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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.
29

%I #59 Aug 26 2024 09:55:24

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C Row n lists a permutation of the first n positive integers.

%C 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.

%C 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.

%C 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

%H Reinhard Zumkeller, <a href="/A130517/b130517.txt">Rows n = 1..120 of triangle, flattened</a>

%H N. Bigss, <a href="https://doi.org/10.1017/CBO9780511608704">Algebraic Graph Theory</a>, Cambridge Univ. Press, Cambridge, 1974.

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a>, 2012, arXiv:1212.2732 [math.CO], 2012.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd graph</a>

%F a(n) = A162630(n)/2. - _Omar E. Pol_, Sep 02 2012

%F 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

%F From _Boris Putievskiy_, Jan 16 2013: (Start)

%F a(n) = |2*A000027(n) - A003056(n)^2 - 2*A003056(n) - 3| + floor((2*A000027(n) - A003056(n)^2 - A003056(n))/(A003056(n)+3)).

%F 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)

%e A geometric model of the atomic nucleus:

%e ......-------------------------------------------------

%e ......|...-----------------------------------------...|

%e ......|...|...---------------------------------...|...|

%e ......|...|...|...-------------------------...|...|...|

%e ......|...|...|...|...-----------------...|...|...|...|

%e ......|...|...|...|...|...---------...|...|...|...|...|

%e ......|...|...|...|...|...|...-...|...|...|...|...|...|

%e ......i...h...g...f...d...p...s...p...d...f...g...h...i

%e ......|...|...|...|...|...|.......|...|...|...|...|...|

%e ......|...|...|...|...|.......1.......|...|...|...|...|

%e ......|...|...|...|.......2.......1.......|...|...|...|

%e ......|...|...|.......3.......1.......2.......|...|...|

%e ......|...|.......4.......2.......1.......3.......|...|

%e ......|.......5.......3.......1.......2.......4.......|

%e ..........6.......4.......2.......1.......3.......5....

%e ......7.......5.......3.......1.......2.......4.......6

%e .......................................................

%e ...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

%e ......|...|...|...|...|...|...|...|...|...|...|...|...|

%e ......|...|...|...|...|...|...-----...|...|...|...|...|

%e ......|...|...|...|...|...-------------...|...|...|...|

%e ......|...|...|...|...---------------------...|...|...|

%e ......|...|...|...-----------------------------...|...|

%e ......|...|...-------------------------------------...|

%e ......|...---------------------------------------------

%e .

%e Triangle begins:

%e 1;

%e 2, 1;

%e 3, 1, 2;

%e 4, 2, 1, 3;

%e 5, 3, 1, 2, 4;

%e 6, 4, 2, 1, 3, 5;

%e 7, 5, 3, 1, 2, 4, 6;

%e 8, 6, 4, 2, 1, 3, 5, 7;

%e 9, 7, 5, 3, 1, 2, 4, 6, 8;

%e 10, 8, 6, 4, 2, 1, 3, 5, 7, 9;

%e ...

%e Also:

%e 1;

%e 2, 1;

%e 3, 1, 2;

%e 4, 2, 1, 3;

%e 5, 3, 1, 2, 4;

%e 6, 4, 2, 1, 3, 5;

%e 7, 5, 3, 1, 2, 4, 6;

%e 8, 6, 4, 2, 1, 3, 5, 7;

%e 9, 7, 5, 3, 1, 2, 4, 6, 8;

%e 10, 8, 6, 4, 2, 1, 3, 5, 7, 9;

%e ...

%e In this view each column contains the same numbers.

%e From _Miquel A. Fiol_, Apr 07 2024: (Start)

%e Eigenvalues of the odd graphs O(n) for n=2..10:

%e 2, -1;

%e 3, 1, -2;

%e 4, 2, -1, -3;

%e 5, 3, 1, -2, -4;

%e 6, 4, 2, -1, -3, -5;

%e 7, 5, 3, 1, -2, -4, -6;

%e 8, 6, 4, 2, -1, -3, -5, -7;

%e 9, 7, 5, 3, 1, -2, -4, -6, -8;

%e 10, 8, 6, 4, 2, -1, -3, -5, -7, -9;

%e ... (End)

%p A130517 := proc(n,k)

%p if k <= (n+1)/2 then

%p n-2*(k-1) ;

%p else

%p 1-n+2*(k-1) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jul 21 2012

%t 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) *)

%o (Haskell)

%o a130517 n k = a130517_tabl !! (n-1) !! (k-1)

%o a130517_row n = a130517_tabl !! (n-1)

%o a130517_tabl = iterate (\row -> (head row + 1) : reverse row) [1]

%o -- _Reinhard Zumkeller_, Dec 03 2012

%o (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

%Y Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.

%Y Cf. A130556, A130598, A130602.

%Y Other versions are A004736, A212121, A213361, A213371.

%Y Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.

%K nonn,tabl,easy

%O 1,2

%A _Omar E. Pol_, Aug 08 2007