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Otto Haxel's guess for magic numbers of nuclear shells.
15

%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_