OFFSET
1,2
COMMENTS
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
LINKS
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
FORMULA
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)
EXAMPLE
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)
MAPLE
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
MATHEMATICA
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) *)
PROG
(Haskell)
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
(PARI) 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
CROSSREFS
KEYWORD
AUTHOR
Omar E. Pol, Aug 08 2007
STATUS
approved