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

1,1

Comments

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

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

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

Example

a(1) = 2: hydrogen and helium.

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 *)

Prog

(PARI) {for (n=1, 20000, a=2*floor((n+2)/2)^2; write("b093907.txt", n, " ", a); )} \\ Harry J. Smith, Jun 17 2009
(Magma) [(2*n+3+(-1)^n)^2/8: n in [1..60]]; // Vincenzo Librandi, Mar 01 2016
(GAP)  List([1..60], n->(2*n+3+(-1)^n)^2/8); # Muniru A Asiru, Mar 18 2019
(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

Crossrefs

Cf. A008794, A116471.

See A269510 for another version.

Sequence in context: A138230 A019240 A269510 * A116471 A364294 A146749

Adjacent sequences: A093904 A093905 A093906 * A093908 A093909 A093910

Keyword

nonn,easy

Author

Guillermo Restrepo, May 26 2004

Extensions

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

Definition clarified by Donghwi Park, Mar 01 2016

Status

approved