login
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.
27
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
1,2
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
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
Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.
Other versions are A004736, A212121, A213361, A213371.
Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.
Sequence in context: A087295 A175344 A056951 * A316715 A130212 A133737
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Aug 08 2007
STATUS
approved