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
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
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