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

%I A167268 #96 Feb 01 2024 04:19:12
%S A167268 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,
%T A167268 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,
%U A167268 10,6,2,30,26,22,18,14,10,6,2,30,26,22,18,14,10,6
%N A167268 Janet's sequence: Number of elements for each successively filled electronic subshell of an atom.
%C A167268 The s-block is placed at
%C A167268 * the left, except row 1 (for Mendeleev-Moseley-Seaborg periodic table)
%C A167268 * the right (for Janet periodic table)
%C A167268 The number of elements in each block of Janet's periodic table are
%C A167268   s-block (1s to 8s): 8*2  = 16 elements;
%C A167268   p-block (2p to 7p): 6*6  = 36 elements;
%C A167268   d-block (3d to 6d): 4*10 = 40 elements;
%C A167268   f-block (4f to 5f): 2*14 = 28 elements.
%C A167268 The atomic numbers of elements in each block of Janet's periodic table are
%C A167268 s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88, 119,120 (cf. A160914)
%C A167268 p-block: 5..10, 13..18, 31..36, 49..54, 81..86, 113..118 (cf. A138469)
%C A167268 d-block: 21..30, 39..48, 71..80, 103..112 (cf. A199934)
%C A167268 f-block: 57..70, 89..102 (cf. A217923).
%C A167268 The number of elements for each successive filled subshell are
%C A167268   s-block (1s): a(0)=2 terms (for H and He, i.e., 1 and 2);
%C A167268   s-block (2s): a(1)=2 terms (for Li and Be, i.e., 3 and 4);
%C A167268   p-block (2p): a(2)=6 terms (for B,C,N,O,F,Ne, i.e., 5 to 10);
%C A167268   s-block (3s): a(3)=2 terms (for Na,Mg, i.e., 11 and 12);
%C A167268   p-block (3p): a(4)=6 terms (for Al,Si,P,S,Cl,Ar, i.e., 13 to 18);
%C A167268   ...
%C A167268 Reference, 2 leaflet 2, with Janet form (5). Extended.
%C A167268 From _Daniel Forgues_, May 09 2011: (Start)
%C A167268 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).
%C A167268 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).
%C A167268 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).
%C A167268 Blocks:
%C A167268   l=0: s (2 electrons) (first subshell of a shell, new period of Mendeleev's table)
%C A167268   l=1: p (6 electrons) (except for first shell, last subshell of a shell)
%C A167268   l=2: d (10 electrons)
%C A167268   l=3: f (14 electrons)
%C A167268   l=4: g (18 electrons)
%C A167268   ...
%C A167268   l=l: ... (2*(2l+1) electrons)
%C A167268 The first subshell of the k-th shell has l = 0, k >= 1.
%C A167268 The second subshell of the k-th shell has l = floor(k/2), k >= 2.
%C A167268 The following subshells of the k-th shell have l decrementing down to 1, k >= 2.
%C A167268 (End)
%C A167268 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
%C A167268 Named after the French mathematician Charles Janet (1849-1932). - _Amiram Eldar_, Jun 22 2021
%D A167268 Charles Janet, Considérations sur la structure du noyau de l'atome, N 5, Décembre 1929, Beauvais, 2+45 pages, 4 leaflets.
%H A167268 Daniel Forgues, <a href="/A167268/b167268.txt">Table of n, a(n) for n = 1..9899</a>
%H A167268 The Chemogenesis web book, <a href="http://www.meta-synthesis.com/webbook/34_qn/qn_to_pt.html">Quantum Numbers to Periodic Tables</a>.
%H A167268 Szymon Łukaszyk and Wawrzyniec Bieniawski, <a href="https://doi.org/10.20944/preprints202401.1113.v1">Assembly Theory of Binary Messages (How to Assemble a Black Hole and Use it to Assemble New Binary Information?)</a>, Preprints (2024).
%F A167268 a(n) = 4*((-n) mod round(sqrt(n))) + 2. - _Jon E. Schoenfield_, Sep 08 2013
%F A167268 a(n) = 4*A216607(n) + 2. - _Szymon Lukaszyk_, Oct 27 2023
%t A167268 row[n_] := Range[2, 4*Floor[n/2] + 2, 4]; Flatten[ Table[ row[n] // Reverse, {n, 0, 15}]] (* _Jean-François Alcover_, Oct 02 2012 *)
%o A167268 (Python)
%o A167268 from mpmath.libmp import sqrtrem
%o A167268 def a(n):
%o A167268     s, r = sqrtrem(n)
%o A167268     return 4 * (-n % (s + (r>s))) + 2
%o A167268 # _Christoph B. Kassir_, Apr 07 2022
%Y A167268 Cf. A016825, A160914, A138469, A199934, A216607.
%K A167268 nonn,easy
%O A167268 1,1
%A A167268 _Paul Curtz_, Oct 31 2009
%E A167268 Edited by _Daniel Forgues_, May 09 2011