A093907 -id:A093907 - cp's OEIS frontend

A269510 Duplicate of A093907.

2, 8, 8, 18, 18, 32, 32, 50, 50

Offset: 1

dead

A018227 Magic numbers: atoms with full shells containing any of these numbers of electrons are considered electronically stable.

2, 10, 18, 36, 54, 86, 118, 168, 218, 290, 362, 460, 558, 686, 814, 976, 1138, 1338, 1538, 1780, 2022, 2310, 2598, 2936, 3274, 3666, 4058, 4508, 4958, 5470, 5982, 6560, 7138, 7786, 8434, 9156, 9878, 10678, 11478, 12360, 13242, 14210, 15178

Offset: 1

John Raithel (raithel(AT)rahul.net)

Atomic numbers of noble elements in the periodic table.
Partial sums of A093907. - Lekraj Beedassy, Mar 24 2006
Comment from Don N. Page (don(AT)phys.ualberta.ca), Dec 12 2006: (Start)
"Relativistic corrections and instabilities to pair creation of electrons and positrons would occur even if one could have stable nuclei of arbitrarily many protons Z for the fixed value of the fine structure constant alpha ~ 1/137 in our universe.
"However, if one considered an imaginary universe with arbitrarily tiny alpha and a fixed point source of charge Z, one could have stable neutral atoms of Z nonrelativistic electrons of mass m for any Z, so long as one takes the limit Z alpha -> 0 by taking alpha -> 0 after fixing Z.
"One could then define noble elements to be given by the integer values of Z such that the ionization energy, in units of m c^2 alpha^2, of any such atom in its ground state with larger Z is less than that of the noble element (which appears to be the case for all the noble elements with the actual nonzero value of alpha).
"This sequence of idealized nonrelativistic noble elements with Z electrons would give an infinite sequence of integers Z, which may or may not be the same as that given by the explicit formula listed for the present sequence. It would likely be a difficult mathematical problem to calculate this infinite sequence." (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324 (p. 312).
Encyclopedia Britannica, magic number
D. Weise, The Pythagorean Approach to Problems of Periodicity in Science
D. Weise, Pythagorean Approach To Problems Of Periodicity In Fermionic System
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A018226 for the magic numbers for nucleons (protons and neutrons).

[n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1: n in [1..50]]; // Vincenzo Librandi, May 03 2011

a(n)=(2*n^3+12*n^2+25*n-6+(-1)^n*(3*n+6))/12 \\ Charles R Greathouse IV, Oct 18 2022

a(n) = a(n-1) + ((2*n + 3 + (-1)^n)^2)/8; a(n) = (2*n^3 + 12*n^2 + 25*n - 6 + (-1)^n*(3*n + 6))/12. - Warut Roonguthai, Jun 20 2005
a(n) = n*((n+3)^2 + 5)/6 for even n, a(n) = n*((n+3)^2 + 2)/6 - 1 [or C(n+3,3) - 2, i.e., A000292(n) - 2] for odd n. - Lekraj Beedassy, Feb 02 2006
Partial sums of A116471. - Lekraj Beedassy, Mar 31 2006
From Daniel Forgues, May 02 2011: (Start)
a(n) = n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1, n >= 1.
a(n) = (n+1)*(n+2)*(n+3)/6 + (n+2)*(1+(-1)^n)/4 - 2, n >= 1.
a(n) = T_{n+1} + (n+2)*(1+(-1)^n)/4 - 2, n >= 1, where T_n is the n-th tetrahedral number.
G.f.: 2*x*(1 + 3*x - 2*x^2 - x^3 + x^4)/((1 - x)^4*(1 + x)^2). (End)

A137583 Number of elements in the n-th period of the Janet periodic table of elements.

Original entry on oeis.org

2, 2, 8, 8, 18, 18, 32, 32

Offset: 1

Paul Curtz, Apr 26 2008

Essentially 2 followed by the first terms of A093907.

This puts Hydrogen and Helium in the first row, Lithium and Beryllium into the second, Boron to Magnesium into the third etc.

Mark R. Leach, The Internet Database of Periodic Tables, Chemogenesis web book.
Albert Tarantola, PSE of Elements (Janet form).
Mark J. Winter, Image of the Janet long format (or 32-column) periodic table, Web Elements Chemistry Nexus.

Cf. A093907, A137508, A138509, A137575.

Flatten@ Table[2 (n #)^2 & /@ {-1, 1}, {n, 4}] (* Michael De Vlieger, Jul 22 2016 *)

Edited by R. J. Mathar, Oct 02 2009

A168380 Row sums of A168281.

Original entry on oeis.org

2, 4, 12, 20, 38, 56, 88, 120, 170, 220, 292, 364, 462, 560, 688, 816, 978, 1140, 1340, 1540, 1782, 2024, 2312, 2600, 2938, 3276, 3668, 4060, 4510, 4960, 5472, 5984, 6562, 7140, 7788, 8436, 9158, 9880, 10680, 11480, 12362, 13244, 14212, 15180, 16238, 17296, 18448, 19600, 20850, 22100

Offset: 1

Paul Curtz, Nov 24 2009

The atomic numbers of the augmented alkaline earth group in Charles Janet's spiral periodic table are 0 and the first eight terms of this sequence (see Stewart reference). - Alonso del Arte, May 13 2011

Maximum number of 123 patterns in an alternating permutation of length n+3. - Lara Pudwell, Jun 09 2019

			From _Lara Pudwell_, Jun 09 2019: (Start)
a(1)=2. The alternating permutation of length 1+3=4 with the maximum number of copies of 123 is 1324.  The two copies are 124 and 134.
a(2)=4.  The alternating permutation of length 2+3=5 with the maximum number of copies of 123 is 13254.  The four copies are 124, 125, 134, and 135.
a(3)=12. The alternating permutation of length 3+3=6 with the maximum number of copies of 123 is 132546.  The twelve copies are 124, 125, 126, 134, 135, 136, 146, 156, 246, 256, 346, and 356. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 4.
Lara Pudwell, Packing patterns in restricted permutations, 2019.
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
Philip Stewart, Charles Janet: unrecognized genius of the Periodic System. Foundations of Chemistry (2010), p. 9.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

[(n+1)*(3+2*n^2+4*n-3*(-1)^n)/12: n in [1..50] ]; // Vincenzo Librandi, Aug 06 2011

LinearRecurrence[{2,1,-4,1,2,-1},{2, 4, 12, 20, 38, 56},50] (* G. C. Greubel, Jul 19 2016 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12, {n, 50}] (* Michael De Vlieger, Jul 20 2016 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,2,1,-4,1,2]^(n-1)*[2;4;12;20;38;56])[1,1] \\ Charles R Greathouse IV, Jul 21 2016

a(n) = 2*A005993(n-1).
a(n) = (n+1)*(3 + 2*n^2 + 4*n - 3*(-1)^n)/12.
a(n+1) - a(n) = A093907(n) = A137583(n+1).
a(2n+1) = A035597(n+1), a(2n) = A002492(n).
a(n) = A099956(n-1), 2 <= n <= 7.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
G.f.: 2*x*(1 + x^2) / ( (1+x)^2*(x-1)^4 ).
a(n) = A000292(n) + A027656(n-1). - Paul Curtz, Oct 26 2012
E.g.f.: (1/12)*(3*(x - 1) + (3 + 15*x + 12*x^2 + 2*x^3)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 19 2016

A168208 Irregular table of the number of electrons of the n-th element of the PSE in atomic shells, read by rows.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 8, 1, 2, 8, 2, 2, 8, 3, 2, 8, 4, 2, 8, 5, 2, 8, 6, 2, 8, 7, 2, 8, 8, 2, 8, 8, 1, 2, 8, 8, 2, 2, 8, 9, 2, 2, 8, 10, 2, 2, 8, 11, 2, 2, 8, 13, 1, 2, 8, 13, 2, 2, 8, 14, 2, 2, 8, 15, 2, 2, 8, 16, 2, 2, 8, 18, 1, 2, 8, 18, 2, 2, 8, 18, 3, 2, 8, 18, 4, 2

Offset: 1

Paul Curtz, Nov 20 2009

For the n-th element in the periodic system of elements, row n of the table shows the occupancy of the K-shell, then the L-shell, then the M-shell etc.
Row sums are A000027(n). A093907(c) is the maximum number that may appear in column c.
How are rows defined when the n-th element has more than one possible electron configuration? For example, element no. 28 (Nickel) has two electron configurations, namely 2, 8, 16, 2 and 2, 8, 17, 1, and it is disputed which of them is the ground state configuration of Nickel. - Felix Fröhlich, Jun 02 2019

			From _Felix Fröhlich_, Jun 02 2019: (Start)
Irregular table starts as follows, where Z denotes the atomic number:
  Z  | Element name | Electrons per shell
  -----------------------------------------
   1 | Hydrogen     | 1
   2 | Helium       | 2
   3 | Lithium      | 2, 1
   4 | Beryllium    | 2, 2
   5 | Boron        | 2, 3
   6 | Carbon       | 2, 4
   7 | Nitrogen     | 2, 5
   8 | Oxygen       | 2, 6
   9 | Fluorine     | 2, 7
  10 | Neon         | 2, 8
  11 | Sodium       | 2, 8,  1
  12 | Magnesium    | 2, 8,  2
  13 | Aluminium    | 2, 8,  3
  14 | Silicon      | 2, 8,  4
  15 | Phosphorus   | 2, 8,  5
  16 | Sulfur       | 2, 8,  6
  17 | Chlorine     | 2, 8,  7
  18 | Argon        | 2, 8,  8
  19 | Potassium    | 2, 8,  8, 1
  20 | Calcium      | 2, 8,  8, 2
  21 | Scandium     | 2, 8,  9, 2
  22 | Titanium     | 2, 8, 10, 2
  23 | Vanadium     | 2, 8, 11, 2
  24 | Chromium     | 2, 8, 13, 1
  25 | Manganese    | 2, 8, 13, 2
  26 | Iron         | 2, 8, 14, 2
  27 | Cobalt       | 2, 8, 15, 2
(End)

Wikipedia, Electron configurations of the elements (data page)

Cf. A173642.

Redefined as an irregular table by R. J. Mathar, Dec 05 2009

Edited by Felix Fröhlich, Jun 02 2019

A116471 Values 2*(n -+ 1)^2 sorted.

Original entry on oeis.org

0, 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

Lekraj Beedassy, Mar 17 2006

For n>2, consists of entries of A001105(n)=2*n^2 (n>1) that appear twice.

The terms a(2)-a(8) give the number of elements in the periods 1-7 of the periodic table of the chemical elements. - Antti Karttunen, Aug 14 2008

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Wikipedia, Atomic electron configuration table
Wikipedia, Periodic table
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001105, A093907, A016742.

a:=[2,8,8,18,18];;  for n in [6..54] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; Concatenation([0],a); # Muniru A Asiru, Oct 25 2018

0,seq(op([2*n^2,2*n^2]),n=1..30); # Muniru A Asiru, Oct 25 2018

Rest@ Flatten@ Table[2 (n #)^2 & /@ {-1, 1}, {n, 0, 27}] (* or *)
Rest@ CoefficientList[Series[-2 x^2 (x^4 - x^3 - 2 x^2 + 3 x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 54}], x] (* Michael De Vlieger, Jul 22 2016 *)

concat(0, Vec(-2*x^2*(x^4-x^3-2*x^2+3*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Oct 06 2014

a(2*n) = A001105(n) for n >= 1.
From Colin Barker, Oct 06 2014: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 6.
G.f.: -2*x^2*(x^4 - x^3 - 2*x^2 + 3*x + 1)/((x - 1)^3*(x + 1)^2). (End)
a(n) = (2*n^2 + 2*n - (2*n + 1)*(-1)^n + 1)/4, with n > 1 and a(1) = 0. - Bruno Berselli, Oct 07 2014
a(n) = A001057(n+1) + A000217(n+1) for n > 1. - Andrew S. Plewe, Sep 24 2018
E.g.f.: (x*(3 + x)*cosh(x) + (1 + x + x^2)*sinh(x) - 4*x)/2. - Stefano Spezia, Aug 13 2022

More terms from Joshua Zucker, May 11 2006

A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

Original entry on oeis.org

2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Paul Curtz, Nov 19 2009

nonn

Janet's extended enumeration of the periodic table of the elements.

The table is read from the right to the left.

Charles Janet, La structure du Noyau de l'atome,consideree dans la Classification periodique des elements chimiques, Nov. 1927, N. 2, Beauvais, 67 pages, 3 leafleats, see page 15.
Charles Janet, Considerations sur la structure du noyau de l'atome, Dec 1929, N 5, Beauvais, 2+45 pp.,4 leaflets, see leaflets 2 and 3.

Cf. A138100, A138101, A093907, A137583, A172002.

Table[Reverse@ Range[2 n^2], {n, 5}] // Flatten (* Michael De Vlieger, Jul 22 2016 *)

Edited by R. J. Mathar, Feb 15 2010

A168388 First number in the n-th row of A172002.

Original entry on oeis.org

1, 3, 5, 13, 21, 39, 57, 89, 121, 171, 221, 293, 365, 463, 561, 689, 817, 979, 1141, 1341, 1541, 1783, 2025, 2313, 2601, 2939, 3277, 3669, 4061, 4511, 4961, 5473, 5985, 6563, 7141, 7789, 8437, 9159, 9881, 10681, 11481, 12363, 13245, 14213, 15181, 16239, 17297

Offset: 1

Paul Curtz, Nov 24 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, -1).

Cf. A168234.

[(12+n+3*(-1)^n*n+2*n^3)/12: n in [1..60]]; // Vincenzo Librandi, Jul 20 2016

LinearRecurrence[{2,1,-4,1,2,-1},{1,3,5,13,21,39},50] (* Harvey P. Dale, Nov 29 2014 *)
Table[(n + 1) (3 + 2 n^2 + 4 n - 3 (-1)^n)/12 + 1, {n, 0, 46}] (* Michael De Vlieger, Jul 19 2016, after Vincenzo Librandi at A168380 *)

a(2*n+1) = A166464(n)        a(2*n) = A166911(n).
a(n+1) - a(n) = A093907(n-1), n>1.
a(n) = 2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6).
G.f.: x*(1 - x^2 + 2*x)*(1 - x + x^2 + x^3)/( (1+x)^2 * (x-1)^4).
a(n+1) = A168380(n)+1.
From G. C. Greubel, Jul 19 2016: (Start)
a(n) = (12 + n + 3*(-1)^n*n + 2*n^3)/12.
E.g.f.: (1/12)*( -3*x - 12*exp(x) + (12 + 3*x + 6*x^2 + 2*x^3)*exp(2*x) )*exp(-x). (End)

Edited and extended by R. J. Mathar, Mar 25 2010

A137508 Successive structures of alkaline earth metals (periodic table elements from 2nd column).

Original entry on oeis.org

2, 2, 2, 8, 2, 2, 8, 8, 2, 2, 8, 18, 8, 2, 2, 8, 18, 18, 8, 2, 2, 8, 18, 32, 18, 8, 2

Offset: 1

Paul Curtz, Apr 23 2008

nonn

Apparently a(n) = A168281(n+1). - Georg Fischer, Nov 11 2021

			27 terms: 2, 2 for beryllium, ... Every structure is palindromic (even and odd mixed). Also 2*A106314.

Cf. A005993, A099956, A168281. Same numbers as in A093907.

A138096 Number of elements in the n-th column of the second extension of the Mendeleyev-Seaborg periodic table of elements.

Original entry on oeis.org

7, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 7

Offset: 1

Paul Curtz, May 03 2008

A variant of A134982, with one term "4" changing place because (relative to the first table) the two elements number 21 and 39 are moved over to the next column of d-metals, placed above element 71.

Cf. A137583, A093907, A134984, A135559.

cp's OEIS Frontend

A269510 Duplicate of A093907.

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A018227 Magic numbers: atoms with full shells containing any of these numbers of electrons are considered electronically stable.

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A137583 Number of elements in the n-th period of the Janet periodic table of elements.

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A168380 Row sums of A168281.

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A168208 Irregular table of the number of electrons of the n-th element of the PSE in atomic shells, read by rows.

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A116471 Values 2*(n -+ 1)^2 sorted.

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A168142 Count downwards from 2, then from 8, then from 18, then from ... 2*k^2, k>=1.

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A168388 First number in the n-th row of A172002.

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