cp's OEIS Frontend

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

Row n lists a permutation of the first n positive integers.

If n is odd then row n lists the first (n+1)/2 odd numbers in decreasing order together with the first (n-1)/2 positive even numbers.

If n is even then row n lists the first n/2 even numbers in decreasing order together with the first n/2 odd numbers.

Row n >= 2, with its floor(n/2) last numbers taken as negative, lists the n different eigenvalues (in decreasing order) of the odd graph O(n). The odd graph O(n) has the (n-1)-subsets of a (2*n-1)-set as vertices, with two (n-1)-subsets adjacent if and only if they are disjoint. For example, O(3) is isomorphic to the Petersen graph. - Miquel A. Fiol, Apr 07 2024

For another version see A213361. First differs from A213361 at a(12).

What defines this sequence? (This appears to be some sort of permutation of A130517 by shifting columns down or upwards in some randomized way.) - R. J. Mathar, Jul 22 2012

First differs from A212124 at a(12).

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

Additional comments from Omar E. Pol, Sep 02 2012: (Start)

Q: What are energy levels?

A: See the link sections of A212122, A213362, A213372. For example, see this link related to A213372: http://www.flickr.com/photos/mitopencourseware/3772864128/in/set-72157621892931990

Q: What defines the order in A212121?

A: The order of A212121 is defined by A212122.

Note that there are at least five versions of the nuclear shell model in the OEIS:

Goeppert-Mayer (1950): A212012, A004736, A212013, A212014.

Goeppert-Mayer, Jensen (1955): A212122, A212121, A212123, A212124.

Talmi (1993): A213362, A213361, A213363, A213364.

Meyerhof: A213372, A213371, A213373, A213374.

For another version: A162630, A130517, A210983, A210984.

Each version is represented by four sequences: the first sequence is the main entry.

(End)

For additional information see A162630.

First differs from A212123 at a(12).

First differs from A212121 at a(14). The illustration of initial terms is essentially the same as the illustration of initial terms of A213372, the main entry for this sequence.

Partial sums of the first seven terms give the "magic numbers" A018226.

Also the partial sums of the first eight terms give the positive first eight terms of A162626 (and possibly more).