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a(n) is the number of edges in the diagram of the symmetric representation of sigma(n) with subparts.
3

%I #34 Feb 03 2021 23:37:04

%S 4,6,8,10,10,14,12,14,16,16,14,24,14,18,24,22,16,28,16,26,26,22,18,36,

%T 24,22,28,30,20,44,20,30

%N a(n) is the number of edges in the diagram of the symmetric representation of sigma(n) with subparts.

%C Since the diagram is symmetric so all terms are even numbers.

%C For another version see A340846 from which first differs at a(6).

%C For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616.

%C Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n).

%F a(n) = A340847(n) + A001227(n) - 1 (Euler's formula).

%e Illustration of initial terms:

%e . _ _ _ _

%e . _ _ _ |_ _ _ |_

%e . _ _ _ |_ _ _| | |_|_

%e . _ _ |_ _ |_ |_ _ |_ _ |

%e . _ _ |_ _|_ |_ | | | | |

%e . _ |_ | | | | | | | | |

%e . |_| |_| |_| |_| |_| |_|

%e .

%e n: 1 2 3 4 5 6

%e a(n): 4 6 8 10 10 14

%e .

%e For n = 6 the diagram has 14 edges so a(6) = 14.

%e On the other hand the diagram has 13 vertices and two subparts or regions, so applying Euler's formula we have that a(6) = 13 + 2 - 1 = 14.

%e . _ _ _ _ _

%e . _ _ _ _ _ |_ _ _ _ _|

%e . _ _ _ _ |_ _ _ _ | |_ _

%e . |_ _ _ _| | |_ |_ |

%e . |_ |_ |_ _ |_|_ _

%e . |_ _ |_ _ | | |

%e . | | | | | |

%e . | | | | | |

%e . | | | | | |

%e . |_| |_| |_|

%e .

%e n: 7 8 9

%e a(n): 12 14 16

%e .

%e For n = 9 the diagram has 16 edges so a(9) = 16.

%e On the other hand the diagram has 14 vertices and three subparts or regions, so applying Euler's formula we have that a(9) = 14 + 3 - 1 = 16.

%e Another way for the illustration of initial terms is as follows:

%e --------------------------------------------------------------------------

%e . n a(n) Diagram

%e --------------------------------------------------------------------------

%e _

%e 1 4 |_| _

%e _| | _

%e 2 6 |_ _| | | _

%e _ _|_| | | _

%e 3 8 |_ _| _| | | | _

%e _ _| _| | | | | _

%e 4 10 |_ _ _| _|_| | | | | _

%e _ _ _| _ _| | | | | | _

%e 5 10 |_ _ _| | _ _| | | | | | | _

%e _ _ _| |_| _|_| | | | | | | _

%e 6 14 |_ _ _ _| _| _ _| | | | | | | | _

%e _ _ _ _| _| _ _| | | | | | | | | _

%e 7 12 |_ _ _ _| | _| _ _|_| | | | | | | | | _

%e _ _ _ _| | _| | _ _| | | | | | | | | | _

%e 8 14 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | | _

%e _ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | |

%e 9 16 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | |

%e _ _ _ _ _| | _| _| _ _ _| | | | | | | | |

%e 10 16 |_ _ _ _ _ _| | _| _| | _ _|_| | | | | | |

%e _ _ _ _ _ _| | _| _| | _ _ _| | | | | |

%e 11 14 |_ _ _ _ _ _| | |_ _| _| | _ _ _| | | | |

%e _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | |

%e 12 24 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| |

%e _ _ _ _ _ _ _| | _| | _ _| | _ _ _|

%e 13 14 |_ _ _ _ _ _ _| | | _| |_| _| |

%e _ _ _ _ _ _ _| | |_ _| _| _|

%e 14 18 |_ _ _ _ _ _ _ _| | _ _| _|

%e _ _ _ _ _ _ _ _| | _ _|

%e 15 24 |_ _ _ _ _ _ _ _| | |

%e _ _ _ _ _ _ _ _| |

%e 16 22 |_ _ _ _ _ _ _ _ _|

%e ...

%Y Cf. A001227 (number of subparts or regions).

%Y Cf. A340847 (number of vertices).

%Y Cf. A340846 (number of edges in the diagram only with parts).

%Y Cf. A317292 (total number of edges in the unified diagram).

%Y Cf. A000203, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340833.

%K nonn,more

%O 1,1

%A _Omar E. Pol_, Jan 24 2021