A212012 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.

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

Offset: 1

Author

Omar E. Pol, Jul 15 2012

Keywords

• nonn
• tabl
• easy

Comments

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

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

Examples

			Illustration of initial terms: one of the views of a three-dimensional shell model of nucleus.
.
.|-------------------------- j --------------------------|
.|                                                       |
.|   |---------------------- i ----------------------|   |
.|   |                                               |   |
.|   |   |------------------ h ------------------|   |   |
.|   |   |                                       |   |   |
.|   |   |   |-------------- g --------------|   |   |   |
.|   |   |   |                               |   |   |   |
.|   |   |   |   |---------- f ----------|   |   |   |   |
.|   |   |   |   |                       |   |   |   |   |
.|   |   |   |   |   |------ d ------|   |   |   |   |   |
.|   |   |   |   |   |               |   |   |   |   |   |
.|   |   |   |   |   |   |-- p --|   |   |   |   |   |   |
.|   |   |   |   |   |   |       |   |   |   |   |   |   |
.|   |   |   |   |   |   |   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   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |1/2|   |   |   |   |   |   |
.|   |   |   |   |   |   |           |   |   |   |   |   |
.|   |   |   |   |   |   |----3/2----|   |   |   |   |   |
.|   |   |   |   |   |                   |   |   |   |   |
.|   |   |   |   |   |--------5/2--------|   |   |   |   |
.|   |   |   |   |                           |   |   |   |
.|   |   |   |   |------------7/2------------|   |   |   |
.|   |   |   |                                   |   |   |
.|   |   |   |----------------9/2----------------|   |   |
.|   |   |                                           |   |
.|   |   |-------------------11/2--------------------|   |
.|   |                                                   |
.|   |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
For another view of the model see the example section of A212122, second part.
Example 1. Triangle begins:
  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, 12, 10,  8,  6, 4, 2;
...
Column 1 gives positive terms of A005843. Right border give positive terms of A007395. Row sums give A002378.
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:
  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, 12, 10,  8,  6,  4,  2, 18;
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).
• M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. I. Empirical evidence, Phys. Rev. 78, 16-21 (1950).
• M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. II. Theoretical considerations, Phys. Rev. 78: 22 (1950).

Crossrefs

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

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

Programs

Mathematica

2*Range[Range[15], 1, -1] (* Paolo Xausa, Mar 14 2025 *)

Formula

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