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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 #29 Jan 23 2023 09:10:24

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

%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 First differs from A212122 at a(12).

%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), 2s_(1/2), 1d_(3/2), 1f_(7/2), 2p_(3/2), 1f_(5/2), 2p_(1/2), 1g_(9/2), 2d_(5/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 5, 1, 9, 5,... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 2, 4, 8, 4, 6, 2, 10, 6,... Note that in the Talmi's table there is a typo: instead 2f_(1/2) should be 2f_(7/2), see references, page 6. Other sequences that arise from this sequence are A213361, A213363, A213364. - _Omar E. Pol_, Sep 02 2012

%D I. Talmi, Simple Models of Complex Nuclei, Hardwood Academic Publishers (1993).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Nuclear_shell_model">Nuclear shell model</a>

%F a(n) = 2*A213361(n).

%e Illustration of initial terms: two 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 .| | | | | | | 2 | | | | | | |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%e 16 | | | | | | | | | | | | | |

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

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

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

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

%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 ..........................................................

%e .

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

%e .* |

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

%e .| * | *

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

%e .* | * | * |

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

%e .| * | * | * | *

%e .| * | * |---------- e ----------| | * | *

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

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

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

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

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

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

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

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

%e 2;

%e 4, 2;

%e 6, 2, 4;

%e 8;

%e 4, 6, 2, 10;

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

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

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

%e ...

%Y Partial sums give A213364. Other versions are A162630, A212012, A212122, A213372.

%Y Cf. A018226, A130517, A213361, A213363.

%K nonn,tabf,hard,more

%O 1,1

%A _Omar E. Pol_, Jun 23 2012