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.

User: Gary Gordon

Gary Gordon's wiki page.

Gary Gordon has authored 4 sequences.

A210507 Number of labeled graphs on [n] with unicyclic components containing a given edge.

Original entry on oeis.org

1, 10, 111, 1468, 22940, 416250, 8626660, 201349672, 5230931454, 149783426470, 4688281021490, 159284662406460, 5838769123729984, 229711022253150382, 9655348958575618320, 431845990498159342000, 20479127764425617465660, 1026429489947790074019978
Offset: 3

Views

Author

Gary Gordon, Jan 25 2013

Keywords

Comments

This gives the number of matroid bases that contain a given element (edge) of the bicircular matroid of K_n.

Examples

			a(4)=10 means that 10 (of the 15) labeled unicyclic graphs on 4 vertices contain a given edge.

References

  • O. Giménez, A. de Mier, M. Noy, On the Number of Bases of Bicircular Matroids, Ann. Comb. 9 (2005), no. 1, 35-45.

Crossrefs

Cf. A137916.

Formula

a(n) = 2*b(n)/(n-1), where b(n) is seq A137916.

A161936 The number of direct isometries that are derangements of the (n-1)-dimensional facets of an n-cube.

Original entry on oeis.org

0, 3, 14, 117, 1164, 13975, 195642, 3130281, 56345048, 1126900971, 24791821350, 595003712413, 15470096522724, 433162702636287, 12994881079088594, 415836194530835025, 14138430614048390832, 508983502105742069971, 19341373080018198658878, 773654923200727946355141
Offset: 1

Views

Author

Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009

Keywords

Comments

a(n) plays the same role as A003221 plays for the derangement numbers A000166.

Examples

			For a square, the 3 rotations are direct edge derangements. For a 3-cube, the 6 edge-centered rotations and the 8 vertex-centered rotations are direct face derangements.

Links

Crossrefs

Cf. A000354, A161937.

Programs

Formula

a(n) = (b(n) + (-1)^n)/2, where b(n) is sequence A000354, i.e., the number of (n-1)-dimensional facet derangements of an n-cube.
From Peter Luschny, May 09 2017: (Start)
a(n) = (-1)^n*(1-n*hypergeom([1,1-n],[],2)).
a(n) = (2^n*Gamma(n+1,-1/2)/exp(1/2)+(-1)^n)/2. (End)

Extensions

More terms from Peter Luschny, May 09 2017

A161937 The number of indirect isometries that are derangements of the (n-1)-dimensional facets of an n-cube.

Original entry on oeis.org

1, 2, 15, 116, 1165, 13974, 195643, 3130280, 56345049, 1126900970, 24791821351, 595003712412, 15470096522725, 433162702636286, 12994881079088595, 415836194530835024, 14138430614048390833, 508983502105742069970, 19341373080018198658879, 773654923200727946355140
Offset: 1

Views

Author

Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009

Keywords

Comments

a(n) plays the same role as A000387 plays for the derangement numbers A000166.

Examples

			For a square, the 2 diagonal reflections are indirect edge derangements. For a 3-cube, the 15 rotary reflections are indirect face derangements.

Links

Crossrefs

Cf. A000354, A000387, A161936.

Programs

  • Maple
    a := n -> (-1)^(n+1)*n*hypergeom([1,1-n],[],2):
seq(simplify(a(n)), n=1..20); # Peter Luschny, May 09 2017
  • Mathematica
    a[n_] := (-1)^(n + 1)*n*HypergeometricPFQ[{1, 1 - n}, {}, 2];
Array[a, 20] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

Formula

a(n) = (b(n) + (-1)^(n+1))/2, where b(n) is sequence A000354, i.e., the number of (n-1)-dimensional facet derangements of an n-cube.
From Peter Luschny, May 09 2017: (Start)
a(n) = (-1)^(n+1)*n*hypergeom([1, 1-n], [], 2).
a(n) = (2^n*Gamma(n+1,-1/2)/exp(1/2)-(-1)^n)/2. (End)
E.g.f.: x*exp(-x) / (1 - 2*x). - Ilya Gutkovskiy, May 23 2020

Extensions

More terms from Peter Luschny, May 09 2017

A138768 For a positive integer n, write the integers 1,2,...,n in the following order: first write 1 (round 0), then all primes less than or equal to n in increasing order (round 1), then 2p for all primes p with 2p<=n, also in increasing order (round 2), then 3p, then 4p and so on. Each number is written down only the first time it is encountered. Let a(n) denote the last number written down.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 8, 8, 8, 12, 12, 12, 12, 16, 16, 16, 16, 16, 16, 16, 16, 24, 24, 24, 27, 27, 27, 27, 27, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 48, 48, 48, 48, 48, 48, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64
Offset: 1

Views

Author

Emeric Deutsch and Gary Gordon, Apr 01 2008

Keywords

Comments

a(1)=1. For a given n>=2, let M be the largest of the numbers in the finite sequence [m/(largest prime dividing m), m=2,3,...,n]. a(n) is defined to be the largest m in (2,3,...,n) for which M is attained. Example: a(14)=12 because the values of m/(largest prime dividing m) for m = 2,3,...,14 are 1,1,2,1,2,1,4,3,2,1,4,1,2. The largest of these is 4 and it is attained for m=8 and m=12; the largest of these is 12.

Examples

			For n=10 we get the ordering 1/ 2, 3, 5, 7/ 4, 6, 10/ 9/ 8 (the rounds are separated by /); so a(10)=8.

Links

Crossrefs

Cf. A052126.

Programs

  • Maple
    with(numtheory): b:=proc(m) local u: if m=1 then 1 else u:=factorset(m): m/max(seq(u[j],j=1..nops(u))) end if end proc: a:=proc(n) local M,i,a: M:=max(seq(b(j),j=1..n)): for i to n do if b(i)=M then a[i]:=i else a[i]:=0 end if end do: max(seq(a[i],i=1..n)) end proc: seq(a(n),n=1..80);
  • Mathematica
    b[m_] := If[m == 1, 1, m/Max[FactorInteger[m][[All, 1]]]];
a[n_] := Module[{M, i, a}, M = Max[Table[b[j], {j, 1, n}]]; For[i = 1, i <= n, i++, If[b[i] == M, a[i] = i, a[i] = 0]]; Max[Table[a[i], {i, 1, n}]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Sep 05 2024, after Maple program *)

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