A210980 - OEIS

A210980

Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

LINKS

Table of n, a(n) for n=0..38.

FORMULA

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

EXAMPLE

For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:

. | Partitions

. A194805(7) = 25 A066186(7) = 105 | of 7

. |

. 1 * * * * * * 1 | 7

. 2 * * * 1 * * 2 | 4+3

. 2 * * * * 1 * 2 | 5+2

. 3 * * 1 * 2 * 3 | 3+2+2

. 1 2 * * * * * 1 2 | 6+1

. 2 3 * * 1 * * 2 3 | 3+3+1

. 2 3 * * * 1 * 2 3 | 4+2+1

. 3 4 * 1 * 2 * 3 4 | 2+2+2+1

. 3 1 * * * * 1 2 3 | 5+1+1

. 4 2 * * 1 * 2 3 4 | 3+2+1+1

. 1 4 * * * 1 2 3 4 | 4+1+1+1

. 2 5 * 1 * 2 3 4 5 | 2+2+1+1+1

. 5 1 * * 1 2 3 4 5 | 3+1+1+1+1

. 1 6 * 1 2 3 4 5 6 | 2+1+1+1+1+1

. 7 1 2 3 4 5 6 7 | 1+1+1+1+1+1+1

. ---------------------------------- |

. |

. * * * * 1 * * * * |

. * * * 1 2 * * * * |

. * 1 * * 2 1 * * * |

. * * 1 2 2 * * 1 * |

. * * * * 2 2 1 * * |

. 1 2 2 3 2 * * * * |

. 2 3 2 2 1 |

. |

. A194804(7) = 59 |

Note that, as a variant, in this case each part is labeled with its position in the partition.

The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

CROSSREFS

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

Sequence in context: A038347 A338089 A294365 * A207380 A268348 A382656

Adjacent sequences: A210977 A210978 A210979 * A210981 A210982 A210983

KEYWORD

nonn

AUTHOR

Omar E. Pol, Apr 21 2012

STATUS

approved