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

1,1

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

Links

Table of n, a(n) for n=1..73.

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

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)

Example

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 -----------------------

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 *)

Crossrefs

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

Other versions are A212012, A212122, A213362, A213372

Sequence in context: A010241 A366839 A278526 * A213362 A212122 A213372

Adjacent sequences: A162627 A162628 A162629 * A162631 A162632 A162633

Keyword

nonn,tabl

Author

Omar E. Pol, Jul 10 2009

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

Status

approved