cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A213361 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, 3, 4, 1, 2, 6, 4, 2, 7, 5, 1, 3, 5, 6, 8, 3, 1, 4, 2
Offset: 1

Views

Author

Omar E. Pol, Jun 23 2012

Keywords

Comments

For another version see A212121. First differs from A212121 at a(12).

Examples

			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   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   3   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   4   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   2   |   |   |   |   |
.|   |   6   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   4   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   2   |   |   |   |   |   |   |   |
.|   7   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   5   |   |
.|   |   |   |   |   |   |   |   1   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   3   |   |   |   |
.|   |   |   5   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   6   |
.8   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   3   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   1   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   4   |   |   |
.|   |   |   |   |   |   |   |   |   2   |   |   |   |   |
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.|   |   |   |   |   |   |   |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;
3, 4, 1, 2, 6;
4, 2, 7, 5, 1, 3;
5, 6, 8, 3, 1, 4, 2;

Crossrefs

Partial sums give A213363.
Cf. A018226, A130517, A212121-A212124, A213362, A213364.

Formula

a(n) = A213362(n)/2.

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

Views

Author

Omar E. Pol, Jun 23 2012

Keywords

Comments

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

Examples

			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;
...

References

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

Links

Crossrefs

Partial sums give A213364. Other versions are A162630, A212012, A212122, A213372.
Cf. A018226, A130517, A213361, A213363.

Formula

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

A213374 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, 76, 80, 82, 90, 100, 114, 118, 124, 126
Offset: 1

Views

Author

Omar E. Pol, Jul 16 2012

Keywords

Comments

First differs from A212124 at a(14). For more information see A213372.

Examples

			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;
58,  64,  76,  80,  82;
90, 100, 114, 118, 124, 126;
...
First seven terms of right border give the "magic numbers" A018226.

Links

Crossrefs

Partial sums of A213372. Other versions are A210984, A212014, A212124, A213364.
Cf. A018226, A213371, A213373.

Formula

a(n) = 2*A213373(n).

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

Views

Author

Omar E. Pol, Jun 23 2012

Keywords

Comments

First differs from A212123 at a(12).

Examples

			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;
...

Crossrefs

Partial sums of A213361. Other versions are A210983, A212013, A213373.
Cf. A018226, A212121-A212124, A213362, A213364.

Formula

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

Views

Author

Omar E. Pol, Jul 16 2012

Keywords

Comments

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

Examples

			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;
...

Links

Crossrefs

Partial sums of A213371. Other versions are A210983, A212013, A212123, A213363.
Cf. A018226, A213372, A213374.

Formula

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

Views

Author

Omar E. Pol, Jul 14 2012

Keywords

Examples

			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.

Crossrefs

Partial sums of A162630. Other versions are A212014, A212124, A213364, A213374.
Cf. A018226, A033547, A007290, A130517, A210983.

Formula

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

A046939 "Magic numbers" for the number of neutrons in nucleus.

Original entry on oeis.org

2, 8, 14, 20, 50, 82, 126, 184
Offset: 1

Views

Author

Jud McCranie

Keywords

References

  • McGraw-Hill Encyclopedia of Science and Technology

Crossrefs

Cf. A046940, A018226.

A046940 "Magic numbers" for the number of protons in nucleus.

Original entry on oeis.org

2, 8, 14, 20, 50, 82, 114, 164
Offset: 1

Views

Author

Jud McCranie

Keywords

References

  • McGraw-Hill Encyclopedia of Science and Technology

Crossrefs

Cf. A046939, A018226.

A162626 If 0 <= n <= 3 then a(n) = n(n+1)(n+2)/3, if n >= 4 then a(n) = n(n^2+5)/3.

Original entry on oeis.org

0, 2, 8, 20, 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
Offset: 0

Views

Author

Omar E. Pol, Jul 07 2009, Jul 13 2009

Keywords

Comments

One way to generalize the magic number sequence in A018226.
See also A130598 and A162630.

Crossrefs

Cf. A007290, A018226, A033547, A130598, A162518, A162519, A162521, A162522, A162523, A162524, A162525, A162630.

Formula

From Daniel Forgues, May 03 2011: (Start)
If 0 <= n <= 3 then a(n) = 2 T_n, otherwise a(n) = 2 (T_n - t_{n-1}), where T_n is the n-th tetrahedral number, t_n the n-th triangular number.
G.f.: (2*x*(1 - 6*x^3 + 14*x^4 - 11*x^5 + 3*x^6))/(1 - x)^4, n >= 0.
(End)
a(n) = n*(n^2+5)/3 + (4*n-6)*A171386(n). - Omar E. Pol, Aug 14 2013

Extensions

Edited by N. J. A. Sloane, Jul 18 2009

A210842 Number of states in the n-th shell of the nuclear shell model.

Original entry on oeis.org

2, 6, 12, 8, 22, 32, 44, 58
Offset: 1

Views

Author

Omar E. Pol, Jun 04 2012

Keywords

Comments

Partial sums of the first seven terms give the "magic numbers" A018226.
Also the partial sums of the first eight terms give the positive first eight terms of A162626 (and possibly more).

References

  • M. Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure, J. Wiley and Sons, Inc. (1955).
  • I. Talmi, Simple Models of Complex Nuclei, Hardwood Academic Publishers (1993).

Crossrefs

Row sums of triangle A212122.
Cf. A018226, A130517, A162522, A162626, A162630, A212121, A212123, A212124, A213361-A213364.
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