109, 1908, 26205, 136935, 94535, 1380, 1, 8, 155, 4085, 110140, 1696407, 7121703, 2774240, 9832, 1, 9, 211, 7992, 384209, 13779220, 209046708, 647596643, 135794730, 90842
Offset: 2

Views

Author

N. J. A. Sloane, Feb 18 2007

Keywords

Comments

This sequence can be derived from A263341 since the number of graphs with clique number <= k is the Euler transform of the number of connected graphs with clique number <= k. - Andrew Howroyd, Feb 19 2020

Examples

			Triangle begins:
n=...1...2...3...4....5....6.....7......8........9........10
k.------------------------------------------------------------
2|...0...1...1...3....6...19....59....267.....1380......9832 = A024607
3|...0...0...1...2...11...63...477...5339....94535...2774240 = A126745
4|...0...0...0...1....3...25...266...4646...136935...7121703 = A126746
5|...0...0...0...0....1....4....45....785....26205...1696407 = A126747
6|...0...0...0...0....0....1.....5.....73.....1908....110140 = A126748
7|...0...0...0...0....0....0.....1......6......109......4085 = A217987
8|...0...0...0...0....0....0.....0......1........7.......155
  ...
From _Andrew Howroyd_, Feb 19 2020: (Start)
As a triangle with columns being clique number >= 2:
     1;
     1,       1;
     3,       2,       1;
     6,      11,       3,       1;
    19,      63,      25,       4,      1;
    59,     477,     266,      45,      5,    1;
   267,    5339,    4646,     785,     73,    6,   1;
  1380,   94535,  136935,   26205,   1908,  109,   7, 1;
  9832, 2774240, 7121703, 1696407, 110140, 4085, 155, 8, 1;
  ...
(End)

Links

Crossrefs

Diagonals give A024607, A126745, A126746, A126747, A126748, A217987.
Row sums are A001349.
Cf. A263341.

Extensions

Terms a(47) and beyond derived from A263341 added by Andrew Howroyd, Feb 19 2020

A294490 Triangle read by rows: T(n,k) is the number of simple connected graphs on n vertices having independence number k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 8, 1, 0, 1, 34, 63, 13, 1, 0, 1, 103, 524, 205, 19, 1, 0, 1, 405, 5863, 4308, 513, 26, 1, 0, 1, 1892, 100702, 135563, 21782, 1105, 34, 1, 0, 1, 12166, 2880002, 7161399, 1576634, 84185, 2140, 43, 1, 0, 1, 105065, 138772607, 652024627, 203380116, 12140094, 274156, 3845, 53, 1, 0
Offset: 1

Views

Author

Andrew Howroyd, Oct 31 2017

Keywords

Comments

Bivariate inverse Euler transform of A263341. This sequence can be derived from A263341 because the independence number of a disconnected graph is the sum of the independence numbers of its components. - Andrew Howroyd, Feb 19 2020

Examples

			Triangle begins:
  1;
  1,   0;
  1,   1,    0;
  1,   4,    1,    0;
  1,  11,    8,    1,   0;
  1,  34,   63,   13,   1,  0;
  1, 103,  524,  205,  19,  1, 0;
  1, 405, 5863, 4308, 513, 26, 1, 0;
  ...

Links

Crossrefs

Columns 2..5 are A243781, A243782, A243783, A243784.
Row sums give A001349.
Cf. A263341 (not necessarily connected).

Extensions

Terms a(56) and beyond derived from A263341 added by Andrew Howroyd, Feb 19 2020

A332402 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with independent domination number k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 11, 16, 5, 1, 1, 34, 90, 25, 5, 1, 1, 156, 668, 188, 25, 5, 1, 1, 1044, 8648, 2394, 228, 25, 5, 1, 1, 12346, 199990, 58578, 3493, 229, 25, 5, 1, 1, 274668, 8776166, 2837118, 113197, 3758, 229, 25, 5, 1, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 11 2020

Keywords

Comments

The independent domination number of a graph is the minimum size of a maximal independent set (sets which are both independent and dominating). For any graph it is greater than or equal to the domination number (A263284) and less than or equal to the independence number (A263341).
The final terms of each row tend to the sequence (1, 1, 5, 25, 229, 3759, ...). This happens because a connected graph on n nodes with n > 1 cannot have an independent domination number > floor(n/2). Similar limits are seen in A263284 and A332404 for the same reason.

Examples

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       4,       5,       1,      1;
      11,      16,       5,      1,    1;
      34,      90,      25,      5,    1,   1;
     156,     668,     188,     25,    5,   1,  1;
    1044,    8648,    2394,    228,   25,   5,  1, 1;
   12346,  199990,   58578,   3493,  229,  25,  5, 1, 1;
  274668, 8776166, 2837118, 113197, 3758, 229, 25, 5, 1, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Column k=1 is A000088(n-1).
Cf. A263284, A263341, A332402, A332403, A332404, A332405.

Formula

T(n,k) = T(n-1,k-1) for 2*(k-1) >= n.

A332403 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes with upper domination number k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 13, 15, 4, 1, 1, 36, 83, 30, 5, 1, 1, 101, 582, 302, 51, 6, 1, 1, 365, 6024, 5025, 843, 80, 7, 1, 1, 1518, 99497, 144370, 27160, 1996, 117, 8, 1, 1, 8002, 2706069, 7441209, 1733211, 112291, 4211, 164, 9, 1
Offset: 1

Views

Author

Andrew Howroyd, Feb 11 2020

Keywords

Comments

First differs from A263341 in row 6.
The upper domination number of a graph is the maximum size of a minimal dominating set (a set that is both dominating and irredundant). For any graph it is greater than or equal to the independence number (A263341) and less than or equal to the upper irredundance number (A332405). The number of graphs where it is strictly greater than is given in A332407.

Examples

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       3,       1;
  1,   13,      15,       4,       1;
  1,   36,      83,      30,       5,      1;
  1,  101,     582,     302,      51,      6,    1;
  1,  365,    6024,    5025,     843,     80,    7,   1;
  1, 1518,   99497,  144370,   27160,   1996,  117,   8, 1;
  1, 8002, 2706069, 7441209, 1733211, 112291, 4211, 164, 9, 1;
  ...

Links

Crossrefs

Row sums are A000088.
Cf. A263284, A263341, A332402, A332404, A332405, A332407.
Showing 1-10 of 18 results. Next

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

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