A213371 -id:A213371 - 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)

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).

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

Omar E. Pol, Jul 16 2012

First differs from A212124 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:
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.

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

Partial sums of A213372. Other versions are A210984, A212014, A212124, A213364.

Cf. A018226, A213371, A213373.

a(n) = 2*A213373(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.

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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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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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.

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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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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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