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Triangle read by rows in which row n lists the number of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.
8

%I #27 Apr 13 2015 16:40:39

%S 2,4,2,6,4,2,8,6,4,2,10,8,6,4,2,12,10,8,6,4,2,14,12,10,8,6,4,2,16,14,

%T 12,10,8,6,4,2,18,16,14,12,10,8,6,4,2,20,18,16,14,12,10,8,6,4,2,22,20,

%U 18,16,14,12,10,8,6,4,2,24,22,20,18,16,14,12

%N Triangle read by rows in which row n lists the number of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

%C Also triangle read by rows in which row i lists the first i positive even numbers in decreasing order.

%C The list of the spin-orbit coupling of this version of the nuclear shell model starts: 1s_(1/2), 1p_(3/2), 1p_(1/2), 1d_(5/2), 1d_(3/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 3,... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 4,... Other sequences that arise from this sequence are both A212013 and A212014. - _Omar E. Pol_, Sep 02 2012

%H M. Goeppert Mayer, <a href="http://dx.doi.org/10.1103/PhysRev.78.16">Nuclear configurations in the spin-orbit coupling model. I. Empirical evidence</a>, Phys. Rev. 78, 16-21 (1950).

%H M. Goeppert Mayer, <a href="http://dx.doi.org/10.1103/PhysRev.78.22">Nuclear configurations in the spin-orbit coupling model. II. Theoretical considerations</a>, Phys. Rev. 78: 22 (1950).

%F a(n) = 2*A004736(n).

%e Illustration of initial terms: one of the views of a three-dimensional shell model of nucleus.

%e .

%e .|-------------------------- j --------------------------|

%e .| |

%e .| |---------------------- i ----------------------| |

%e .| | | |

%e .| | |------------------ h ------------------| | |

%e .| | | | | |

%e .| | | |-------------- g --------------| | | |

%e .| | | | | | | |

%e .| | | | |---------- f ----------| | | | |

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

%e .| | | | | |------ d ------| | | | | |

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

%e .| | | | | | |-- p --| | | | | | |

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

%e .| | | | | | | s | | | | | | |

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

%e .| | | | | | | 2 | | | | | | |

%e .| | | | | | 4 | | | | | | | |

%e .| | | | | | | | 2 | | | | | |

%e .| | | | | 6 | | | | | | | | |

%e .| | | | | | | | | 4 | | | | |

%e .| | | | | | | 2 | | | | | | |

%e .| | | | 8 | | | | | | | | | |

%e .| | | | | | | | | | 6 | | | |

%e .| | | | | | 4 | | | | | | | |

%e .| | | | | | | | 2 | | | | | |

%e .| | | 10 | | | | | | | | | | |

%e .| | | | | | | | | | | 8 | | |

%e .| | | | | 6 | | | | | | | | |

%e .| | | | | | | | | 4 | | | | |

%e .| | | | | | | 2 | | | | | | |

%e .| | 12 | | | | | | | | | | | |

%e .| | | | | | | | | | | | 10 | |

%e .| | | | 8 | | | | | | | | | |

%e .| | | | | | | | | | 6 | | | |

%e .| | | | | | 4 | | | | | | | |

%e .| | | | | | | | 2 | | | | | |

%e .| 14 | | | | | | | | | | | | |

%e .| | | | | | | | | | | | | 12 |

%e .| | | 10 | | | | | | | | | | |

%e .| | | | | | | | | | | 8 | | |

%e .| | | | | 6 | | | | | | | | |

%e .| | | | | | | | | 4 | | | | |

%e .| | | | | | | 2 | | | | | | |

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

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

%e .| | | | | | | |1/2| | | | | | |

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

%e .| | | | | | |----3/2----| | | | | |

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

%e .| | | | | |--------5/2--------| | | | |

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

%e .| | | | |------------7/2------------| | | |

%e .| | | | | | |

%e .| | | |----------------9/2----------------| | |

%e .| | | | |

%e .| | |-------------------11/2--------------------| |

%e .| | |

%e .| |-----------------------13/2------------------------|

%e .|

%e .|---------------------------15/2-------------------------

%e .

%e For another view of the model see the example section of A212122, second part.

%e Example 1. Triangle begins:

%e 2;

%e 4, 2;

%e 6, 4, 2;

%e 8, 6, 4, 2;

%e 10, 8, 6, 4, 2;

%e 12, 10, 8, 6, 4, 2;

%e 14, 12, 10, 8, 6, 4, 2;

%e 16, 14, 12, 10, 8, 6, 4, 2;

%e ...

%e Column 1 gives positive terms of A005843. Right border give positive terms of A007395. Row sums give A002378.

%e Example 2. Written as an irregular triangle in which row j represents the j-th shell of nucleus. Note that row 4 has only one term. Triangle begins:

%e 2;

%e 4, 2;

%e 6, 4, 2;

%e 8;

%e 6, 4, 2, 10;

%e 8, 6, 4, 2, 12;

%e 10, 8, 6, 4, 2, 14;

%e 12, 10, 8, 6, 4, 2, 16;

%e 14, 12, 10, 8, 6, 4, 2, 18;

%Y Partial sums give A212014. Other versions are A162630, A212122, A213362, A213372.

%Y Cf. A002378, A004736, A005843, A007395, A212013, A212014.

%K nonn,tabf,easy

%O 1,1

%A _Omar E. Pol_, Jul 15 2012