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%I #37 Feb 03 2024 13:37:45
%S 0,2,6,14,28,50,82,126,184,258,350,462,596,754,938,1150,1392,1666,
%T 1974,2318,2700,3122,3586,4094,4648,5250,5902,6606,7364,8178,9050,
%U 9982,10976,12034,13158,14350,15612,16946,18354,19838,21400,23042,24766,26574,28468
%N Otto Haxel's guess for magic numbers of nuclear shells.
%C O. Haxel gave a construction procedure. The formulas are due to _Wolfdieter Lang_.
%H G. C. Greubel, <a href="/A033547/b033547.txt">Table of n, a(n) for n = 0..1000</a>
%H O. Haxel, <a href="https://doi.org/10.1002/phbl.19940500410">Die Entstehung des Schalenmodells der Atomkerne</a>, Physikalische Blätter, vol. 50, p. 339, 1994.
%H O. Haxel et al., <a href="http://dx.doi.org/10.1103/PhysRev.75.1766.2">On the "Magic Numbers" in Nuclear Structure</a>, Phys. Rev., 75 (1949), 1766.
%H V. Ladma, <a href="http://www.sweb.cz/vladimir_ladma/english/notes/texts/magicn.htm">Magic Numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = n*(n^2 + 5)/3.
%F G.f.: 2*x*(1 - x + x^2)/(1-x)^4.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Apr 05 2015
%F E.g.f.: x*(6 + 3*x + x^2)*exp(x)/3. - _G. C. Greubel_, Oct 12 2019
%F a(n) = A046127(n+1) - 2. - _Jianing Song_, Feb 03 2024
%p A033547:=n->n*(n^2+5)/3: seq(A033547(n), n=0..50); # _Wesley Ivan Hurt_, Apr 05 2015
%t Table[n(n^2+5)/3, {n,0,50}] (* _Harvey P. Dale_, Apr 07 2011 *)
%t LinearRecurrence[{4, -6, 4, -1}, {0, 2, 6, 14}, 50] (* _Vincenzo Librandi_, Apr 06 2015 *)
%o (Magma) [n*(n^2+5)/3 : n in [0..50]]; // _Wesley Ivan Hurt_, Apr 05 2015
%o (PARI) a(n)=n*(n^2+5)/3 \\ _Charles R Greathouse IV_, Jun 25 2017
%o (Sage) [n*(n^2+5)/3 for n in range(50)] # _G. C. Greubel_, Oct 12 2019
%o (GAP) List([0..50], n-> n*(n^2+5)/3); # _G. C. Greubel_, Oct 12 2019
%Y Equals 2*A004006, partial sums of A014206, 2*(partial sums of A000124).
%Y Cf. A018226, A046127.
%K easy,nonn,nice
%O 0,2
%A _Wolfdieter Lang_
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