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a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n) with subparts.
4

%I #35 Feb 03 2021 23:36:54

%S 4,6,7,10,9,13,11,14,14,15,13,23,13,17,21,22,15,26,15,25,23,21,27,35,

%T 22,21,25,29,19,41,19,30

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

%C Theorem: Indices of even terms give A028982. Indices of odd terms give A028983.

%C If A001227(n) is odd then a(n) is even.

%C If A001227(n) is even then a(n) is odd.

%C The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.

%C For another version see A340833 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) = A340848(n) - A001227(n) + 1 (Euler's formula).

%p Illustration of initial terms:

%p . _ _ _ _

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%p . _ _ |_ _|_ |_ | | | | |

%p . _ |_ | | | | | | | | |

%p . |_| |_| |_| |_| |_| |_|

%p .

%p n: 1 2 3 4 5 6

%p a(n): 4 6 7 10 9 13

%p .

%p For n = 6 the diagram has 13 vertices so a(6) = 13.

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

%p .

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%p .

%p n: 7 8 9

%p a(n): 11 14 14

%p .

%p For n = 9 the diagram has 14 vertices so a(9) = 14.

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

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

%p --------------------------------------------------------------------------

%p . n a(n) Diagram

%p --------------------------------------------------------------------------

%p _

%p 1 4 |_| _

%p _| | _

%p 2 6 |_ _| | | _

%p _ _|_| | | _

%p 3 7 |_ _| _| | | | _

%p _ _| _| | | | | _

%p 4 10 |_ _ _| _|_| | | | | _

%p _ _ _| _ _| | | | | | _

%p 5 9 |_ _ _| | _ _| | | | | | | _

%p _ _ _| |_| _|_| | | | | | | _

%p 6 13 |_ _ _ _| _| _ _| | | | | | | | _

%p _ _ _ _| _| _ _| | | | | | | | | _

%p 7 11 |_ _ _ _| | _| _ _|_| | | | | | | | | _

%p _ _ _ _| | _| | _ _| | | | | | | | | | _

%p 8 14 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | | _

%p _ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | |

%p 9 14 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | |

%p _ _ _ _ _| | _| _| _ _ _| | | | | | | | |

%p 10 15 |_ _ _ _ _ _| | _| _| | _ _|_| | | | | | |

%p _ _ _ _ _ _| | _| _| | _ _ _| | | | | |

%p 11 13 |_ _ _ _ _ _| | |_ _| _| | _ _ _| | | | |

%p _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | |

%p 12 23 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| |

%p _ _ _ _ _ _ _| | _| | _ _| | _ _ _|

%p 13 13 |_ _ _ _ _ _ _| | | _| |_| _| |

%p _ _ _ _ _ _ _| | |_ _| _| _|

%p 14 17 |_ _ _ _ _ _ _ _| | _ _| _|

%p _ _ _ _ _ _ _ _| | _ _|

%p 15 21 |_ _ _ _ _ _ _ _| | |

%p _ _ _ _ _ _ _ _| |

%p 16 22 |_ _ _ _ _ _ _ _ _|

%p ...

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

%Y Cf. A340848 (number of edges).

%Y Cf. A340833 (numer of vertices in the diagram only with parts).

%Y Cf. A317293 (total number of vertices in the unified diagram).

%Y Cf. A000203, A028982, A028983, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340846.

%K nonn,more

%O 1,1

%A _Omar E. Pol_, Jan 24 2021