This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212012 #34 Mar 14 2025 17:11:30 %S A212012 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 A212012 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 A212012 18,16,14,12,10,8,6,4,2,24,22,20,18,16,14,12,10,8,6,4,2 %N 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. %C A212012 Also triangle read by rows in which row i lists the first i positive even numbers in decreasing order. %C A212012 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 A212012 Paolo Xausa, <a href="/A212012/b212012.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened). %H A212012 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 A212012 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 A212012 a(n) = 2*A004736(n). %e A212012 Illustration of initial terms: one of the views of a three-dimensional shell model of nucleus. %e A212012 . %e A212012 .|-------------------------- j --------------------------| %e A212012 .| | %e A212012 .| |---------------------- i ----------------------| | %e A212012 .| | | | %e A212012 .| | |------------------ h ------------------| | | %e A212012 .| | | | | | %e A212012 .| | | |-------------- g --------------| | | | %e A212012 .| | | | | | | | %e A212012 .| | | | |---------- f ----------| | | | | %e A212012 .| | | | | | | | | | %e A212012 .| | | | | |------ d ------| | | | | | %e A212012 .| | | | | | | | | | | | %e A212012 .| | | | | | |-- p --| | | | | | | %e A212012 .| | | | | | | | | | | | | | %e A212012 .| | | | | | | s | | | | | | | %e A212012 .| | | | | | | | | | | | | | | %e A212012 .| | | | | | | 2 | | | | | | | %e A212012 .| | | | | | 4 | | | | | | | | %e A212012 .| | | | | | | | 2 | | | | | | %e A212012 .| | | | | 6 | | | | | | | | | %e A212012 .| | | | | | | | | 4 | | | | | %e A212012 .| | | | | | | 2 | | | | | | | %e A212012 .| | | | 8 | | | | | | | | | | %e A212012 .| | | | | | | | | | 6 | | | | %e A212012 .| | | | | | 4 | | | | | | | | %e A212012 .| | | | | | | | 2 | | | | | | %e A212012 .| | | 10 | | | | | | | | | | | %e A212012 .| | | | | | | | | | | 8 | | | %e A212012 .| | | | | 6 | | | | | | | | | %e A212012 .| | | | | | | | | 4 | | | | | %e A212012 .| | | | | | | 2 | | | | | | | %e A212012 .| | 12 | | | | | | | | | | | | %e A212012 .| | | | | | | | | | | | 10 | | %e A212012 .| | | | 8 | | | | | | | | | | %e A212012 .| | | | | | | | | | 6 | | | | %e A212012 .| | | | | | 4 | | | | | | | | %e A212012 .| | | | | | | | 2 | | | | | | %e A212012 .| 14 | | | | | | | | | | | | | %e A212012 .| | | | | | | | | | | | | 12 | %e A212012 .| | | 10 | | | | | | | | | | | %e A212012 .| | | | | | | | | | | 8 | | | %e A212012 .| | | | | 6 | | | | | | | | | %e A212012 .| | | | | | | | | 4 | | | | | %e A212012 .| | | | | | | 2 | | | | | | | %e A212012 .| | | | | | | | | | | | | | | %e A212012 .| | | | | | | | | | | | | | | %e A212012 .| | | | | | | |1/2| | | | | | | %e A212012 .| | | | | | | | | | | | | %e A212012 .| | | | | | |----3/2----| | | | | | %e A212012 .| | | | | | | | | | | %e A212012 .| | | | | |--------5/2--------| | | | | %e A212012 .| | | | | | | | | %e A212012 .| | | | |------------7/2------------| | | | %e A212012 .| | | | | | | %e A212012 .| | | |----------------9/2----------------| | | %e A212012 .| | | | | %e A212012 .| | |-------------------11/2--------------------| | %e A212012 .| | | %e A212012 .| |-----------------------13/2------------------------| %e A212012 .| %e A212012 .|---------------------------15/2------------------------- %e A212012 . %e A212012 For another view of the model see the example section of A212122, second part. %e A212012 Example 1. Triangle begins: %e A212012 2; %e A212012 4, 2; %e A212012 6, 4, 2; %e A212012 8, 6, 4, 2; %e A212012 10, 8, 6, 4, 2; %e A212012 12, 10, 8, 6, 4, 2; %e A212012 14, 12, 10, 8, 6, 4, 2; %e A212012 16, 14, 12, 10, 8, 6, 4, 2; %e A212012 ... %e A212012 Column 1 gives positive terms of A005843. Right border give positive terms of A007395. Row sums give A002378. %e A212012 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 A212012 2; %e A212012 4, 2; %e A212012 6, 4, 2; %e A212012 8; %e A212012 6, 4, 2, 10; %e A212012 8, 6, 4, 2, 12; %e A212012 10, 8, 6, 4, 2, 14; %e A212012 12, 10, 8, 6, 4, 2, 16; %e A212012 14, 12, 10, 8, 6, 4, 2, 18; %t A212012 2*Range[Range[15], 1, -1] (* _Paolo Xausa_, Mar 14 2025 *) %Y A212012 Partial sums give A212014. Other versions are A162630, A212122, A213362, A213372. %Y A212012 Cf. A002378, A004736, A005843, A007395, A212013, A212014. %K A212012 nonn,tabl,easy %O A212012 1,1 %A A212012 _Omar E. Pol_, Jul 15 2012