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%I #59 Nov 10 2023 17:45:44
%S 2,8,8,18,18,32,32,50,50,72,72,98,98,128,128,162,162,200,200,242,242,
%T 288,288,338,338,392,392,450,450,512,512,578,578,648,648,722,722,800,
%U 800,882,882,968,968,1058,1058,1152,1152,1250,1250,1352,1352,1458,1458
%N Number of elements in the n-th period of the periodic table as predicted by the Aufbau principle.
%C Maximum number of electrons in the n-th shell of an atom. - _Daniel Forgues_, May 09 2011
%D Restrepo, G. and Pachon, L., Pythagoras and the Periodic Table, Journal of Chemical Education, submitted, 2004.
%H Harry J. Smith, <a href="/A093907/b093907.txt">Table of n, a(n) for n = 1..20000</a>
%H Guillermo Restrepo and Leonardo A. Pachon, <a href="https://doi.org/10.1007/s10698-006-9026-6">Mathematical Aspects of the Periodic Law</a>, Foundations of Chemistry volume 9, pages 189-214 (2007); <a href="https://arxiv.org/abs/math/0611410">arXiv:math/0611410 [math.GM]</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Aufbau_principle">Aufbau principle</a>
%H M. Winter, <a href="http://www.webelements.com/webelements/scholar/index.html">WebElements Periodic Table</a>
%F a(n) = 2*floor((n+2)/2)^2. - Leonardo Pachon (leaupaco(AT)yahoo.es), Jul 31 2004
%F From _R. J. Mathar_, Oct 04 2009: (Start)
%F a(n) = 2*A008794(n+2).
%F G.f.: 2*x*(1 + 3*x - x^3 - 2*x^2 + x^4)/((1 + x)^2*(1 - x)^3). (End)
%F a(n) = (2*n+3+(-1)^n)^2/8, from _Luce ETIENNE_. - _Bruno Berselli_, Jun 03 2014
%F E.g.f.: ((4 + 3*x + x^2)*cosh(x) + (1 + 5*x + x^2)*sinh(x))/2. - _Stefano Spezia_, Aug 13 2022
%e a(1) = 2: hydrogen and helium.
%p A093907:=n->(2*n+3+(-1)^n)^2/8: seq(A093907(n), n=1..100); # _Wesley Ivan Hurt_, Jan 10 2017
%t Table[(2 n + 3 + (-1)^n)^2/8, {n, 60}] (* _Bruno Berselli_, Jun 03 2014 *)
%o (PARI) {for (n=1, 20000, a=2*floor((n+2)/2)^2; write("b093907.txt", n, " ", a); )} \\ _Harry J. Smith_, Jun 17 2009
%o (Magma) [(2*n+3+(-1)^n)^2/8: n in [1..60]]; // _Vincenzo Librandi_, Mar 01 2016
%o (GAP) List([1..60],n->(2*n+3+(-1)^n)^2/8); # _Muniru A Asiru_, Mar 18 2019
%o (Python)
%o from math import floor
%o a = lambda n : 2*floor((n+2)/2)**2
%o for i in range(1, 60):
%o print("{}, ".format(a(i)), end="")
%o # _Christoph B. Kassir_, Apr 06 2022
%Y Cf. A008794, A116471.
%Y See A269510 for another version.
%K nonn,easy
%O 1,1
%A Guillermo Restrepo, May 26 2004
%E More terms added by _Harry J. Smith_, Jun 17 2009
%E Definition clarified by _Donghwi Park_, Mar 01 2016
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