A130556 -id:A130556 - 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)

A130598 A shell geometric model of the nucleus. The location of the magic numbers. A triangle.

Original entry on oeis.org

10, 1111, 10, 111111, 11, 1110, 11111110, 1111, 11, 111111, 1111111110, 111111, 11, 1111, 11111111, 111111111110, 11111111, 1111, 11, 111111, 1111111111, 11111111111110, 1111111111, 111111, 11, 1111, 11111111, 111111111111

Offset: 1

Omar E. Pol, Aug 10 2007

The magic numbers of the atomic nucleus: 2, 8, 20, 28, 50, 82, 126. 0 is the location of a magic number. 10 or 11 is equal to 2 protons (or neutrons). 1110 or 1111 is equal to 2+2 protons (or neutrons). 111111 is equal to 2+2+2 protons (or neutrons)... The 2D model is a triangle and a spiral. The 3D model is a double tetrahedron and a double spiral.

			......|----------------------- h -------------------|.....
......|.....|----------------- g --------------|....|.....
......|.....|.....|----------- f ---------|....|....|.....
......|.....|.....|....|------ d -----|...|....|....|.....
......|.....|.....|....|...|-- p -|...|...|....|....|.....
......|.....|.....|....|...|.. s .|...|...|....|....|.....
......|.....|.....|....|...|......|...|...|....|....|.....
......|.....|.....|....|......10......|...|....|....|.....
......|.....|.....|......1111....10.......|....|....|.....
......|.....|.......111111....11....1110.......|....|.....
......|........11111110..1111....11....111111.......|.....
.......1111111110...111111....11....1111...11111111.|.....
111111111110...11111111..1111....11....111111...1111111111
......|.....|.....|....|...|..|...|...|...|....|....|.....
......|.....|.....|....|...|..|1/2|...|...|....|....|.....
......|.....|.....|....|...|-- 3/2 ---|...|....|....|.....
......|.....|.....|....|------ 5/2 -------|....|....|.....
......|.....|.....|----------- 7/2 ------------|....|.....
......|.....|----------------- 9/2 -----------------|.....
......|---------------------- 11/2 -----------------------

Cf. A056951, A130517, A130556, A130602.

A130602 A shell geometric model of the atomic nucleus.

Original entry on oeis.org

11, 1111, 11, 111111, 11, 1111, 11111111, 1111, 11, 111111, 1111111111, 111111, 11, 1111, 11111111, 111111111111, 11111111, 1111, 11, 111111, 1111111111, 11111111111111, 1111111111, 111111, 11, 1111, 11111111, 111111111111

Offset: 1

Omar E. Pol, Aug 10 2007, Aug 12 2007

nonn

11 is equal to 2 protons. 1111 is equal 2+2 protons. 111111 is equal 2+2+2 protons...

Repunit numbers represent the subshells.

			See the model in the entry: A130517, A130556.

Cf. A056951, A130517, A130556.

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.

Original entry on oeis.org

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

A162522 First differences of magic numbers A018226.

Original entry on oeis.org

6, 12, 8, 22, 32, 44

Offset: 1

Omar E. Pol, Jul 06 2009

Sequence related to atomic nuclei.

Cf. A018226, A130517, A130556, A130598, A130602, A162521, A162523, A162524, A162525.

A162521 Magic numbers A018226 divided by 2.

Original entry on oeis.org

1, 4, 10, 14, 25, 41, 63

Offset: 1

Omar E. Pol, Jul 06 2009

Sequence related to atomic nuclei.

Cf. A018226, A130517, A130556, A130598, A130602, A162522, A162523, A162524, A162525.

A162523 First differences of magic numbers A018226, divided by 2.

Original entry on oeis.org

3, 6, 4, 11, 16, 22

Offset: 1

Omar E. Pol, Jul 06 2009

Sequence related to atomic nucleus.

Cf. A018226, A130517, A130556, A130598, A130602, A162521, A162522, A162524, A162525.

a(n) = A162522(n)/2.

A162524 Partial sums of magic numbers A018226.

Original entry on oeis.org

2, 10, 30, 58, 108, 190, 316

Offset: 1

Omar E. Pol, Jul 06 2009

Sequence related to atomic nuclei.

Cf. A018226, A130517, A130556, A130598, A130602, A162521, A162522, A162523, A162525.

Edited by Omar E. Pol, Jul 16 2009

A162525 Partial sums of magic numbers A018226, divided by 2.

Original entry on oeis.org

1, 5, 15, 29, 54, 95, 158

Offset: 1

Omar E. Pol, Jul 06 2009

Sequence related to atomic nuclei.

Cf. A018226, A130517, A130556, A130598, A130602, A162521, A162522, A162523, A162524.

a(n) = A162524(n)/2.

A162518 Characteristic function of magic numbers A018226: 1 if n is a magic number else 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jul 07 2009

nonn

Sequence related to atomic nuclei.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A018226, A130517, A130556, A130598, A130602, A162519, A162521, A162522, A162523, A162524, A162525.

A162518(n) = if(n<1, 0, polcoeff( x^126+x^82+x^50+x^28+x^20+x^8+x^2, n)); \\ Antti Karttunen, Dec 24 2018, after code in A089011

Data section extended up to a(126) by Antti Karttunen, Dec 24 2018

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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A130598 A shell geometric model of the nucleus. The location of the magic numbers. A triangle.

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A130602 A shell geometric model of the atomic nucleus.

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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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A162522 First differences of magic numbers A018226.

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A162521 Magic numbers A018226 divided by 2.

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A162523 First differences of magic numbers A018226, divided by 2.

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A162524 Partial sums of magic numbers A018226.

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A162525 Partial sums of magic numbers A018226, divided by 2.

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A162518 Characteristic function of magic numbers A018226: 1 if n is a magic number else 0.

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