A212012 -id:A212012 - cp's OEIS frontend

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.

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

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).

A212013 Triangle read by rows: 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, 9, 10, 14, 17, 19, 20, 25, 29, 32, 34, 35, 41, 46, 50, 53, 55, 56, 63, 69, 74, 78, 81, 83, 84, 92, 99, 105, 110, 114, 117, 119, 120, 129, 137, 144, 150, 155, 159, 162, 164, 165, 175, 184, 192, 199, 205, 210, 214, 217, 219, 220, 231, 241, 250, 258, 265, 271, 276, 280, 283, 285, 286

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:
    1;
    3,   4;
    7,   9,  10;
   14,  17,  19,  20;
   25,  29,  32,  34,  35;
   41,  46,  50,  53,  55,  56;
   63,  69,  74,  78,  81,  83,  84;
   92,  99, 105, 110, 114, 117, 119, 120;
  129, 137, 144, 150, 155, 159, 162, 164, 165;
  175, 184, 192, 199, 205, 210, 214, 217, 219, 220;
  ...
Column 1 gives positive terms of A004006. Right border gives positive terms of A000292. Row sums give positive terms of A006325.
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,   9,  10;
   14;
   17,  19,  20,  25;
   29,  32,  34,  35,  41;
   46,  50,  53,  55,  56,  63;
   69,  74,  78,  81,  83,  84,  92;
   99, 105, 110, 114, 117, 119, 120, 129;
  137, 144, 150, 155, 159, 162, 164, 165, 175;
  184, 192, 199, 205, 210, 214, 217, 219, 220, 231;
  ...

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

Partial sums of A004736. Other versions are A210983, A212123, A213363, A213373.

Cf. A000292, A004006, A006325, A212012, A212014, A014370.

row =: monad define
    d=.>y
    < |. (+/d)-d
)
;}. row"0 <\ +/\ 1+i.11 NB. Vanessa McHale (vamchale(AT)gmail.com), Mar 01 2025

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

row(n) = vector(n, k, n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2); \\ Michel Marcus, Mar 10 2025

a(n) = A212014(n)/2.
Let R = floor(sqrt(8*n+1)) and S = floor(R/2) + R mod 2; then a(n) = binomial(S,3) + n + (n-binomial(S,2))*(S*(S+3)-2*n-2)/4. - Gerald Hillier, Jan 16 2018
T(n,k) = n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2. - Davide Rotondo, Mar 10 2025
G.f.: x*y*(1 - x + x^2*(1 - 3*y) - x^5*y^3 + x^3*y*(1 + y) - x^4*y*(1 - 2*y))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Mar 10 2025

More terms from Michel Marcus, Mar 10 2025

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).

A213362 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, 6, 8, 2, 4, 12, 8, 4, 14, 10, 2, 6, 10, 12, 16, 6, 2, 8, 4

Offset: 1

Omar E. Pol, Jun 23 2012

First differs from A212122 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), 2d_(5/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 5, 1, 9, 5,... then we add 1 to every numerator, so we have this sequence: 2, 4, 2, 6, 2, 4, 8, 4, 6, 2, 10, 6,... Note that in the Talmi's table there is a typo: instead 2f_(1/2) should be 2f_(7/2), see references, page 6. Other sequences that arise from this sequence are A213361, A213363, A213364. - 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   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
.|   |  12   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   8   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   4   |   |   |   |   |   |   |   |
.|  14   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |  10   |   |
.|   |   |   |   |   |   |   |   2   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   6   |   |   |   |
.|   |   |  10   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |  12   |
16   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   6   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   2   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   8   |   |   |
.|   |   |   |   |   |   |   |   |   4   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |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;
6,   8,  2,  4, 12;
8,   4, 14, 10,  2,  6;
10, 12, 16,  6,  2,  8,  4;
...

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

Wikipedia, Nuclear shell model

Partial sums give A213364. Other versions are A162630, A212012, A212122, A213372.

Cf. A018226, A130517, A213361, A213363.

a(n) = 2*A213361(n).

A213372 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, 12, 4, 2, 8, 10, 14, 4, 6, 2

Offset: 1

Omar E. Pol, Jul 16 2012

First differs from A212122 at a(14).

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), 1h_(11/2), etc. (see link section). The numerators of the fractions are 1, 3, 1, 5, 1, 3, 7, 3, 5, 1, 9, 7, 5, 11,... 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, 12,... Other sequences that arise from this sequence are A213371, A213373, A213374. - Omar E. Pol, Sep 02 2012

			Illustration of initial terms on one of views of a three-dimensional shell model of nucleus.
.
.  |---------------------- i ----------------------|
.  |                                               |
.  |   |------------------ h ------------------|   |
.  |   |                                       |   |
.  |   |   |-------------- g --------------|   |   |
.  |   |   |                               |   |   |
.  |   |   |   |---------- f ----------|   |   |   |
.  |   |   |   |                       |   |   |   |
.  |   |   |   |   |------ d ------|   |   |   |   |
.  |   |   |   |   |               |   |   |   |   |
.  |   |   |   |   |   |-- p --|   |   |   |   |   |
.  |   |   |   |   |   |       |   |   |   |   |   |
.  |   |   |   |   |   |   s   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   2   |   |   |   |   |   |
.  |   |   |   |   |   4   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   2   |   |   |   |   |
.  |   |   |   |   6   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   2   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   4   |   |   |   |
.  |   |   |   8   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   4   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   6   |   |   |
.  |   |   |   |   |   |   |   2   |   |   |   |   |
.  |   |  10   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   8   |   |
.  |   |   |   |   6   |   |   |   |   |   |   |   |
.  |  12   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   4   |   |   |   |
.  |   |   |   |   |   |   2   |   |   |   |   |   |
.  |   |   |   8   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   |  10   |
. 14   |   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   4   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   6   |   |   |
.  |   |   |   |   |   |   |   2   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |   |   |   |   |   |   |
.  |   |   |   |   |   |   |1/2|   |   |   |   |   |
.  |   |   |   |   |   |           |   |   |   |   |
.  |   |   |   |   |   |----3/2----|   |   |   |   |
.  |   |   |   |   |                   |   |   |   |
.  |   |   |   |   |--------5/2--------|   |   |   |
.  |   |   |   |                           |   |   |
.  |   |   |   |------------7/2------------|   |   |
.  |   |   |                                   |   |
.  |   |   |----------------9/2----------------|   |
.  |   |                                       |   |
.  |   |-------------------11/2--------------------|
.  |
.  |-----------------------13/2------------------------
.
For another view of the model see the example section of A212122, second part.
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, 12, 4, 2;
8, 10, 14, 4, 6, 2;

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Chap. 2.

HyperPhysics, G.S.U., Shell Model of Nucleus
W. E. Meyerhof, Energy Levels of Nucleons in a Smoothly-Varying Potential Well, MIT OpenCourseWare

Partial sums give A213374. Other versions are A162630, A212012, A212122, A213362.

Cf. A213371, A213373.

a(n) = 2*A213371(n).

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.

A355564 Triangle read by rows: T(n,k) = n(1+2k) - k*(1+k), n >= 1, 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 4, 3, 7, 9, 4, 10, 14, 16, 5, 13, 19, 23, 25, 6, 16, 24, 30, 34, 36, 7, 19, 29, 37, 43, 47, 49, 8, 22, 34, 44, 52, 58, 62, 64, 9, 25, 39, 51, 61, 69, 75, 79, 81, 10, 28, 44, 58, 70, 80, 88, 94, 98, 100, 11, 31, 49, 65, 79, 91, 101, 109, 115, 119, 121

Offset: 1

Lucas B. Vieira, Jul 07 2022

T(n,k) is the number of potentially nonzero elements in a square, n X n band matrix of bandwidth k, i.e., the number of matrix elements (i,j) for which |i-j| <= k.

T(n,0) = n, as a zero-bandwidth matrix is diagonal, and T(n,n-1) = n^2, as the band encompasses the entire matrix.

			Triangle starts:
  1;
  2,  4;
  3,  7,  9;
  4, 10, 14, 16;
  5, 13, 19, 23, 25;
  6, 16, 24, 30, 34, 36;
  7, 19, 29, 37, 43, 47, 49;
  ...
Example: For n = 6 and k = 2, we have a band matrix of the form
    [. . . 0 0 0]
    [. . . . 0 0]
    [. . . . . 0]
    [0 . . . . .],
    [0 0 . . . .]
    [0 0 0 . . .]
where dots represent the entries which may have nonzero values. The number of such entries is T(6,2) = 24.

The sequence is complementary to A095832, i.e.: T(n,k) = n^2 - A095832(n,k).

Differences within a row T(n,k+1) - T(n,k) give A212012.

Flatten[Table[n (1 + 2 k) - k (1 + k), {n, 1, 10}, {k, 0, n - 1}]]

cp's OEIS Frontend

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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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.

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A212013 Triangle read by rows: 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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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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A213362 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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A213372 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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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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A355564 Triangle read by rows: T(n,k) = n*(1+2*k) - k*(1+k), n >= 1, 0 <= k <= n-1.

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A355564 Triangle read by rows: T(n,k) = n(1+2k) - k*(1+k), n >= 1, 0 <= k <= n-1.