A093907 Number of elements in the n-th period of the periodic table as predicted by the Aufbau principle.

2, 8, 8, 18, 18, 32, 32, 50, 50, 72, 72, 98, 98, 128, 128, 162, 162, 200, 200, 242, 242, 288, 288, 338, 338, 392, 392, 450, 450, 512, 512, 578, 578, 648, 648, 722, 722, 800, 800, 882, 882, 968, 968, 1058, 1058, 1152, 1152, 1250, 1250, 1352, 1352, 1458, 1458

Offset: 1

Author

Guillermo Restrepo, May 26 2004

Keywords

• nonn
• easy

Comments

Maximum number of electrons in the n-th shell of an atom. - Daniel Forgues, May 09 2011

Examples

			a(1) = 2: hydrogen and helium.
		

References

• Restrepo, G. and Pachon, L., Pythagoras and the Periodic Table, Journal of Chemical Education, submitted, 2004.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Guillermo Restrepo and Leonardo A. Pachon, Mathematical Aspects of the Periodic Law, Foundations of Chemistry volume 9, pages 189-214 (2007); arXiv:math/0611410 [math.GM]
• Wikipedia, Aufbau principle
• M. Winter, WebElements Periodic Table

Crossrefs

Cf. A008794, A116471.

See A269510 for another version.

Programs

GAP

List([1..60],n->(2*n+3+(-1)^n)^2/8); # Muniru A Asiru, Mar 18 2019

Magma

[(2*n+3+(-1)^n)^2/8: n in [1..60]]; // Vincenzo Librandi, Mar 01 2016

Maple

A093907:=n->(2*n+3+(-1)^n)^2/8: seq(A093907(n), n=1..100); # Wesley Ivan Hurt, Jan 10 2017

Mathematica

Table[(2 n + 3 + (-1)^n)^2/8, {n, 60}] (* Bruno Berselli, Jun 03 2014 *)

PARI

{for (n=1, 20000, a=2*floor((n+2)/2)^2; write("b093907.txt", n, " ", a); )} \\ Harry J. Smith, Jun 17 2009

Python

from math import floor
a = lambda n : 2*floor((n+2)/2)**2
for i in range(1, 60):
    print("{}, ".format(a(i)), end="")
# Christoph B. Kassir, Apr 06 2022

Formula

a(n) = 2*floor((n+2)/2)^2. - Leonardo Pachon (leaupaco(AT)yahoo.es), Jul 31 2004

From R. J. Mathar, Oct 04 2009: (Start)

a(n) = 2*A008794(n+2).

G.f.: 2*x*(1 + 3*x - x^3 - 2*x^2 + x^4)/((1 + x)^2*(1 - x)^3). (End)

a(n) = (2*n+3+(-1)^n)^2/8, from Luce ETIENNE. - Bruno Berselli, Jun 03 2014

E.g.f.: ((4 + 3*x + x^2)*cosh(x) + (1 + 5*x + x^2)*sinh(x))/2. - Stefano Spezia, Aug 13 2022

Extensions

More terms added by Harry J. Smith, Jun 17 2009

Definition clarified by Donghwi Park, Mar 01 2016