A210983 -id:A210983 - 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

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

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.

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.

A210984 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, 34, 40, 50, 56, 58, 62, 70, 82, 90, 94, 96, 102, 112, 126, 136, 142, 144, 148, 156, 168, 184, 196, 204, 208, 210, 216, 226, 240, 258, 272, 282, 288, 290, 294, 302, 314, 330, 350, 366, 378, 386, 390, 392, 398, 408, 422, 440, 462

Offset: 1

Omar E. Pol, Jul 14 2012

			Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus the sequence begins:
2;
6,     8;
14,   16,  20;
28,   32,  34,  40;
50,   56,  58,  62,  70;
82,   90,  94,  96, 102, 112;
126, 136, 142, 144, 148, 156, 168;
184, 196, 204, 208, 210, 216, 226, 240;
258, 272, 282, 288, 290, 294, 302, 314, 330;
350, 366, 378, 386, 390, 392, 398, 408, 422, 440;
...
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,   16,  20;
28;
32,   34,  40;  50;
56,   58,  62,  70;  82;
90,   94,  96, 102, 112; 126;
136, 142, 144, 148, 156, 168; 184;
196, 204, 208, 210, 216, 226, 240; 258;
272, 282, 288, 290, 294, 302, 314, 330, 350;
366, 378, 386, 390, 392, 398, 408, 422, 440, 462;
...
First seven terms of right border give the "magic numbers" A018226.

Partial sums of A162630. Other versions are A212014, A212124, A213364, A213374.

Cf. A018226, A033547, A007290, A130517, A210983.

a(n) = 2*A210983(n).

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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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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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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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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A210984 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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