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 1-4 of 4 results.

A324968 Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 22, 26, 29, 31, 41, 58, 62, 79, 82, 101, 109, 127, 158, 179, 202, 218, 254, 271, 293, 358, 401, 421, 542, 547, 586, 599, 709, 802, 842, 929, 1063, 1094, 1198, 1231, 1361, 1418, 1609, 1741, 1858, 1913, 2126, 2411, 2462, 2722, 2749
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence ranks rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.

Examples

			The sequence of trees together with the Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    5: (((o)))
    6: (o(o))
   10: (o((o)))
   11: ((((o))))
   13: ((o(o)))
   22: (o(((o))))
   26: (o(o(o)))
   29: ((o((o))))
   31: (((((o)))))
   41: (((o(o))))
   58: (o(o((o))))
   62: (o((((o)))))
   79: ((o(((o)))))
   82: (o((o(o))))
  101: ((o(o(o))))
  109: (((o((o)))))
  127: ((((((o))))))

Links

Crossrefs

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324969, A324970, A324978.

Programs

  • Mathematica
    mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Formula

Intersection of A324935 and A276625.

A324971 Number of rooted identity trees with n vertices whose non-leaf terminal subtrees are not all different.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 12, 31, 79, 192, 459, 1082, 2537, 5922, 13816, 32222, 75254, 176034, 412667, 969531, 2283278
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.

Examples

			The a(6) = 1 through a(8) = 12 trees:
  ((o)((o)))  ((o)(o(o)))   (o(o)(o(o)))
              (o(o)((o)))   (((o))(o(o)))
              (((o)((o))))  (((o)(o(o))))
              ((o)(((o))))  ((o)((o(o))))
                            ((o)(o((o))))
                            ((o(o)((o))))
                            (o((o)((o))))
                            (o(o)(((o))))
                            ((((o)((o)))))
                            (((o))(((o))))
                            (((o)(((o)))))
                            ((o)((((o)))))

Crossrefs

The Matula-Goebel numbers of these trees are given by A324970.
Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324979.

Programs

  • Mathematica
    rits[n_]:=Join@@Table[Select[Union[Sort/@Tuples[rits/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[rits[n],!UnsameQ@@Cases[#,{},{0,Infinity}]&]],{n,10}]

A324978 Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

4, 7, 8, 12, 14, 16, 17, 19, 20, 21, 24, 28, 32, 34, 35, 37, 38, 40, 42, 43, 44, 48, 51, 52, 53, 56, 57, 59, 64, 67, 68, 70, 71, 73, 74, 76, 77, 80, 84, 85, 86, 88, 89, 91, 95, 96, 102, 104, 106, 107, 112, 114, 116, 118, 124, 128, 129, 131, 133, 134, 136, 139
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.

Examples

			The sequence of trees together with the Matula-Goebel numbers begins:
   4: (oo)
   7: ((oo))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  34: (o((oo)))
  35: (((o))(oo))
  37: ((oo(o)))
  38: (o(ooo))
  40: (ooo((o)))
  42: (o(o)(oo))
  43: ((o(oo)))

Links

Crossrefs

Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324968, A324969, A324970, A324971, A324979.

Programs

  • Mathematica
    mgtree[n_]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[!And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{},{0,Infinity}]]&]

Formula

Complement of A276625 in A324935.

A324979 Number of rooted trees with n vertices that are not identity trees but whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 29, 70, 168, 402, 959, 2284, 5434, 12923, 30727, 73055, 173678, 412830
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.

Examples

			The a(3) = 1 through a(6) = 12 trees:
  (oo)  (ooo)   (oooo)    (ooooo)
        ((oo))  ((ooo))   ((oooo))
                (o(oo))   (o(ooo))
                (oo(o))   (oo(oo))
                (((oo)))  (ooo(o))
                          (((ooo)))
                          ((o)(oo))
                          ((o(oo)))
                          ((oo(o)))
                          (o((oo)))
                          (oo((o)))
                          ((((oo))))

Crossrefs

The Matula-Goebel numbers of these trees are given by A324978.
Cf. A000081, A004111, A290689, A317713, A324850, A324922, A324923, A324924, A324931, A324935, A324936, A324970, A324971.

Programs

  • Mathematica
    rits[n_]:=Join@@Table[Union[Sort/@Tuples[rits/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[rits[n],And[UnsameQ@@Cases[#,{},{0,Infinity}],!And@@Cases[mgtree[#],q:{}:>UnsameQ@@q,{0,Infinity}]]&]],{n,10}]
Showing 1-4 of 4 results.

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