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 1-4 of 4 results.

A084268 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 12, 16, 4, 1, 1, 34, 84, 31, 5, 1, 1, 87, 579, 318, 52, 6, 1, 1, 302, 5721, 5366, 867, 81, 7, 1, 1, 1118, 87381, 155291, 28722, 2028, 118, 8, 1, 1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1, 1, 32302, 78315231, 675054876, 250530482, 14662562, 393963, 8214, 222, 10, 1
Offset: 1

Views

Author

Eric W. Weisstein, May 24 2003

Keywords

Comments

T(n,1) = T(n,n) = 1 (here we count the empty graph and the complete graph). T(n,n-1) = n-1 (here we count the graphs with clique number equal to n-1). - Geoffrey Critzer, Oct 12 2016
Row sums give A000088. - Joerg Arndt, Oct 13 2016

Examples

			Triangle begins:
  1;
  1,    1;
  1,    2,       1;
  1,    6,       3,       1;
  1,   12,      16,       4,       1;
  1,   34,      84,      31,       5,      1;
  1,   87,     579,     318,      52,      6,    1;
  1,  302,    5721,    5366,     867,     81,    7,   1;
  1, 1118,   87381,  155291,   28722,   2028,  118,   8, 1;
  1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1;
  ...

Links

Crossrefs

Columns k=1..9 are A057427, A076278, A076279, A076280, A076281, A076282, A076283, A205567, A205568.
Partial row sums include A033995, A076315, A076316, A076317, A076318, A076319, A076320, A076321.
Row sums are A000088.
Cf. A084269 (connected), A115597 (essentially the same sequence).

Programs

  • Sage
    # prints triangle with a leading zero in each row
for n in range(1, 8) :
    st = [0 for j in range(n+1)]
    G = graphs(n)
    for g in G :
        st[ g.chromatic_number() ] += 1
    print(st)
# Joerg Arndt, Oct 13 2016

Extensions

Offset corrected by Joerg Arndt, Oct 13 2016
a(36)-a(55) from Joerg Arndt, Oct 15 2016
a(56)-a(66) from Andrew Howroyd, Dec 02 2018

A076278 Number of 2-chromatic (i.e., chromatic number equals 2) simple graphs on n nodes.

Original entry on oeis.org

0, 1, 2, 6, 12, 34, 87, 302, 1118, 5478, 32302, 251134, 2527711, 33985852, 611846939, 14864650923, 488222721991, 21712049275197, 1308300679611468, 106897965189674290, 11852113048215107821, 1784730721403509209214, 365323537513403184463272
Offset: 1

Views

Author

Eric W. Weisstein, Oct 06 2002

Keywords

Links

Crossrefs

Column k=2 of A084268.
Cf. A076279, A076280, A076281, A076282, A115597.

Programs

  • Mathematica
    A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
a = etr[A005142[[# + 1]]&][#] - 1&;
Array[a, 23] (* Jean-François Alcover, Sep 03 2019 *)

Formula

a(n) = A033995(n)-1.

Extensions

More terms from Vladeta Jovovic, Jul 31 2003
Terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

A076279 Number of 3-chromatic (i.e., chromatic number equals 3) simple graphs on n nodes.

Original entry on oeis.org

0, 0, 1, 3, 16, 84, 579, 5721, 87381, 2104349, 78315231, 4383566676, 362178638727
Offset: 1

Views

Author

Eric W. Weisstein, Oct 06 2002

Keywords

Links

Crossrefs

Column k=3 of A084268.
Cf. A076278, A076280, A076281, A076282, A115597.

Extensions

a(10) from Keith Briggs, Mar 14 2006
a(11) from Michael Sollami, Jan 28 2012
a(12)-a(13) from Brendan McKay, Nov 08 2022

A084269 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 5, 12, 3, 1, 0, 17, 64, 26, 4, 1, 0, 44, 475, 282, 46, 5, 1, 0, 182, 5036, 5009, 809, 74, 6, 1, 0, 730, 80947, 149551, 27794, 1940, 110, 7, 1, 0, 4032, 2010328, 7694428, 1890221, 113272, 4125, 156, 8, 1, 0, 25598, 76115143, 667036310, 248580644, 14545025, 389583, 8040, 212, 9, 1
Offset: 0

Views

Author

Eric W. Weisstein, May 24 2003

Keywords

Examples

			Triangle begins:
  1;
  0,   1;
  0,   1,     1;
  0,   3,     2,      1;
  0,   5,    12,      3,     1;
  0,  17,    64,     26,     4,    1;
  0,  44,   475,    282,    46,    5,   1;
  0, 182,  5036,   5009,   809,   74,   6, 1;
  0, 730, 80947, 149551, 27794, 1940, 110, 7, 1;
  ...

Links

Crossrefs

Row sums are A001349.
Columns k=3..7 are A126737, A126738, A126739, A126740, A241702.
Partial row sums include A005142, A076322, A076323, A076324, A076325, A076326, A076327, A076328.
Essentially the same table as A126736.
Cf. A084268 (not necessarily connected), A115597.

Extensions

a(37)-a(66) from Andrew Howroyd, Dec 02 2018
Showing 1-4 of 4 results.

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