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.

Previous Showing 11-17 of 17 results.

A185203 Number of disconnected 10-regular graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 11, 550, 806174, 2585947720, 9802278927562, 42709859521915286, 214798119408798346811, 1251607430636395979871600, 8463468717232507491862780325, 66406919318277846825588474735084
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

10-regular simple graphs: A014382 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), this sequence (k=10), A185213 (k=11).

Extensions

Terms a(29) and beyond from Andrew Howroyd, May 20 2020

A185213 Number of disconnected 11-regular graphs with 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 8037887, 945095928322681, 187549741420313256356540, 66398446859255608487987488813721, 43100445877221052008718432480116589483823
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

11-regular simple graphs: A014384 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), A185293 (k=9), A185203 (k=10), this sequence (k=11).

Extensions

a(15)-a(18) from Andrew Howroyd, May 20 2020

A185293 Number of disconnected 9-regular graphs with 2n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 88238, 113315027550, 281342192047999912, 1251394783006077652496450, 9854615127100313024544239975139, 134283364935428822131144679491097123786
Offset: 0

Views

Author

Jason Kimberley, Jan 26 2012

Keywords

Links

Crossrefs

9-regular simple graphs: A014381 (connected), this sequence (disconnected).
Disconnected regular simple graphs: A068932 (any degree), A068933 (triangular array), specified degree k: A157928 (k=0), A157928 (k=1), A165652 (k=2), A165653 (k=3), A033483 (k=4), A165655 (k=5), A165656 (k=6), A165877 (k=7), A165878 (k=8), this sequence (k=9), A185203 (k=10), A185213 (k=11).

Extensions

a(14)-a(17) from Andrew Howroyd, May 20 2020

A210703 Triangular array D(n,k) counting disconnected k-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 0, 3, 0, 1, 0, 0, 4, 8, 3, 1, 0, 0, 5, 0, 8, 0, 0, 0, 6, 29, 25, 3, 1, 0, 0, 9, 0, 88, 0, 1, 0, 0, 10, 138, 377, 66, 5, 1, 0, 0, 13, 0, 2026, 0, 25, 0, 0, 0, 17, 774, 13349, 8029, 297, 5, 1, 0, 0, 21, 0, 104593, 0, 8199, 0, 1, 0, 0, 25, 5678, 930571, 3484759, 377004, 1562, 7, 1
Offset: 2

Views

Author

Jason Kimberley, Jan 21 2013

Keywords

Examples

			2: 0;
3: 0;
4: 0, 0;
5: 0, 0;
6: 0, 0, 1;
7: 0, 0, 1;
8: 0, 0, 1, 1;
9: 0, 0, 2, 0;
10: 0, 0, 2, 2, 1;
11: 0, 0, 3, 0, 1;
12: 0, 0, 4, 8, 3, 1;
13: 0, 0, 5, 0, 8, 0;
14: 0, 0, 6, 29, 25, 3, 1;
15: 0, 0, 9, 0, 88, 0, 1;
16: 0, 0, 10, 138, 377, 66, 5, 1;
17: 0, 0, 13, 0, 2026, 0, 25, 0;
18: 0, 0, 17, 774, 13349, 8029, 297, 5, 1;
19: 0, 0, 21, 0, 104593, 0, 8199, 0, 1;
20: 0, 0, 25, 5678, 930571, 3484759, 377004, 1562, 7, 1;
21: 0, 0, 33, 0, 9124627, 0, 22014143, 0, 100, 0;
22: 0, 0, 39, 53324, 96699740, 2595985769, 1493574756, 21617036, 10901, 9, 1;
23: 0, 0, 49, 0, 1095467916, 0, 114880777582, 0, 3470736, 0, 1;
24: 0, 0, 60, 622716, 13175254799, 2815099031409, 9919463450854, 733460349818, 1473822243, 88238, 11, 1;

Links

Crossrefs

The sum of the n-th row is A210713(n).
Disconnected k-regular simple graphs with girth exactly 3: A210713 (any k), this sequence (triangle); for a fixed k: A185033 (k=3), A185043 (k=4), A185053 (k=5), A185063 (k=6).

Formula

D(n,k) = A068933(n,k) - A185204(n,k) [the former is padded to be a tabl but the latter is a tabf].
D(n,k) = A185643(n,k) - A186733(n,k) [both are tabl but the result is tabf].

A184324 The number of disconnected k-regular simple graphs on 2k+4 vertices.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 13, 18, 21, 26, 33, 40, 49, 61, 73, 89, 110, 131, 158, 192, 230, 274, 331, 392, 468, 557, 660, 780, 927, 1088, 1284, 1511, 1775, 2076, 2438, 2843, 3323, 3873, 4510, 5238, 6095, 7057, 8182, 9466, 10945, 12626, 14578, 16780, 19323, 22211
Offset: 0

Views

Author

Jason Kimberley, Jan 11 2011

Keywords

Examples

			The a(0)=1 graph is 4K_1. The a(1)=1 graph is 3K_2. The a(2)=2 graphs are C_3+C_5 and C_4+C_4.

Crossrefs

This sequence is the third highest diagonal of D=A068933: that is a(n)=D(2k+4, k).
Cf. A184325(k) = D(4k+5, 2k) and A184326(k) = D(2k+6, k).

Programs

  • Magma
    A184324 := func< n | n eq 0 select 1 else (n+1)mod 2 + A008483(n+3) >; // see A008483 for its MAGMA code.

Formula

a(0)=1. For k>0, a(k) = (k+1) mod 2 + A008483(k+3).
For k>=0, a(k) = A040001(k) + A165652(k+3).
Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+4, k) = C(k+1, k) C(k+3, k) + A000217(C(k+2,k)), from the disconnected Euler transform. C(k+1, k)=1 because K_{k+1} is connected and the unique k-regular graph on k+1 vertices. For k > 1, since D(k+3,k)=0, then C(k+3,k) = E(k+3,k) = E(k+3,2) = A008483(k + 3). Also, for k >0, since D(k+2,k)=0, then C(k+2,k) = E(k+2,k) = E(k+2,1) = (k+1) mod 2. With the examples below and A165652(n)=0 for n < 6 = offset, QED.

A184325 The number of disconnected 2k-regular simple graphs on 4k+5 vertices.

Original entry on oeis.org

1, 3, 8, 25, 100, 550, 4224, 42135, 516383, 7373984, 118573680, 2103205868, 40634185593, 847871397697, 18987149095396, 454032821689310, 11544329612486760, 310964453836199398, 8845303172513782781
Offset: 0

Views

Author

Jason Kimberley, Jan 11 2011

Keywords

Examples

			The a(0)=1 graph is 5K_1. The a(1)=3 graphs are 3C_3, C_3+C_6, and C_4+C_5.

Crossrefs

This sequence is the (even indices of the) fourth highest diagonal of D=A068933: that is a(n) = D(4k+5, 2k).
Cf. A184324(k) = D(2k+4, k) and A184326(k) = D(2k+6, k).

Formula

a(0)=1. For n > 0, a(n) = A051031(2k+4,3) + A051031(2k+3,2) = A005638(k+2) + A008483(2k+3).
Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(4k+5,2k) = C(2k+1, 2k) C(2k+4,2k) + C(2k+2,2k) C(2k+3,2k), from the disconnected Euler transform. For n > 1, D(2k+1,2k) = D(2k+2,2k) = D(2k+3,2k) = D(2k+4,2k) = 0. Therefore, a(n) = E(2k+1, 2k) E(2k+4,2k) + E(2k+2,2k) E(2k+3,2k) = E(2k+1,0) E(2k+4,3) + E(2k+2,1) E(2k+3,2). Note that E(2k+1,0) = E(2k+2,1) = 1. Checking a(1) = E(6,3) + E(5,2), QED.

A184326 The number of disconnected k-regular simple graphs on 2k+6 vertices.

Original entry on oeis.org

1, 1, 4, 9, 25, 66, 297, 1562, 10901, 88238, 806174, 8037887, 86228020, 985884104, 11946634677, 152808994328, 2056701656260
Offset: 0

Views

Author

Jason Kimberley, Jan 15 2011

Keywords

Examples

			The a(0)=1 graph is 6K_1. The a(1)=1 graph is 4K_2. The a(2)=4 graphs are 2C_3+C_4, 2C_5, C_4+C_6, and C_3+C_7.

Crossrefs

This sequence is the fifth highest diagonal of D=A068933: that is a(n)=D(2k+6, k).
Cf. A184324(k) = D(2k+4, k) and A184325(k) = D(4k+5, 2k).

Formula

a(0)=1, a(1)=1, a(2)=4, a(3)=9. For n>3, a(n) = A033301(k+5) + ((k+1)mod 2)*A005638(k div 2 + 2) + A000217(A008483(k+3)).
Proof: Let C=A068934, D=A068933, and E=A051031. Now a(n) = D(2k+6,k) = C(k+1,k)C(k+5,k) + C(k+2,k)C(k+4,k) + A000217(C(k+3,k)), from the disconnected Euler transform. Notice that D(k+i,k)=0 provided k+i < 2k+2; that is k > i-2. So if i <= 5 and k > 3, then D(k+i,k)=0. Hence for k > 3, a(n) = E(k+1,k)E(k+5,k) + E(k+2,k)E(k+4,k) + A000217(E(k+3,k)) = E(k+1,0)E(k+5,4) + E(k+2,1)E(k+4,3) + A000217(E(k+3,2)). We have E(k+1,0)=1, and E(k+2,1)=(k+1)mod 2. For even k, E(k+4,3)=A005638(k div 2 + 2); for odd k, E(k+2,1)=0. QED.
Previous Showing 11-17 of 17 results.

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

Content is available under The OEIS End-User License Agreement.