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%I #53 Apr 15 2024 17:52:35
%S 2,10,18,36,54,86,118,168,218,290,362,460,558,686,814,976,1138,1338,
%T 1538,1780,2022,2310,2598,2936,3274,3666,4058,4508,4958,5470,5982,
%U 6560,7138,7786,8434,9156,9878,10678,11478,12360,13242,14210,15178
%N Magic numbers: atoms with full shells containing any of these numbers of electrons are considered electronically stable.
%C Atomic numbers of noble elements in the periodic table.
%C Partial sums of A093907. - _Lekraj Beedassy_, Mar 24 2006
%C Comment from Don N. Page (don(AT)phys.ualberta.ca), Dec 12 2006: (Start)
%C "Relativistic corrections and instabilities to pair creation of electrons and positrons would occur even if one could have stable nuclei of arbitrarily many protons Z for the fixed value of the fine structure constant alpha ~ 1/137 in our universe.
%C "However, if one considered an imaginary universe with arbitrarily tiny alpha and a fixed point source of charge Z, one could have stable neutral atoms of Z nonrelativistic electrons of mass m for any Z, so long as one takes the limit Z alpha -> 0 by taking alpha -> 0 after fixing Z.
%C "One could then define noble elements to be given by the integer values of Z such that the ionization energy, in units of m c^2 alpha^2, of any such atom in its ground state with larger Z is less than that of the noble element (which appears to be the case for all the noble elements with the actual nonzero value of alpha).
%C "This sequence of idealized nonrelativistic noble elements with Z electrons would give an infinite sequence of integers Z, which may or may not be the same as that given by the explicit formula listed for the present sequence. It would likely be a difficult mathematical problem to calculate this infinite sequence." (End)
%H Vincenzo Librandi, <a href="/A018227/b018227.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Bjornholm, <a href="https://doi.org/10.1080/00107519008213781">Clusters, condensed matter in embryonic form</a>, Contemp. Phys. 31 1990 pp. 309-324 (p. 312).
%H Encyclopedia Britannica, <a href="https://www.britannica.com/science/magic-number-atomic-structure">magic number</a>
%H D. Weise, <a href="http://www.metaelmelet.hu/egyelm/konferencia_2008/en/weise.ppt">The Pythagorean Approach to Problems of Periodicity in Science</a>
%H D. Weise, <a href="http://www.mi.sanu.ac.rs/vismath/visbook/weise2">Pythagorean Approach To Problems Of Periodicity In Fermionic System</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F a(n) = a(n-1) + ((2*n + 3 + (-1)^n)^2)/8; a(n) = (2*n^3 + 12*n^2 + 25*n - 6 + (-1)^n*(3*n + 6))/12. - _Warut Roonguthai_, Jun 20 2005
%F a(n) = n{(n+3)^2 + 5}/6 for even n, a(n) = n{(n+3)^2 + 2)/6 - 1 [or C(n+3,3) - 2, i.e., A000292(n) - 2] for odd n. - _Lekraj Beedassy_, Feb 02 2006
%F Partial sums of A116471. - _Lekraj Beedassy_, Mar 31 2006
%F From _Daniel Forgues_, May 02 2011: (Start)
%F a(n) = n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1, n >= 1.
%F a(n) = (n+1)*(n+2)*(n+3)/6 + (n+2)*(1+(-1)^n)/4 - 2, n >= 1.
%F a(n) = T_{n+1} + (n+2)*(1+(-1)^n)/4 - 2, n >= 1, where T_n is the n-th tetrahedral number.
%F G.f.: 2*x*(1 + 3*x - 2*x^2 - x^3 + x^4)/((1 - x)^4*(1 + x)^2). (End)
%o (Magma) [n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1: n in [1..50]]; // _Vincenzo Librandi_, May 03 2011
%o (PARI) a(n)=(2*n^3+12*n^2+25*n-6+(-1)^n*(3*n+6))/12 \\ _Charles R Greathouse IV_, Oct 18 2022
%Y Cf. A018226 for the magic numbers for nucleons (protons and neutrons).
%K nonn,easy,changed
%O 1,1
%A John Raithel (raithel(AT)rahul.net)
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