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 1-3 of 3 results.

A317787 Number of locally nonintersecting rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 40, 95, 227, 557, 1382, 3485, 8865, 22790, 59022, 153972, 404066, 1066236, 2826885, 7527411, 20121154
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

Examples

			The a(6) = 18 locally nonintersecting rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  (((ooo)))
  ((o(oo)))
  ((oo(o)))
  (o((oo)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (o(ooo))
  (oo(oo))
  (ooo(o))
  (ooooo)
Missing from this list are (((o)(o))) and ((o)(oo)).

Crossrefs

Cf. A000081, A276625, A301700, A316473, A316475, A316501, A316502, A317708, A317785, A317789.

Programs

  • Mathematica
    rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Intersection@@#=={}]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

Extensions

a(16)-a(21) from Robert Price, Sep 16 2018

A317786 Matula-Goebel numbers of locally connected rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

Examples

			The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

Crossrefs

A subset of A184155.
Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

Programs

  • Mathematica
    multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[1000],rupQ[#]&]

A317789 Matula-Goebel numbers of rooted trees that are not locally nonintersecting.

Original entry on oeis.org

9, 21, 23, 25, 27, 39, 46, 49, 57, 63, 65, 69, 73, 81, 83, 87, 91, 92, 97, 103, 111, 115, 117, 121, 125, 129, 133, 138, 146, 147, 159, 161, 166, 167, 169, 171, 183, 184, 185, 189, 194, 199, 203, 206, 207, 213, 219, 227, 230, 235, 237, 243, 247, 249, 253, 259
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

Examples

			The sequence of rooted trees that are not locally nonintersecting together with their Matula-Goebel numbers begins:
   9: ((o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  39: ((o)(o(o)))
  46: (o((o)(o)))
  49: ((oo)(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  65: (((o))(o(o)))
  69: ((o)((o)(o)))
  73: (((o)(oo)))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  87: ((o)(o((o))))
  91: ((oo)(o(o)))
  92: (oo((o)(o)))
  97: ((((o))((o))))

Crossrefs

Cf. A000081, A184155, A276625, A316470, A316495, A316502, A317785, A317786, A317787.

Programs

  • Mathematica
    rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!rupQ[#]&]
Showing 1-3 of 3 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.