A212013 -id:A212013 - cp's OEIS frontend

A212012 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.

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

Offset: 1

Omar E. Pol, Jul 15 2012

Also triangle read by rows in which row i lists the first i positive even numbers in decreasing order.

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), 1d_(3/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 3,... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 4,... Other sequences that arise from this sequence are both A212013 and A212014. - Omar E. Pol, Sep 02 2012

			Illustration of initial terms: one of the views of a three-dimensional shell model of nucleus.
.
.|-------------------------- j --------------------------|
.|                                                       |
.|   |---------------------- i ----------------------|   |
.|   |                                               |   |
.|   |   |------------------ h ------------------|   |   |
.|   |   |                                       |   |   |
.|   |   |   |-------------- g --------------|   |   |   |
.|   |   |   |                               |   |   |   |
.|   |   |   |   |---------- f ----------|   |   |   |   |
.|   |   |   |   |                       |   |   |   |   |
.|   |   |   |   |   |------ d ------|   |   |   |   |   |
.|   |   |   |   |   |               |   |   |   |   |   |
.|   |   |   |   |   |   |-- p --|   |   |   |   |   |   |
.|   |   |   |   |   |   |       |   |   |   |   |   |   |
.|   |   |   |   |   |   |   s   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
.|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |   |   |   8   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   6   |   |   |   |
.|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
.|   |   |  10   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
.|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |  12   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |  10   |   |
.|   |   |   |   8   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   6   |   |   |   |
.|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
.|  14   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |  12   |
.|   |   |  10   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
.|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |1/2|   |   |   |   |   |   |
.|   |   |   |   |   |   |           |   |   |   |   |   |
.|   |   |   |   |   |   |----3/2----|   |   |   |   |   |
.|   |   |   |   |   |                   |   |   |   |   |
.|   |   |   |   |   |--------5/2--------|   |   |   |   |
.|   |   |   |   |                           |   |   |   |
.|   |   |   |   |------------7/2------------|   |   |   |
.|   |   |   |                                   |   |   |
.|   |   |   |----------------9/2----------------|   |   |
.|   |   |                                           |   |
.|   |   |-------------------11/2--------------------|   |
.|   |                                                   |
.|   |-----------------------13/2------------------------|
.|
.|---------------------------15/2-------------------------
.
For another view of the model see the example section of A212122, second part.
Example 1. Triangle begins:
  2;
  4,   2;
  6,   4,  2;
  8,   6,  4,  2;
  10,  8,  6,  4,  2;
  12, 10,  8,  6,  4,  2;
  14, 12, 10,  8,  6,  4, 2;
  16, 14, 12, 10,  8,  6, 4, 2;
...
Column 1 gives positive terms of A005843. Right border give positive terms of A007395. Row sums give A002378.
Example 2. Written as an irregular triangle in which row j represents the j-th shell of nucleus. Note that row 4 has only one term. Triangle begins:
  2;
  4,   2;
  6,   4,  2;
  8;
  6,   4,  2, 10;
  8,   6,  4,  2, 12;
  10,  8,  6,  4,  2, 14;
  12, 10,  8,  6,  4,  2, 16;
  14, 12, 10,  8,  6,  4,  2, 18;

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).
M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. I. Empirical evidence, Phys. Rev. 78, 16-21 (1950).
M. Goeppert Mayer, Nuclear configurations in the spin-orbit coupling model. II. Theoretical considerations, Phys. Rev. 78: 22 (1950).

Partial sums give A212014. Other versions are A162630, A212122, A213362, A213372.

Cf. A002378, A004736, A005843, A007395, A212013, A212014.

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

a(n) = 2*A004736(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).

A212123 Total number of pairs 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

1, 3, 4, 7, 8, 10, 14, 16, 19, 20, 25, 29, 32, 34, 35, 41, 46, 50, 53, 55, 56, 63, 68, 71, 77, 81, 82, 84, 92

Offset: 1

Omar E. Pol, Jun 03 2012

First differs from A213363 at a(12).

			Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
1;
3,    4;
7,    8,  10;
14,  16,  19,  20;
25,  29,  32,  34,  35;
41,  46,  50,  53,  55,  56;
63,  68,  71,  77,  81,  82,  84;
...
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. Note that in this case row 4 has only one term. Triangle begins:
1;
3,   4;
7,   8, 10;
14,
16, 19, 20, 25;
29, 32, 34, 35, 41;
46, 50, 53, 55, 56, 63;
68, 71, 77, 81, 82, 84, 92;
...

Partial sums of A212121. Other versions are A210983, A212013, A213363, A213373.

Cf. A000292, A004006, A212122, A212124.

a(n) = A212124(n)/2.

A210983 Total number of pairs 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

1, 3, 4, 7, 8, 10, 14, 16, 17, 20, 25, 28, 29, 31, 35, 41, 45, 47, 48, 51, 56, 63, 68, 71, 72, 74, 78, 84, 92, 98, 102, 104, 105, 108, 113, 120, 129, 136, 141, 144, 145, 147, 151, 157, 165, 175, 183, 189, 193, 195, 196, 199, 204, 211, 220, 231

Offset: 1

Omar E. Pol, Jul 14 2012

Additional comments from Omar E. Pol, Sep 02 2012: (Start)
Q: What are energy levels?
A: See the link sections of A212122, A213362, A213372. For example, see this link related to A213372: http://www.flickr.com/photos/mitopencourseware/3772864128/in/set-72157621892931990
Q: What defines the order in A212121?
A: The order of A212121 is defined by A212122.
Note that there are at least five versions of the nuclear shell model in the OEIS:
Goeppert-Mayer (1950): A212012, A004736, A212013, A212014.
Goeppert-Mayer, Jensen (1955): A212122, A212121, A212123, A212124.
Talmi (1993): A213362, A213361, A213363, A213364.
Meyerhof: A213372, A213371, A213373, A213374.
For another version: A162630, A130517, A210983, A210984.
Each version is represented by four sequences: the first sequence is the main entry.
(End)
For additional information see A162630.

			Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus, the sequence begins:
1;
3,     4;
7,     8,  10;
14,   16,  17,  20;
25,   28,  29,  31,  35;
41,   45,  47,  48,  51,  56;
63,   68,  71,  72,  74,  78,  84;
92,   98, 102, 104, 105, 108, 113, 120;
129, 136, 141, 144, 145, 147, 151, 157, 165;
175, 183, 189, 193, 195, 196, 199, 204, 211, 220;
...
Column 1 gives positive terms of A004006. Right border gives positives terms of A000292.
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. Note that in this case row 4 has only one term. Triangle begins:
1;
3,     4;
7,     8,  10;
14;
16,   17,  20,  25;
28,   29,  31,  35,  41;
45,   47,  48,  51,  56,  63;
68,   71,  72,  74,  78,  84,  92;
98,  102, 104, 105, 108, 113, 120, 129;
136, 141, 144, 145, 147, 151, 157, 165, 175;
183, 189, 193, 195, 196, 199, 204, 211, 220, 231;
...

Partial sums of A130517 (when that sequence is regarded as a flattened triangle). Other versions are A212013, A212123, A213363, A213373.

Cf. A000292, A004006, A162630, A210984

a(n) = A210984(n)/2.

A213363 Total number of pairs 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

1, 3, 4, 7, 8, 10, 14, 16, 19, 20, 25, 28, 32, 33, 35, 41, 45, 47, 54, 59, 60, 63, 68, 74, 82, 85, 86, 90, 92

Offset: 1

Omar E. Pol, Jun 23 2012

First differs from A212123 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:
1;
3, 4;
7, 8, 10;
14;
16, 19, 20, 25;
28, 32, 33, 35, 41;
45, 47, 54, 59, 60, 63;
68, 74, 82, 85, 86, 90, 92;
...

Partial sums of A213361. Other versions are A210983, A212013, A213373.

Cf. A018226, A212121-A212124, A213362, A213364.

a(n) = A213364(n)/2.

A213373 Total number of pairs 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

1, 3, 4, 7, 8, 10, 14, 16, 19, 20, 25, 29, 32, 38, 40, 41, 45, 50, 57, 59, 62, 63

Offset: 1

Omar E. Pol, Jul 16 2012

First differs from A212123 at a(14). For more information see A213372.

			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:
1;
3, 4;
7, 8, 10;
14;
16, 19, 20, 25;
29, 32, 38, 40, 41;
45, 50, 57, 59, 62, 63;
...

MIT OpenCourseWare, Energy Levels of Nucleons in a Smoothly-Varying Potential Well

Partial sums of A213371. Other versions are A210983, A212013, A212123, A213363.

Cf. A018226, A213372, A213374.

a(n) = A213374(n)/2.

A014370 If n = binomial(b,2) + binomial(c,1), b > c >= 0 then a(n) = binomial(b+1,3) + binomial(c+1,2).

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 11, 13, 16, 20, 21, 23, 26, 30, 35, 36, 38, 41, 45, 50, 56, 57, 59, 62, 66, 71, 77, 84, 85, 87, 90, 94, 99, 105, 112, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 166, 168, 171, 175, 180, 186, 193, 201, 210, 220, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 286

Offset: 1

N. J. A. Sloane

			The triangle starts:
  1
  2 4
  5 7 10
  11 13 16 20
  21 23 26 30 35

W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, 1993, p. 159.

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

Cf. A002260, A000292 (main diagonal), A000217, A014368, A014369, A006046, A050407 (1st column), A005581 (subdiagonal), A071239 (row sums), A212013.

a := 0: for i from 1 to 15 do for j from 1 to i do a := a+j: printf(`%d,`,a); od:od:

A014370[n_, k_] := Binomial[n + 1, 3] + Binomial[k + 1, 2];
Table[A014370[n, k], {n, 12}, {k, n}] (* Paolo Xausa, Mar 11 2025 *)

a(n) = Sum_{m = 1..n} b(m), b(m) = 1,1,2,1,2,3,1,2,3,4,... = A002260. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
a(n*(n+1)/2+m) = n*(n+1)*(n+2)/6 + m*(m+1)/2 = A000292(n)+ A000217(m), m = 0...n+1, n = 1, 2, 3.. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
a(n) = a(n-1) + A002260(n). As a triangle with n >= k >= 1: a(n, k) = a(n-1, k) + (n-1)*n/2 = a(n, k-1) + k = (n^3-n+3k^2+3k)/6. - Henry Bottomley, Nov 14 2001
a(n) = b(n) * (b(n) + 1) * (b(n) + 2) / 6 + c(n) * (c(n) + 1) / 2, where b(n) = [sqrt(2 * n) - 1/2] and c(n) = n - b(n) * (b(n) + 1) / 2. - Robert A. Stump (bee_ess107(AT)msn.com), Sep 20 2002
As a triangle, T(n,k) = binomial(n+1, 3) + binomial(k+1,2). - Franklin T. Adams-Watters, Jan 27 2014

More terms from James Sellers, Feb 05 2000

cp's OEIS Frontend

A212012 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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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.

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A212123 Total number of pairs 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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A210983 Total number of pairs 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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A213363 Total number of pairs 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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A213373 Total number of pairs 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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A014370 If n = binomial(b,2) + binomial(c,1), b > c >= 0 then a(n) = binomial(b+1,3) + binomial(c+1,2).

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