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%I #40 Nov 29 2023 06:58:18
%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.
%H Reinhard Zumkeller, <a href="/A130517/b130517.txt">Rows n = 1..120 of triangle, flattened</a>
%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.
%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*t-2*t-3| + floor((2*n-t*t-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.
%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
%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).
%K nonn,tabl,easy
%O 1,2
%A _Omar E. Pol_, Aug 08 2007
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