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.

A121941 Number of unlabeled connected simple graphs with n nodes of degree 4 or less.

Original entry on oeis.org

1, 1, 1, 2, 6, 21, 78, 353, 1929, 12207, 89402, 739335, 6800637, 68531618, 748592936, 8788983173, 110201690911, 1468157196474, 20695559603921, 307590282700915, 4805537369573319, 78710267083015571, 1348394635886684901, 24109112440149231355, 449050443283294835914
Offset: 0

Views

Author

Parthasarathy Nambi, Sep 03 2006

Keywords

Comments

Number of graphs of hydrogen bonded water clusters.
This counts the connected graphs where each vertex has degree 4 or less. - Charles R Greathouse IV, Jul 07 2017
Also number of saturated hydrocarbons and allotropes with n carbon, or valence 4, atoms (excluding stereoisomers) following the octet rule. - Natan Arie Consigli, Jul 07 2017

Examples

			With 4 carbons, n-butane, i-butane, cyclobutane, bicyclobutane, methylcyclopropane and tetrahedrane are the 6 isomers satisfying the property above, so a(4)=6. - _Natan Arie Consigli_, Jul 07 2017
If n=5 then the number of graphs of hydrogen bonded water clusters is 21.

Links

Crossrefs

Cf. A121942, A243393 (degree 3 or less), A287424 (excludes allotropes), A289157, A289158, A303032 (degree 5 or less).

Programs

  • nauty
    geng -c -D4 ${n} -q | multig -m1 -D4 -u

Extensions

More terms sent by Natan Arie Consigli, Jul 07 2017
Renamed by Andrew Howroyd, Mar 19 2020 based on comment by Charles R Greathouse IV.
a(16)-a(24) from Andrew Howroyd, Mar 19 2020

A134818 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less and with at most triple edges.

Original entry on oeis.org

1, 3, 9, 37, 146, 772, 4449, 30307, 228605, 1921464, 17652327, 176162548, 1893738334, 21806975279, 267636988052, 3486370839295, 48029272657002, 697542580286159, 10649954607360119, 170508064788069346, 2856122791685125616, 49951625299057923405
Offset: 1

Views

Author

David Consiglio, Jr., Jan 28 2008

Keywords

Comments

From Natan Arie Consigli, May 29 2017: (Start)
Original name was "Number of hydrocarbon structures that can be drawn (excluding stereoisomers)" but this has been replaced with a mathematical definition which is more consistent with the terms of the sequence and the program.
In chemical terms this counts the following, given n carbon atoms:
- carbon allotropes;
- aliphatic hydrocarbons;
- resonance structures of graphically non-equivalent anti-aromatic and aromatic hydrocarbons.
Some molecules are theoretical and may or may not exist.
(End)
Computed over a period of several years and confirmed using the Molgen program.
Terms for n = 8,9,10 calculated using an exhaustive algorithm and Nauty. The algorithm correctly found the 7 known terms and the known acyclic hydrocarbons (up to n=10, see A002986) were extracted from the results correctly. - Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
Except for a(2), the same as A289157. The extra graph in A289157 is the 4-regular graph on 2 nodes. - Andrew Howroyd, Mar 20 2020

Examples

			For n = 2 there are a(2) = 3 structures that can be drawn with 2 carbons (ethane, ethene, and ethyne).
For n = 7 there are a(7) = 4449 structures that can be drawn with 7 carbons.

Links

Crossrefs

Cf. A134819 gives the number of possible structures, broken down by units of unsaturation.
Cf. A002986 (non-cyclic hydrocarbons).
Cf. A121941, A289157, A289158, A303033.

Programs

  • nauty
    geng -c -D4 ${n} -q | multig -m3 -D4 -u

Formula

a(n) = A289157(n) for n > 2. - Andrew Howroyd, Mar 20 2020

Extensions

a(8)-a(10) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 17 2008
a(11) from Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 24 2008
a(12) sent by David Consiglio, Jr., Apr 23 2008
a(12) corrected, a(13) and a(14) added - David Consiglio, Jr. Nov 03 2011
a(15)-a(17) computed using nauty by Sean A. Irvine, Jan 19 2015
New name from Natan Arie Consigli, May 29 2016
a(18)-a(22) from Andrew Howroyd, Mar 20 2020

A289157 Number of unlabeled connected loopless multigraphs with n nodes of degree 4 or less.

Original entry on oeis.org

1, 1, 4, 9, 37, 146, 772, 4449, 30307, 228605, 1921464, 17652327, 176162548, 1893738334, 21806975279, 267636988052, 3486370839295, 48029272657002, 697542580286159, 10649954607360119, 170508064788069346, 2856122791685125616, 49951625299057923405
Offset: 0

Views

Author

Natan Arie Consigli, Jul 04 2017

Keywords

Crossrefs

Column k=4 of A334546.
Cf. A121941 (single edges only), A134818 (with no more than triple edges), A289158 (with no more than double edges).
Cf. A243391 (degree 3 or less).

Programs

  • nauty
    geng -c -D4 ${n} -q | multig -D4 -u

Extensions

a(18)-a(22) from Andrew Howroyd, Mar 20 2020

A303031 Number of unlabeled connected loopless multigraphs with n nodes of degree 5 or less and with single or double edges.

Original entry on oeis.org

1, 1, 2, 7, 43, 282, 2708, 31175, 451701, 7731154, 154264825, 3515514725, 90381251065, 2594105950453, 82437061616923, 2880328250160638, 109987239823116870, 4566153786442575091, 205144850920195457266, 9933105076082553631262, 516439104253062357469829
Offset: 0

Views

Author

Natan Arie Consigli, Jun 04 2018

Keywords

Crossrefs

Cf. A289158, A303032, A303033.

Programs

  • nauty
    for n in {1..11}; do geng -c -D5 ${n}  -q | multig -m2 -D5 -u; done

Extensions

a(12)-a(20) from Andrew Howroyd, Mar 20 2020

A289988 Number of unlabeled connected loopless multigraphs with n nodes of degree n or less.

Original entry on oeis.org

1, 1, 2, 4, 37, 602, 34126, 6021463, 3616906549, 7361925161868, 51324462383008758, 1240420936122453106498, 105141919479926837860474091, 31581183353539008502807807352728
Offset: 0

Views

Author

Natan Arie Consigli, Aug 19 2017

Keywords

Comments

Multigraphs are loopless.

Crossrefs

Main diagonal of A334546.
Cf. A007719, A076864, A134818, A289157, A289158, A289987.

Programs

  • nauty
    for n in {1..8}; do geng -c -D${n} ${n} -q | multig -m$[${n}-1] -D$[${n}-1] -u; done

Extensions

a(0) corrected and a(9)-a(13) from Andrew Howroyd, May 05 2020

A303030 Number of unlabeled connected loopless multigraphs with n nodes of degree 3 or less and with single or double edges.

Original entry on oeis.org

1, 1, 2, 4, 12, 22, 68, 166, 534, 1589, 5464, 18579, 68320, 255424, 1000852, 4018156, 16671976, 70890940, 309439942, 1381815168, 6310880471, 29428287639, 140012980007, 678970863717, 3353545264060, 16857749613964, 86191265140699, 447951112379963, 2365177154077186
Offset: 0

Views

Author

Natan Arie Consigli, Apr 17 2018

Keywords

Comments

For n >= 1, a(n) is also the number of hydronitrogen molecules containing only n nitrogen trivalent (octet rule satisfying) atoms. So for example, diazene is counted but hydrazoic acid is not because the former has only trivalent nitrogens and the latter has two non-trivalent nitrogens.
Some of the molecules are theoretical and may or may not exist due to their strained geometries.
Apparently the same as A243391 for n > 2. - Georg Fischer, Oct 16 2018
This is the case since A243391 gives the number of loopless multigraphs with nodes of degree 3 or less. The extra graph in A243391 is the 3-regular graph on 2 nodes. - Andrew Howroyd, Mar 20 2020

Examples

			a(3) = 4 because there are 4 molecules satisfying the above condition: triazane, triazene, triazirine, triazidirine.
Note: hydrazoic acid is not counted because there are 2 nitrogens not satisfying the octet rule (one has a positive formal charge and the other one has a negative one).
Graphically, a(3) = 4 because there are 4 graphs satisfying the above condition: the linear graph, the linear graph with one double edge, the triangle graph, and the triangle graph with one double edge. - _Michael B. Porter_, Apr 28 2018

Crossrefs

Cf. A134818, A243391, A289158, A303031.

Programs

  • nauty
    for n in {1..18}; do geng -c -D3 ${n}  -q | multig -m2 -D3 -u;done

Formula

a(n) = A243391(n) for n > 2. - Andrew Howroyd, Mar 20 2020

Extensions

a(20)-a(28) from Andrew Howroyd, Mar 20 2020
Showing 1-6 of 6 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.