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.

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

Offset: 1

Author

Omar E. Pol, Jul 10 2009

Keywords

• nonn
• tabl

Comments

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

Examples

			A geometric shell model of the atomic nucleus:
   +---------------------- i ----------------------+
   |   +------------------ h ------------------+   |
   |   |   +-------------- g --------------+   |   |
   |   |   |   +---------- f ----------+   |   |   |
   |   |   |   |   +------ d ------+   |   |   |   |
   |   |   |   |   |   +-- p --+   |   |   |   |   |
   |   |   |   |   |   |   s   |   |   |   |   |   |
   |   |   |   |   |   |   |   |   |   |   |   |   |
   |   |   |   |   |   |       |   |   |   |   |   |
   |   |   |   |   |       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
   |   |   |   |   |   |   |   |   |   |   |   |   |
   |   |   |   |   |   |   +1/2+   |   |   |   |   |
   |   |   |   |   |   +--- 3/2 ---+   |   |   |   |
   |   |   |   |   +------- 5/2 -------+   |   |   |
   |   |   |   +----------- 7/2 -----------+   |   |
   |   |   +--------------- 9/2 ---------------+   |
   |   +------------------ 11/2 -------------------+
   +---------------------- 13/2 -----------------------
		

Links

• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A018226, A130517, A130556, A130598, A130602, A162626.

Other versions are A212012, A212122, A213362, A213372

Programs

Mathematica

t[n_, 1] := n; t[n_, n_] := n-1;
t[n_, k_] := Abs[2k - n - If[2k <= n+1, 2, 1]];
2 Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

Formula

a(n) = 2*A130517(n).

From Boris Putievskiy, Jan 16 2013: (Start)

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))).

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)

Extensions

Corrected by Omar E. Pol, Jul 13 2009

More terms from Omar E. Pol, Jul 14 2012

New name from Omar E. Pol, Sep 02 2012