A018226 -id:A018226 - cp's OEIS frontend

A137306 Related to A018226 and A018227: due to the stable element 118 (last term in A018227) this is submitted as a suggested new observed list with 459 as a probable next new atomic weight stable island.

2, 8, 20, 28, 50, 83, 118, 126, 168, 194, 298

Offset: 1

Roger L. Bagula, Apr 20 2008

"These are crucial to the notion of an 'island of stability', which Glenn Seaborg espoused."

Y. Oganessian, J. Phys. G: Nucl. Part. Phys, 2007, iop.org
Rosenfeld, L. (1948). Nuclear Forces. Interscience Publishers, New York, xvii.
G. T. Seaborg - Contemporary Physics, 2004, informaworld.com

(* nuclear radius visualization from Rosenfeld*) Clear[r, A, r0, p, n, m] A[p_, n_] := If[p == 0, 1, p + n] r0 = 1.3214405*10^(-13); r[p_, n_] := r0*( 1 + 0.8*(n/A[p, n])^2 - 0.3/A[p, n]^(1/3) + 0.010*p^2/A[p, n]^(4/3)) a0 = Table[Table[r[p, n], {n, p, Floor[2*p]}], {p, 0, 120}]; ListPlot[Flatten[a0]]

Roughly a(n) = Floor[1.53*a(n-1)]

Definition not clear to me. - N. J. A. Sloane, Apr 25 2008

A230434 Magic numbers of nucleons. Another version of A018226, with 34 inserted.

Original entry on oeis.org

2, 8, 20, 28, 34, 50, 82, 126

Offset: 1

Omar E. Pol, Oct 22 2013

Union of 34 and A018226.

D. Steppenbeck et al., Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature, 502 (2013), 207-210, DOI: 10.1038/nature12522.

Cf. A018226, A018227, A033547, A046939, A046940, A110856, A117612.

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

Original entry on oeis.org

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)

A033547 Otto Haxel's guess for magic numbers of nuclear shells.

Original entry on oeis.org

0, 2, 6, 14, 28, 50, 82, 126, 184, 258, 350, 462, 596, 754, 938, 1150, 1392, 1666, 1974, 2318, 2700, 3122, 3586, 4094, 4648, 5250, 5902, 6606, 7364, 8178, 9050, 9982, 10976, 12034, 13158, 14350, 15612, 16946, 18354, 19838, 21400, 23042, 24766, 26574, 28468

Offset: 0

Wolfdieter Lang

O. Haxel gave a construction procedure. The formulas are due to Wolfdieter Lang.

G. C. Greubel, Table of n, a(n) for n = 0..1000
O. Haxel, Die Entstehung des Schalenmodells der Atomkerne, Physikalische Blätter, vol. 50, p. 339, 1994.
O. Haxel et al., On the "Magic Numbers" in Nuclear Structure, Phys. Rev., 75 (1949), 1766.
V. Ladma, Magic Numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Equals 2*A004006, partial sums of A014206, 2*(partial sums of A000124).

Cf. A018226, A046127.

List([0..50], n-> n*(n^2+5)/3); # G. C. Greubel, Oct 12 2019

[n*(n^2+5)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 05 2015

A033547:=n->n*(n^2+5)/3: seq(A033547(n), n=0..50); # Wesley Ivan Hurt, Apr 05 2015

Table[n(n^2+5)/3, {n,0,50}] (* Harvey P. Dale, Apr 07 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 2, 6, 14}, 50] (* Vincenzo Librandi, Apr 06 2015 *)

a(n)=n*(n^2+5)/3 \\ Charles R Greathouse IV, Jun 25 2017

[n*(n^2+5)/3 for n in range(50)] # G. C. Greubel, Oct 12 2019

a(n) = n*(n^2 + 5)/3.
G.f.: 2*x*(1 - x + x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 05 2015
E.g.f.: x*(6 + 3*x + x^2)*exp(x)/3. - G. C. Greubel, Oct 12 2019
a(n) = A046127(n+1) - 2. - Jianing Song, Feb 03 2024

A162630 Triangle read by rows in which row n lists the number of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 8, 4, 2, 6, 10, 6, 2, 4, 8, 12, 8, 4, 2, 6, 10, 14, 10, 6, 2, 4, 8, 12, 16, 12, 8, 4, 2, 6, 10, 14, 18, 14, 10, 6, 2, 4, 8, 12, 16, 20, 16, 12, 8, 4, 2, 6, 10, 14, 18, 22, 18, 14, 10, 6, 2, 4, 8, 12, 16, 20, 24, 20, 16, 12, 8, 4, 2

Offset: 1

Omar E. Pol, Jul 10 2009

The list of the spin-orbit coupling of this version of the nuclear shell model starts: 1s_(1/2), 1p_(3/2), 1p_(1/2), 1d_(5/2), 2s_(1/2), 1d_(3/2), 1f_(7/2), 2p_(3/2), 2p_(1/2), etc. The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 1, ... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 2, 4, 8, 4, 2, ... Other sequences that arise from this sequence are A A130517, A210983, A210984. - Omar E. Pol, Sep 02 2012

			A geometric shell model of the atomic nucleus:
   +---------------------- i ----------------------+
   |   +------------------ h ------------------+   |
   |   |   +-------------- g --------------+   |   |
   |   |   |   +---------- f ----------+   |   |   |
   |   |   |   |   +------ d ------+   |   |   |   |
   |   |   |   |   |   +-- p --+   |   |   |   |   |
   |   |   |   |   |   |   s   |   |   |   |   |   |
   |   |   |   |   |   |   |   |   |   |   |   |   |
   |   |   |   |   |   |       |   |   |   |   |   |
   |   |   |   |   |       2       |   |   |   |   |
   |   |   |   |       4       2       |   |   |   |
   |   |   |       6       2       4       |   |   |
   |   |       8       4       2       6       |   |
   |      10       6       2       4       8       |
      12       8       4       2       6      10
  14      10       6       2       4       8      12
   |   |   |   |   |   |   |   |   |   |   |   |   |
   |   |   |   |   |   |   +1/2+   |   |   |   |   |
   |   |   |   |   |   +--- 3/2 ---+   |   |   |   |
   |   |   |   |   +------- 5/2 -------+   |   |   |
   |   |   |   +----------- 7/2 -----------+   |   |
   |   |   +--------------- 9/2 ---------------+   |
   |   +------------------ 11/2 -------------------+
   +---------------------- 13/2 -----------------------

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.

Cf. A018226, A130517, A130556, A130598, A130602, A162626.

Other versions are A212012, A212122, A213362, A213372

t[n_, 1] := n; t[n_, n_] := n-1;
t[n_, k_] := Abs[2k - n - If[2k <= n+1, 2, 1]];
2 Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

a(n) = 2*A130517(n).
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = 2*(|2*A000027(n) - A003056(n)^2 - 2*A003056(n) - 3| + floor((2*A000027(n) - A003056(n)^2 - A003056(n))/(A003056(n) + 3))).
a(n) = 2*(|2*n - t*t - 2*t - 3| + floor((2*n - t*t - t)/(t+3))) where t = floor((-1 + sqrt(8*n-7))/2). (End)

Corrected by Omar E. Pol, Jul 13 2009
More terms from Omar E. Pol, Jul 14 2012
New name from Omar E. Pol, Sep 02 2012

A212124 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

Original entry on oeis.org

2, 6, 8, 14, 16, 20, 28, 32, 38, 40, 50, 58, 64, 68, 70, 82, 92, 100, 106, 110, 112, 126, 136, 142, 154, 162, 164, 168, 184

Offset: 1

Omar E. Pol, Jun 03 2012

First differs from A213364 at a(12).

			Example 1: written as a triangle in which apparently row i is related to the (i-1)st level of nucleus. Triangle begins:
2;
6,     8;
14,   16,  20;
28,   32,  38,  40;
50,   58,  64,  68,  70;
82,   92, 100, 106, 110, 112;
126, 136, 142, 154, 162, 164, 168;
...
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. In this case note that row 4 has only one term. Triangle begins:
2;
6,     8;
14,   16,  20;
28,
32,   38,  40,  50;
58,   64,  68,  70,  82;
92,  100, 106, 110, 112, 126;
136, 142, 154, 162, 164, 168, 184;
...
First seven terms of right border give the "magic numbers" A018226.

M. Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure, J. Wiley and Sons, Inc. (1955).

Library of Halexandria, Nuclear shell model, Table 1.

Partial sums of A212122. Other versions are A210984, A212014, A213364, A213374.

Cf. A007290, A018226, A033547, A210842, A212121, A212123.

a(n) = 2*A212123(n).

A212121 Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 2, 3, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 5, 3, 6, 4, 1, 2, 8

Offset: 1

Omar E. Pol, Jun 03 2012

For another version see A213361. First differs from A213361 at a(12).

What defines this sequence? (This appears to be some sort of permutation of A130517 by shifting columns down or upwards in some randomized way.) - R. J. Mathar, Jul 22 2012

			Illustration of initial terms: two views of a three-dimensional shell model of nucleus.
.
.|-------------------------- j --------------------------|
.|                                                       |
.|   |---------------------- i ----------------------|   |
.|   |                                               |   |
.|   |   |------------------ h ------------------|   |   |
.|   |   |                                       |   |   |
.|   |   |   |-------------- g --------------|   |   |   |
.|   |   |   |                               |   |   |   |
.|   |   |   |   |---------- f ----------|   |   |   |   |
.|   |   |   |   |                       |   |   |   |   |
.|   |   |   |   |   |------ d ------|   |   |   |   |   |
.|   |   |   |   |   |               |   |   |   |   |   |
.|   |   |   |   |   |   |-- p --|   |   |   |   |   |   |
.|   |   |   |   |   |   |       |   |   |   |   |   |   |
.|   |   |   |   |   |   |   s   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   |   |   |   |   2   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   1   |   |   |   |   |   |
.|   |   |   |   |   3   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   2   |   |   |   |   |
.|   |   |   |   4   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   2   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   3   |   |   |   |
.|   |   |   |   |   |   |   |   1   |   |   |   |   |   |
.|   |   |   5   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   4   |   |   |
.|   |   |   |   |   3   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   2   |   |   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   6   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   5   |   |
.|   |   |   |   4   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   3   |   |   |   |
.|   |   |   |   |   |   2   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   1   |   |   |   |   |   |
.|   7   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   5   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   3   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   6   |
.|   |   |   |   |   |   |   |   |   |   |   4   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   2   |   |   |   |   |
.8   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |1/2|   |   |   |   |   |   |
.|   |   |   |   |   |   |           |   |   |   |   |   |
.|   |   |   |   |   |   |----3/2----|   |   |   |   |   |
.|   |   |   |   |   |                   |   |   |   |   |
.|   |   |   |   |   |--------5/2--------|   |   |   |   |
.|   |   |   |   |                           |   |   |   |
.|   |   |   |   |------------7/2------------|   |   |   |
.|   |   |   |                                   |   |   |
.|   |   |   |----------------9/2----------------    |   |
.|   |   |                                           |   |
.|   |   |-------------------11/2--------------------|   |
.|   |                                                   |
.|   |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
..........................................................
.
.|-------------------------- j --------------------------|
.|                                                       |
.*   |---------------------- i ----------------------|   |
.|   |                                               |   |
.|   *   |------------------ h ------------------|   |   *
.|   |   |                                       |   |   |
.*   |   *   |-------------- f --------------|   |   *   |
.|   |   |   |                               |   |   |   |
.|   *   |   *   |---------- e ----------|   |   *   |   *
.|   |   |   |   |                       |   |   |   |   |
.*   |   *   |   *   |------ d ------|   |   *   |   *   |
.|   |   |   |   |   |               |   |   |   |   |   |
.|   *   |   *   |   *   |-- p --|   |   *   |   *   |   *
.|   |   |   |   |   |   |       |   |   |   |   |   |   |
.*   |   *   |   *   |   *   s   |   *   |   *   |   *   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   *   |   *   |   *   |   *   *   |   *   |   *   |   *
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.*   |   *   |   *   |   *   |1/2|   *   |   *   |   *   |
.|   |   |   |   |   |   |           |   |   |   |   |   |
.|   *   |   *   |   *   |----3/2----|   *   |   *   |   *
.|   |   |   |   |   |                   |   |   |   |   |
.*   |   *   |   *   |--------5/2--------|   *   |   *   |
.|   |   |   |   |                           |   |   |   |
.|   *   |   *   |------------7/2------------|   *   |   *
.|   |   |   |                                   |   |   |
.*   |   *   |----------------9/2----------------|   *   |
.|   |   |                                           |   |
.|   *   |-------------------11/2--------------------|   *
.|   |                                                   |
.*   |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
Written as an irregular triangle in which row n represents the n-th shell of nucleus. Note that row 4 has only one term. Triangle begins:
1;
2, 1;
3, 1, 2;
4;
2, 3, 1, 5;
4, 3, 2, 1, 6;
5, 4, 3, 2, 1, 7;
5, 3, 6, 4, 1, 2, 8;
...

Partial sums give A212123.

Cf. A018226, A130517, A212122, A212124, A213361-A213364.

a(n) = A212122(n)/2.

A213364 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

Original entry on oeis.org

2, 6, 8, 14, 16, 20, 28, 32, 38, 40, 50, 56, 64, 66, 70, 82, 90, 94, 108, 118, 120, 126, 136, 148, 164, 170, 172, 180, 184

Offset: 1

Omar E. Pol, Jun 23 2012

First differs from A212124 at a(12).

			Written as an irregular triangle in which row j is related to the j-th shell of nucleus. Note that row 4 has only one term. Triangle begins:
2;
6,     8;
14,   16,  20;
28;
32,   38,  40,  50;
56,   64,  66,  70,  82;
90,   94, 108, 118, 120, 126;
136, 148, 164, 170, 172, 180, 184;
...
First seven terms of right border give the "magic numbers" A018226

I. Talmi, Simple Models of Complex Nuclei, Hardwood Academic Publishers (1993).

Partial sums of A213362. Other versions are A210984, A212014, A212124, A213374.

Cf. A018226, A212121-A212124, A213361, A213363.

a(n) = 2*A213363(n).

A212122 Triangle read by rows in which row n lists the number of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 8, 4, 6, 2, 10, 8, 6, 4, 2, 12, 10, 8, 6, 4, 2, 14, 10, 6, 12, 8, 2, 4, 16

Offset: 1

Omar E. Pol, Jun 03 2012

First differs from A213362 at a(12).

The list of the spin-orbit coupling of this version of the nuclear shell model starts: 1s_(1/2), 1p_(3/2), 1p_(1/2), 1d_(5/2), 2s_(1/2), 1d_(3/2), 1f_(7/2), 2p_(3/2), 1f_(5/2), 2p_(1/2), 1g_(9/2), 1g_(7/2), 2d_(5/2), 2d_(3/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 5, 1, 9, 7, 5, 3,... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 2, 4, 8, 4, 6, 2, 10, 8, 6, 4,... Other sequences that arise from this sequence are A212121, A212123, A212124. - Omar E. Pol, Sep 02 2012

			Illustration of initial terms: two views of a three-dimensional shell model of nucleus.
|-------------------------- j --------------------------|
|                                                       |
|   |---------------------- i ----------------------|   |
|   |                                               |   |
|   |   |------------------ h ------------------|   |   |
|   |   |                                       |   |   |
|   |   |   |-------------- g --------------|   |   |   |
|   |   |   |                               |   |   |   |
|   |   |   |   |---------- f ----------|   |   |   |   |
|   |   |   |   |                       |   |   |   |   |
|   |   |   |   |   |------ d ------|   |   |   |   |   |
|   |   |   |   |   |               |   |   |   |   |   |
|   |   |   |   |   |   |-- p --|   |   |   |   |   |   |
|   |   |   |   |   |   |       |   |   |   |   |   |   |
|   |   |   |   |   |   |   s   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
|   |   |   |   8   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   6   |   |   |   |
|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
|   |   |  10   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
|   |  12   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |  10   |   |
|   |   |   |   8   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   6   |   |   |   |
|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
|  14   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |  10   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |   |  12   |
|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
16  |   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |1/2|   |   |   |   |   |   |
|   |   |   |   |   |   |           |   |   |   |   |   |
|   |   |   |   |   |   |----3/2----|   |   |   |   |   |
|   |   |   |   |   |                   |   |   |   |   |
|   |   |   |   |   |--------5/2--------|   |   |   |   |
|   |   |   |   |                           |   |   |   |
|   |   |   |   |------------7/2------------|   |   |   |
|   |   |   |                                   |   |   |
|   |   |   |----------------9/2----------------|   |   |
|   |   |                                           |   |
|   |   |-------------------11/2--------------------|   |
|   |                                                   |
|   |-----------------------13/2------------------------|
|
|---------------------------15/2-------------------------
.
..........................................................
.
|-------------------------- j --------------------------|
*                                                       |
*   |---------------------- i ----------------------|   |
|   *                                               |   *
|   *   |------------------ h ------------------|   |   *
*   |   *                                       |   *   |
*   |   *   |-------------- f --------------|   |   *   |
|   *   |   *                               |   *   |   *
|   *   |   *   |---------- e ----------|   |   *   |   *
*   |   *   |   *                       |   *   |   *   |
*   |   *   |   *   |------ d ------|   |   *   |   *   |
|   *   |   *   |   *               |   *   |   *   |   *
|   *   |   *   |   *   |-- p --|   |   *   |   *   |   *
*   |   *   |   *   |   *       |   *   |   *   |   *   |
*   |   *   |   *   |   *   s   |   *   |   *   |   *   |
|   *   |   *   |   *   |   *   *   |   *   |   *   |   *
|   *   |   *   |   *   |   *   *   |   *   |   *   |   *
*   |   *   |   *   |   *   |   |   *   |   *   |   *   |
*   |   *   |   *   |   *   |1/2|   *   |   *   |   *   |
|   *   |   *   |   *   |           |   *   |   *   |   *
|   *   |   *   |   *   |----3/2----|   *   |   *   |   *
*   |   *   |   *   |                   |   *   |   *   |
*   |   *   |   *   |--------5/2--------|   *   |   *   |
|   *   |   *   |                           |   *   |   *
|   *   |   *   |------------7/2------------|   *   |   *
*   |   *   |                                   |   *   |
*   |   *   |----------------9/2----------------|   *   |
|   *   |                                           |   *
|   *   |-------------------11/2--------------------|   *
*   |                                                   |
*   |-----------------------13/2------------------------|
|
|---------------------------15/2-------------------------
.
Written as an irregular triangle in which row n represents the n-th shell of nucleus. Note that row 4 has only one term. Triangle begins:
2;
4,   2;
6,   2,  4;
8;
4,   6,  2, 10;
8,   6,  4,  2, 12;
10,  8,  6,  4,  2, 14;
10,  6, 12,  8,  2,  4, 16;
...

M. Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure, J. Wiley and Sons, Inc. (1955).

HyperPhysics, G.S.U., Shell Model of Nucleus
Library of Halexandria, Nuclear shell model, Table 1.

Row sums give A210842. Partial sums give A212124.
Other versions are A162630, A212012, A213362, A213372.
Cf. A018226, A002378, A130517, A130598, A210842, A212121, A212123.

a(n) = 2*A212121(n).

A212014 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

Original entry on oeis.org

2, 6, 8, 14, 18, 20, 28, 34, 38, 40, 50, 58, 64, 68, 70, 82, 92, 100, 106, 110, 112, 126, 138, 148, 156, 162, 166, 168, 184, 198, 210, 220, 228, 234, 238, 240, 258, 274, 288, 300, 310, 318, 324, 328, 330, 350, 368, 384, 398, 410, 420, 428, 434, 438, 440, 462, 482, 500, 516, 530, 542, 552, 560, 566, 570, 572

Offset: 1

Omar E. Pol, Jul 15 2012

			Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
    2;
    6,   8;
   14,  18,  20;
   28,  34,  38,  40;
   50,  58,  64,  68,  70;
   82,  92, 100, 106, 110, 112;
  126, 138, 148, 156, 162, 166, 168;
  ...
Column 1 gives positive terms of A033547. Right border gives positive terms of A007290.
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. In this case note that row 4 has only one term. Triangle begins:
    2;
    6,   8;
   14,  18,  20;
   28;
   34,  38,  40,  50;
   58,  64,  68,  70,  82;
   92, 100, 106, 110, 112, 126;
  138, 148, 156, 162, 166, 168, 184;
  ...
First seven terms of right border give the "magic numbers" A018226.

M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. I. Empirical evidence, Phys. Rev. 78: 16 (1950). II. Theoretical considerations, Phys. Rev. 78: 22 (1950).

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Partial sums of A212012. Other versions are A210984, A212124, A213364, A213374.

Cf. A004736, A007290, A033547, A018226, A212013.

2*Accumulate[Flatten[Range[Range[15], 1, -1]]] (* Paolo Xausa, Mar 14 2025 *)

a(n) = 2*A212013(n).

cp's OEIS Frontend

A137306 Related to A018226 and A018227: due to the stable element 118 (last term in A018227) this is submitted as a suggested new observed list with 459 as a probable next new atomic weight stable island.

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A230434 Magic numbers of nucleons. Another version of A018226, with 34 inserted.

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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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A033547 Otto Haxel's guess for magic numbers of nuclear shells.

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A162630 Triangle read by rows in which row n lists the number of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

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A212124 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

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A212121 Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

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A213364 Total number of states of the first n subshells of the nuclear shell model in which the subshells are ordered by energy level in increasing order.

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