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A162630 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.

%I A162630 #24 Nov 18 2018 09:05:34
%S A162630 2,4,2,6,2,4,8,4,2,6,10,6,2,4,8,12,8,4,2,6,10,14,10,6,2,4,8,12,16,12,
%T A162630 8,4,2,6,10,14,18,14,10,6,2,4,8,12,16,20,16,12,8,4,2,6,10,14,18,22,18,
%U A162630 14,10,6,2,4,8,12,16,20,24,20,16,12,8,4,2
%N A162630 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 A162630 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), 2p_(1/2), etc. The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 1, ... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 2, 4, 8, 4, 2, ... Other sequences that arise from this sequence are A A130517, A210983, A210984. - _Omar E. Pol_, Sep 02 2012
%H A162630 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a>, 2012, arXiv:1212.2732 [math.CO], 2012.
%F A162630 a(n) = 2*A130517(n).
%F A162630 From _Boris Putievskiy_, Jan 16 2013: (Start)
%F A162630 a(n) = 2*(|2*A000027(n) - A003056(n)^2 - 2*A003056(n) - 3| + floor((2*A000027(n) - A003056(n)^2 - A003056(n))/(A003056(n) + 3))).
%F A162630 a(n) = 2*(|2*n - t*t - 2*t - 3| + floor((2*n - t*t - t)/(t+3))) where t = floor((-1 + sqrt(8*n-7))/2). (End)
%e A162630 A geometric shell model of the atomic nucleus:
%e A162630    +---------------------- i ----------------------+
%e A162630    |   +------------------ h ------------------+   |
%e A162630    |   |   +-------------- g --------------+   |   |
%e A162630    |   |   |   +---------- f ----------+   |   |   |
%e A162630    |   |   |   |   +------ d ------+   |   |   |   |
%e A162630    |   |   |   |   |   +-- p --+   |   |   |   |   |
%e A162630    |   |   |   |   |   |   s   |   |   |   |   |   |
%e A162630    |   |   |   |   |   |   |   |   |   |   |   |   |
%e A162630    |   |   |   |   |   |       |   |   |   |   |   |
%e A162630    |   |   |   |   |       2       |   |   |   |   |
%e A162630    |   |   |   |       4       2       |   |   |   |
%e A162630    |   |   |       6       2       4       |   |   |
%e A162630    |   |       8       4       2       6       |   |
%e A162630    |      10       6       2       4       8       |
%e A162630       12       8       4       2       6      10
%e A162630   14      10       6       2       4       8      12
%e A162630    |   |   |   |   |   |   |   |   |   |   |   |   |
%e A162630    |   |   |   |   |   |   +1/2+   |   |   |   |   |
%e A162630    |   |   |   |   |   +--- 3/2 ---+   |   |   |   |
%e A162630    |   |   |   |   +------- 5/2 -------+   |   |   |
%e A162630    |   |   |   +----------- 7/2 -----------+   |   |
%e A162630    |   |   +--------------- 9/2 ---------------+   |
%e A162630    |   +------------------ 11/2 -------------------+
%e A162630    +---------------------- 13/2 -----------------------
%t A162630 t[n_, 1] := n; t[n_, n_] := n-1;
%t A162630 t[n_, k_] := Abs[2k - n - If[2k <= n+1, 2, 1]];
%t A162630 2 Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 17 2018 *)
%Y A162630 Cf. A018226, A130517, A130556, A130598, A130602, A162626.
%Y A162630 Other versions are A212012, A212122, A213362, A213372
%K A162630 nonn,tabl
%O A162630 1,1
%A A162630 _Omar E. Pol_, Jul 10 2009
%E A162630 Corrected by _Omar E. Pol_, Jul 13 2009
%E A162630 More terms from _Omar E. Pol_, Jul 14 2012
%E A162630 New name from _Omar E. Pol_, Sep 02 2012