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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. 25
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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO].

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*t-2*t-3| + floor((2*n-t*t-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.

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-Fran├žois 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

CROSSREFS

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).

Sequence in context: A087295 A175344 A056951 * A316715 A130212 A133737

Adjacent sequences:  A130514 A130515 A130516 * A130518 A130519 A130520

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol, Aug 08 2007

STATUS

approved

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Last modified April 20 04:42 EDT 2019. Contains 322294 sequences. (Running on oeis4.)