A167268 Janet's sequence: Number of elements for each successively filled electronic subshell of an atom.

2, 2, 6, 2, 6, 2, 10, 6, 2, 10, 6, 2, 14, 10, 6, 2, 14, 10, 6, 2, 18, 14, 10, 6, 2, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, 2, 30, 26, 22, 18, 14, 10, 6, 2, 30, 26, 22, 18, 14, 10, 6

Offset: 1

Author

Paul Curtz, Oct 31 2009

Keywords

• nonn
• easy

Comments

The s-block is placed at

* the left, except row 1 (for Mendeleev-Moseley-Seaborg periodic table)

* the right (for Janet periodic table)

The number of elements in each block of Janet's periodic table are

s-block (1s to 8s): 8*2 = 16 elements;

p-block (2p to 7p): 6*6 = 36 elements;

d-block (3d to 6d): 4*10 = 40 elements;

f-block (4f to 5f): 2*14 = 28 elements.

The atomic numbers of elements in each block of Janet's periodic table are

s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88, 119,120 (cf. A160914)

p-block: 5..10, 13..18, 31..36, 49..54, 81..86, 113..118 (cf. A138469)

d-block: 21..30, 39..48, 71..80, 103..112 (cf. A199934)

f-block: 57..70, 89..102 (cf. A217923).

The number of elements for each successive filled subshell are

s-block (1s): a(0)=2 terms (for H and He, i.e., 1 and 2);

s-block (2s): a(1)=2 terms (for Li and Be, i.e., 3 and 4);

p-block (2p): a(2)=6 terms (for B,C,N,O,F,Ne, i.e., 5 to 10);

s-block (3s): a(3)=2 terms (for Na,Mg, i.e., 11 and 12);

p-block (3p): a(4)=6 terms (for Al,Si,P,S,Cl,Ar, i.e., 13 to 18);

...

Reference, 2 leaflet 2, with Janet form (5). Extended.

From Daniel Forgues, May 09 2011: (Start)

Janet also envisaged an 'element zero' ('neutronium'?) - whose 'atom' would consist of two neutrons (and thus zero electrons, which would give a(0) = 0) and he speculated that this would be the link to a mirror-image table of elements with negative atomic numbers - in effect anti-matter (which would give a(-n) = -a(n), since positrons are negated electrons).

Maximum number of electrons for successive subshells of each shell of an atom, in the building up order (per aufbau principle and Madelung's rule).

Every term is twice an odd number since each filled subshell in block l has m going from -l to + l (2l+1 values,) each with 2 electrons (spin +1/2 and spin -1/2).

Blocks:

l=0: s (2 electrons) (first subshell of a shell, new period of Mendeleev's table)

l=1: p (6 electrons) (except for first shell, last subshell of a shell)

l=2: d (10 electrons)

l=3: f (14 electrons)

l=4: g (18 electrons)

...

l=l: ... (2*(2l+1) electrons)

The first subshell of the k-th shell has l = 0, k >= 1.

The second subshell of the k-th shell has l = floor(k/2), k >= 2.

The following subshells of the k-th shell have l decrementing down to 1, k >= 2.

(End)

Concatenation of finite arithmetic sequences, each followed by 2: { }, 2, { }, 2, {6}, 2, {6}, 2, {10, 6}, 2, {10, 6}, 2, {14, 10, 6}, 2, {14, 10, 6}, 2, {18, 14, 10, 6}, 2, {18, 14, 10, 6}, 2, ... - Daniel Forgues, May 15 2011

Named after the French mathematician Charles Janet (1849-1932). - Amiram Eldar, Jun 22 2021

References

• Charles Janet, Considérations sur la structure du noyau de l'atome, N 5, Décembre 1929, Beauvais, 2+45 pages, 4 leaflets.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..9899
• The Chemogenesis web book, Quantum Numbers to Periodic Tables.
• Szymon Łukaszyk and Wawrzyniec Bieniawski, Assembly Theory of Binary Messages (How to Assemble a Black Hole and Use it to Assemble New Binary Information?), Preprints (2024).

Crossrefs

Cf. A016825, A160914, A138469, A199934, A216607.

Programs

Mathematica

row[n_] := Range[2, 4*Floor[n/2] + 2, 4]; Flatten[ Table[ row[n] // Reverse, {n, 0, 15}]] (* Jean-François Alcover, Oct 02 2012 *)

Python

from mpmath.libmp import sqrtrem
def a(n):
    s, r = sqrtrem(n)
    return 4 * (-n % (s + (r>s))) + 2
# Christoph B. Kassir, Apr 07 2022

Formula

a(n) = 4*((-n) mod round(sqrt(n))) + 2. - Jon E. Schoenfield, Sep 08 2013

a(n) = 4*A216607(n) + 2. - Szymon Lukaszyk, Oct 27 2023

Extensions

Edited by Daniel Forgues, May 09 2011