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A213362
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
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
OFFSET
1,1
COMMENTS
First differs from A212122 at a(12).
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
REFERENCES
I. Talmi, Simple Models of Complex Nuclei, Hardwood Academic Publishers (1993).
FORMULA
a(n) = 2*A213361(n).
EXAMPLE
Illustration of initial terms: two views of a three-dimensional shell model of nucleus.
.
.|-------------------------- j --------------------------|
.| |
.| |---------------------- i ----------------------| |
.| | | |
.| | |------------------ h ------------------| | |
.| | | | | |
.| | | |-------------- g --------------| | | |
.| | | | | | | |
.| | | | |---------- f ----------| | | | |
.| | | | | | | | | |
.| | | | | |------ d ------| | | | | |
.| | | | | | | | | | | |
.| | | | | | |-- p --| | | | | | |
.| | | | | | | | | | | | | |
.| | | | | | | 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 | | | | |
.| | | | | | | | | | | | | | |
.| | | | | | | |1/2| | | | | | |
.| | | | | | | | | | | | |
.| | | | | | |----3/2----| | | | | |
.| | | | | | | | | | |
.| | | | | |--------5/2--------| | | | |
.| | | | | | | | |
.| | | | |------------7/2------------| | | |
.| | | | | | |
.| | | |----------------9/2----------------| | |
.| | | | |
.| | |-------------------11/2--------------------| |
.| | |
.| |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
..........................................................
.
.|-------------------------- j --------------------------|
.* |
.* |---------------------- i ----------------------| |
.| * | *
.| * |------------------ h ------------------| | *
.* | * | * |
.* | * |-------------- f --------------| | * |
.| * | * | * | *
.| * | * |---------- e ----------| | * | *
.* | * | * | * | * |
.* | * | * |------ d ------| | * | * |
.| * | * | * | * | * | *
.| * | * | * |-- p --| | * | * | *
.* | * | * | * | * | * | * |
.* | * | * | * s | * | * | * |
.| * | * | * | * * | * | * | *
.| * | * | * | * * | * | * | *
.* | * | * | * | | * | * | * |
.* | * | * | * |1/2| * | * | * |
.| * | * | * | | * | * | *
.| * | * | * |----3/2----| * | * | *
.* | * | * | | * | * |
.* | * | * |--------5/2--------| * | * |
.| * | * | | * | *
.| * | * |------------7/2------------| * | *
.* | * | | * |
.* | * |----------------9/2----------------| * |
.| * | | *
.| * |-------------------11/2--------------------| *
.* | |
.* |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
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:
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;
...
CROSSREFS
Partial sums give A213364. Other versions are A162630, A212012, A212122, A213372.
Sequence in context: A366839 A278526 A162630 * A212122 A213372 A286601
KEYWORD
nonn,tabf,hard,more
AUTHOR
Omar E. Pol, Jun 23 2012
STATUS
approved