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 1-9 of 9 results.

A135388 Number of (directed) Eulerian circuits on the complete graph K_{2n+1}.

Original entry on oeis.org

2, 264, 129976320, 911520057021235200, 257326999238092967427785160130560, 6705710151431658873046319662156165939200000000000000, 32132958735643556926111996291480203406145819659840760945049600000000000000000
Offset: 1

Views

Author

Max Alekseyev, Dec 10 2007

Keywords

References

  • B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.

Links

Crossrefs

Bisection of A350028.
Cf. A007082, A232545, A357856, A357857, A356366, A357855, A357885, A357886, A357887.
Cf. A369820 (undirected Eulerian circuits).

Programs

  • Mathematica
    Table[2 Length[FindEulerianCycle[CompleteGraph[2 n + 1], All]], {n, 3}] (* Eric W. Weisstein, Jan 09 2018 *)
  (* a(3) requires a very large amount of memory *)

Formula

a(n) = A007082(n) * (n-1)!^(2*n+1).
a(n) = A350028(2n+1) = A357887(2n+1,n(2n+1)). - Max Alekseyev, Oct 19 2022

A007082 Number of Eulerian circuits on the complete graph K_{2n+1}, divided by (n-1)!^(2n+1).

Original entry on oeis.org

2, 264, 1015440, 90449251200, 169107043478365440, 6267416821165079203599360, 4435711276305905572695127676467200, 58393052751308545653929138771580386824519680, 14021772793551297695593332913856884153315254190271692800, 60498832138791357698014788383803842810832836262245623803123983974400
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Examples

			From _Günter Rote_, Dec 09 2021: (Start)
For n=2, in the graph K5, if we fix the Euler tour to start with the edge 12, we get 132 Euler tours. Here are the first and the last few in lexicographic order.
  12314253451
  12314254351
  12314352451
  12314354251
  12314524351
  ...
  12543153241
  12543241351
  12543241531
  12543513241
  12543514231.
To get all 264*1!^5 = 264 Euler tours, the number must be multiplied by 2 to include the reversed tours. (End)

References

  • D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, p. 745, Problem 107.
  • B. D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A135388, A232545, A350028, A356366, A357855, A357856, A357857, A357885, A357886, A357887.

Programs

  • Python
    # for n <= 4
def A(n,w="12"):
    if len(w) > (2*n+1)*n: return 2
    return sum(A(n, w+t) for t in "123456789"[:2*n+1]
        if t!=w[-1] and t+w[-1] not in w and w[-1]+t not in w)

Formula

a(n) = A135388(n) / (n-1)!^(2n+1) = A350028(2n+1) / (n-1)!^(2n+1) = A357887(2n+1,n(2n+1)) / (n-1)!^(2n+1). - Max Alekseyev, Oct 19 2022

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

A357857 Number of (open and closed) trails in the complete undirected graph on n labeled vertices.

Original entry on oeis.org

1, 4, 21, 232, 14425, 3653196, 17705858989, 261353065517776, 241809117107232026097
Offset: 1

Views

Author

Max Alekseyev, Oct 16 2022

Keywords

Comments

Trails are directed and pass through each (undirected) edge at most once in either of the two directions.

Crossrefs

Cf. A007082, A135388, A232545, A350028, A356366, A357855, A357856, A357885, A357886, A357887.

Formula

a(n) = n * A357855(n) + n * (n-1) * A357856(n).

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022

A357885 Triangle read by rows: T(n,k) = number of closed trails of length k starting and ending at a fixed vertex in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 6, 6, 0, 0, 1, 0, 0, 12, 24, 24, 72, 168, 0, 0, 528, 1, 0, 0, 20, 60, 120, 480, 1680, 3120, 5760, 15840, 29040, 22320, 0, 0, 0, 1, 0, 0, 30, 120, 360, 1800, 8280, 27360, 88560, 310320, 934560, 2296800, 5541120, 12965760, 21837600, 27740160, 58752000, 101882880, 0, 0, 389928960
Offset: 1

Views

Author

Max Alekseyev, Oct 18 2022

Keywords

Examples

			Triangle starts:
 n=1: 1
 n=2: 1, 0
 n=3: 1, 0, 0, 2
 n=4: 1, 0, 0, 6, 6, 0, 0
 n=5: 1, 0, 0, 12, 24, 24, 72, 168, 0, 0, 528
 ...

Links

Crossrefs

Cf. A007082, A135388, A232545, A297383, A350028, A356366, A357855 (row sums), A357856, A357857, A357886, A357887.

Formula

For k >= 1, T(n,k) = A357887(n,k) * k / n.
Last nonzero element in row n:
T(2n+1,n(2n+1)) = A135388(n) * n = A350028(2n+1) * n = A007082(n) * n * (n-1)!^(2*n+1);
T(2n,2n(n-1)) = A350028(2n) * (n-1) * (2n-1)!! = A297383(n) * (2n-2) * (2n-1)!!.

A357886 Triangle read by rows: T(n,k) = number of open trails of length k starting and ending at fixed distinct vertices in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n*(n-1)/2.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 2, 4, 6, 0, 0, 1, 3, 6, 18, 48, 78, 96, 132, 132, 0, 0, 1, 4, 12, 48, 180, 528, 1392, 3600, 7920, 13680, 21840, 31872, 25008, 0, 0, 0, 1, 5, 20, 100, 480, 1980, 7680, 29040, 100920, 316320, 923520, 2502000, 6011760, 12584880, 23417280, 38196480, 50112000, 53667840, 64988160, 64988160, 0
Offset: 1

Views

Author

Max Alekseyev, Oct 19 2022

Keywords

Examples

			Triangle T(n,k) starts with:
n=1: 0,
n=2: 0, 1,
n=3: 0, 1, 1, 0,
n=4: 0, 1, 2, 2, 4, 6, 0,
n=5: 0, 1, 3, 6, 18, 48, 78, 96, 132, 132, 0,
...

Links

Crossrefs

Cf. A007082, A135388, A232545, A350028, A357855, A357856 (row sums), A357857, A356366, A357885, A357887.

A357887 Triangle read by rows: T(n,k) = number of circuits of length k in the complete undirected graph on n labeled vertices, for n >= 1 and k = 0 .. n(n-1)/2.

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 2, 4, 0, 0, 8, 6, 0, 0, 5, 0, 0, 20, 30, 24, 60, 120, 0, 0, 264, 6, 0, 0, 40, 90, 144, 480, 1440, 2340, 3840, 9504, 15840, 11160, 0, 0, 0, 7, 0, 0, 70, 210, 504, 2100, 8280, 23940, 68880, 217224, 594720, 1339800, 2983680, 6482880, 10190880, 12136320, 24192000, 39621120, 0, 0, 129976320
Offset: 1

Views

Author

Max Alekseyev, Oct 19 2022

Keywords

Comments

A circuit of length k is viewed as a sequence of k+1 vertices it visits modulo cyclic rotations. Hence T(n,0) = n enumerates individual vertices.

Examples

			Triangle T(n,k) starts with:
 n=1: 1,
 n=2: 2, 0,
 n=3: 3, 0, 0, 2,
 n=4: 4, 0, 0, 8, 6, 0, 0,
 n=5: 5, 0, 0, 20, 30, 24, 60, 120, 0, 0, 264,
 ...

Links

Crossrefs

Cf. A007082, A135388, A232545, A297383, A350028, A356366 (row sums), A357855, A357856, A357857, A357885, A357886.

Formula

For k >= 1, T(n,k) = A357885(n,k) * n / k.
Last nonzero element in row n:
T(2n+1,n(2n+1)) = A135388(n) = A350028(2n+1) = A007082(n) * (n-1)!^(2*n+1);
T(2n,2n(n-1)) = A350028(2n) * (2n-1)!! = A297383(n) * 2 * (2n-1)!!.

A356366 Number of (directed) circuits in the complete undirected graph on n labeled vertices.

Original entry on oeis.org

1, 2, 5, 18, 523, 44884, 227838935, 1086696880188, 1566338449874827101, 694432397394116143569646
Offset: 1

Views

Author

Max Alekseyev, Oct 16 2022

Keywords

Comments

In other words, number of closed trails up to cyclic rotations (cf. A357855).

Examples

			For n = 3, we have 5 circuits: 3 of length 0 (singleton vertices), and 2 of length 3 (1->2->3->1 and 1->3->2->1).

Crossrefs

Cf. A007082, A135388, A232545, A350028, A357855, A357856, A357857, A357885, A357886, A357887.

Formula

a(n) = Sum_{k = 0..n(n-1)/2} A357887(n,k) = n + n * Sum_{k = 1..n(n-1)/2} A357885(n,k) / k.

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022
a(10) from Max Alekseyev, Jul 17 2025

A357855 Number of closed trails starting and ending at a fixed vertex in the complete undirected graph on n labeled vertices.

Original entry on oeis.org

1, 1, 3, 13, 829, 78441, 622316671, 3001764349333, 5926347237626029593, 2616519370820267981798929
Offset: 1

Views

Author

Max Alekseyev, Oct 16 2022

Keywords

Comments

Trails are directed and pass through each (undirected) edge at most once in either of the two directions.

Examples

			For n = 3, we have a(3) = 3 trails starting and ending at vertex 1: 1 (single vertex), 1->2->3->1, and 1->3->2->1.

Crossrefs

Cf. A007082, A135388, A232545, A350028, A356366, A357856, A357857, A357885, A357886, A357887.

Formula

a(n) = Sum_{k = 0..n(n-1)/2} A357885(n,k) = 1 + Sum_{k = 1..n(n-1)/2} A357887(n,k) * k / n.

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022
a(10) from Max Alekseyev, Jul 17 2025

A357856 Number of trails between two fixed distinct vertices in the complete undirected graph on n labeled vertices.

Original entry on oeis.org

0, 1, 2, 15, 514, 106085, 317848626, 4238195548627, 2617666555119413330
Offset: 1

Views

Author

Max Alekseyev, Oct 16 2022

Keywords

Comments

Trails are directed and pass through each (undirected) edge at most once in either of the two directions.

Crossrefs

Cf. A007082, A135388, A232545, A350028, A356366, A357855, A357857, A357885, A357886, A357887.

Formula

a(n) = Sum_{k = 0..n(n-1)/2} A357886(n,k).

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022
Showing 1-9 of 9 results.

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

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