A162630 -id:A162630 - cp's OEIS frontend

A130517 Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n}, again in steps of 2.

1, 2, 1, 3, 1, 2, 4, 2, 1, 3, 5, 3, 1, 2, 4, 6, 4, 2, 1, 3, 5, 7, 5, 3, 1, 2, 4, 6, 8, 6, 4, 2, 1, 3, 5, 7, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 13, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14, 12, 10

Offset: 1

Omar E. Pol, Aug 08 2007

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.
Row n lists a permutation of the first n positive integers.
If n is odd then row n lists the first (n+1)/2 odd numbers in decreasing order together with the first (n-1)/2 positive even numbers.
If n is even then row n lists the first n/2 even numbers in decreasing order together with the first n/2 odd numbers.
Row n >= 2, with its floor(n/2) last numbers taken as negative, lists the n different eigenvalues (in decreasing order) of the odd graph O(n). The odd graph O(n) has the (n-1)-subsets of a (2*n-1)-set as vertices, with two (n-1)-subsets adjacent if and only if they are disjoint. For example, O(3) is isomorphic to the Petersen graph. - Miquel A. Fiol, Apr 07 2024

			A geometric model of the atomic nucleus:
......-------------------------------------------------
......|...-----------------------------------------...|
......|...|...---------------------------------...|...|
......|...|...|...-------------------------...|...|...|
......|...|...|...|...-----------------...|...|...|...|
......|...|...|...|...|...---------...|...|...|...|...|
......|...|...|...|...|...|...-...|...|...|...|...|...|
......i...h...g...f...d...p...s...p...d...f...g...h...i
......|...|...|...|...|...|.......|...|...|...|...|...|
......|...|...|...|...|.......1.......|...|...|...|...|
......|...|...|...|.......2.......1.......|...|...|...|
......|...|...|.......3.......1.......2.......|...|...|
......|...|.......4.......2.......1.......3.......|...|
......|.......5.......3.......1.......2.......4.......|
..........6.......4.......2.......1.......3.......5....
......7.......5.......3.......1.......2.......4.......6
.......................................................
...13/2.11/2.9/2.7/2.5/2.3/2.1/2.1/2.3/2.5/2.7/2.9/2.11/2
......|...|...|...|...|...|...|...|...|...|...|...|...|
......|...|...|...|...|...|...-----...|...|...|...|...|
......|...|...|...|...|...-------------...|...|...|...|
......|...|...|...|...---------------------...|...|...|
......|...|...|...-----------------------------...|...|
......|...|...-------------------------------------...|
......|...---------------------------------------------
.
Triangle begins:
   1;
   2, 1;
   3, 1, 2;
   4, 2, 1, 3;
   5, 3, 1, 2, 4;
   6, 4, 2, 1, 3, 5;
   7, 5, 3, 1, 2, 4, 6;
   8, 6, 4, 2, 1, 3, 5, 7;
   9, 7, 5, 3, 1, 2, 4, 6, 8;
  10, 8, 6, 4, 2, 1, 3, 5, 7, 9;
  ...
Also:
                     1;
                   2,  1;
                 3,  1,  2;
               4,  2,  1,  3;
             5,  3,  1,  2,  4;
           6,  4,  2,  1,  3,  5;
         7,  5,  3,  1,  2,  4,  6;
       8,  6,  4,  2,  1,  3,  5,  7;
     9,  7,  5,  3,  1,  2,  4,  6,  8;
  10,  8,  6,  4,  2,  1,  3,  5,  7,  9;
  ...
In this view each column contains the same numbers.
From _Miquel A. Fiol_, Apr 07 2024: (Start)
Eigenvalues of the odd graphs O(n) for n=2..10:
   2, -1;
   3,  1, -2;
   4,  2, -1, -3;
   5,  3,  1, -2, -4;
   6,  4,  2, -1, -3, -5;
   7,  5,  3,  1, -2, -4, -6;
   8,  6,  4,  2, -1, -3, -5, -7;
   9,  7,  5,  3,  1, -2, -4, -6, -8;
  10,  8,  6,  4,  2, -1, -3, -5, -7, -9;
... (End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
N. Bigss, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Odd graph

Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.
Cf. A130556, A130598, A130602.
Other versions are A004736, A212121, A213361, A213371.
Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.

a130517 n k = a130517_tabl !! (n-1) !! (k-1)
a130517_row n = a130517_tabl !! (n-1)
a130517_tabl = iterate (\row -> (head row + 1) : reverse row) [1]
-- Reinhard Zumkeller, Dec 03 2012

A130517 := proc(n,k)
     if k <= (n+1)/2 then
        n-2*(k-1) ;
    else
        1-n+2*(k-1) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2012

t[n_, 1] := n; t[n_, n_] := n-1; t[n_, k_] := Abs[2*k-n - If[2*k <= n+1, 2, 1]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013, from abs(A056951) *)

a130517_row(n) = my(v=vector(n), s=1, n1=0, n2=n+1); forstep(k=n, 1,-1, s=-s; if(s>0, n2--; v[n2]=k, n1++; v[n1]=k)); v \\ Hugo Pfoertner, Aug 26 2024

a(n) = A162630(n)/2. - Omar E. Pol, Sep 02 2012
T(1,1) = 1; for n > 1: T(n,1) = T(n-1,1)+1 and T(n,k) = T(n-1,n-k+1), 1 < k <= n. - Reinhard Zumkeller, Dec 03 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = |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*n - t^2 - 2*t - 3| + floor((2*n - t^2 - t)/(t+3)) where t = floor((-1+sqrt(8*n-7))/2). (End)

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.

Original entry on oeis.org

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).

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).

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.

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).

A072785 Differences between A072781 and A072738.

Original entry on oeis.org

0, 0, -1, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Jun 12 2002

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

Cf. A130517, A162630.

(define (A072785 n) (- (A072781 n) (A072738 n)))

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

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

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

One way to generalize the magic number sequence in A018226.

A377137 Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd; ; see Comments.

Original entry on oeis.org

1, 4, 2, 3, 6, 5, 12, 10, 8, 7, 9, 11, 15, 13, 14, 24, 22, 20, 18, 16, 17, 19, 21, 23, 28, 26, 25, 27, 40, 38, 36, 34, 32, 30, 29, 31, 33, 35, 37, 39, 45, 43, 41, 42, 44, 60, 58, 56, 54, 52, 50, 48, 46, 47, 49, 51, 53, 55, 57, 59, 66, 64, 62, 61, 63, 65, 84, 82, 80, 78, 76, 74, 72, 70, 68, 67, 69, 71, 73, 75, 77, 79, 81, 83, 91, 89, 87, 85, 86

Offset: 1

Boris Putievskiy, Oct 17 2024

Row n has length A064455(n). The sequence A064455 is non-monotonic.
The array consists of two triangular arrays alternating row by row.
For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).
These permutations are generated by the algorithm described A130517.
The sequence is an intra-block permutation of the positive integers.

			Array begins:
     k =  1   2   3   4   5   6
  n=1:    1;
  n=2:    4,  2,  3;
  n=3:    6,  5;
  n=4:   12, 10,  8,  7,  9, 11;
The triangular arrays alternate by row: n=1 and n=3 comprise one, and n=2 and n=4 comprise the other.
Subtracting (n^2 - 1)/2 if n is odd from each term in row n produces a permutation of 1 .. (n+1)/2. Subtracting (n^2 - n)/2 if n is even from each term in row n produces a permutation of 1 .. 3n/2:
  1,
  3, 1, 2,
  2, 1,
  6, 4, 2, 1, 3, 5,
  ...

Boris Putievskiy, Table of n, a(n) for n = 1..9940
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A064455, A001844, A130517, A130883, A162630, A265225, A375602, A376180.

a[n_]:=Module[{L,R, P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]]; R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2]; P[3]=2; P= Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]]; Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; Result=Res; Result] Nmax= 12; Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + B(L(n)-1), where L(n) = ceiling(x(n)), x(n) is largest real root of the equation B(x) - n = 0. B(n) = (n+1)*(2*n-(-1)^n+5)/4 = A265225(n). P(n) = A162630(n)/2.
Array T(n,k) (see Example):
T(n, k) = P(n, k) + (n^2 - n)/2 if n is even, T(n, k) = P(n, k) + (n^2 - 1)/2 if n is odd, T(n, k) = P(n, k) +  A265225(n-1). P(n, k) = |2k - 3n / 2 - 2| if n is even and if 2k <= 3n / 2 + 1, P(n, k) = |2k - 3n / 2 - 1| if n is even and if 2k > 3n / 2 + 1. P(n, k) = |2k - (n + 1) / 2 - 2| if n is odd and if 2k <=  (n + 1) / 2 + 1, P(n, k) = |2k - (n + 1) / 2 - 1| if n is odd and if 2k > (n + 1) / 2 + 1. There are several special cases: P(n, 1) = 3n/2 if n is even, P(n, 1) = (n+1)/2 if n is odd. P(2, 2) = 1. P(n, n) = n/2 - 1 if n is even, P(n, n) = (n-3)/2 if n is odd.

cp's OEIS Frontend

A130517 Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n}, again in steps of 2.

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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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Mathematica

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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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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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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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A072785 Differences between A072781 and A072738.

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