This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A130517 #59 Aug 26 2024 09:55:24
%S A130517 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 A130517 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 A130517 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 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.
%C A130517 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 A130517 Row n lists a permutation of the first n positive integers.
%C A130517 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 A130517 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 A130517 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 A130517 Reinhard Zumkeller, <a href="/A130517/b130517.txt">Rows n = 1..120 of triangle, flattened</a>
%H A130517 N. Bigss, <a href="https://doi.org/10.1017/CBO9780511608704">Algebraic Graph Theory</a>, Cambridge Univ. Press, Cambridge, 1974.
%H A130517 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 A130517 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd graph</a>
%F A130517 a(n) = A162630(n)/2. - _Omar E. Pol_, Sep 02 2012
%F A130517 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 A130517 From _Boris Putievskiy_, Jan 16 2013: (Start)
%F A130517 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 A130517 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 A130517 A geometric model of the atomic nucleus:
%e A130517 ......-------------------------------------------------
%e A130517 ......|...-----------------------------------------...|
%e A130517 ......|...|...---------------------------------...|...|
%e A130517 ......|...|...|...-------------------------...|...|...|
%e A130517 ......|...|...|...|...-----------------...|...|...|...|
%e A130517 ......|...|...|...|...|...---------...|...|...|...|...|
%e A130517 ......|...|...|...|...|...|...-...|...|...|...|...|...|
%e A130517 ......i...h...g...f...d...p...s...p...d...f...g...h...i
%e A130517 ......|...|...|...|...|...|.......|...|...|...|...|...|
%e A130517 ......|...|...|...|...|.......1.......|...|...|...|...|
%e A130517 ......|...|...|...|.......2.......1.......|...|...|...|
%e A130517 ......|...|...|.......3.......1.......2.......|...|...|
%e A130517 ......|...|.......4.......2.......1.......3.......|...|
%e A130517 ......|.......5.......3.......1.......2.......4.......|
%e A130517 ..........6.......4.......2.......1.......3.......5....
%e A130517 ......7.......5.......3.......1.......2.......4.......6
%e A130517 .......................................................
%e A130517 ...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 A130517 ......|...|...|...|...|...|...|...|...|...|...|...|...|
%e A130517 ......|...|...|...|...|...|...-----...|...|...|...|...|
%e A130517 ......|...|...|...|...|...-------------...|...|...|...|
%e A130517 ......|...|...|...|...---------------------...|...|...|
%e A130517 ......|...|...|...-----------------------------...|...|
%e A130517 ......|...|...-------------------------------------...|
%e A130517 ......|...---------------------------------------------
%e A130517 .
%e A130517 Triangle begins:
%e A130517 1;
%e A130517 2, 1;
%e A130517 3, 1, 2;
%e A130517 4, 2, 1, 3;
%e A130517 5, 3, 1, 2, 4;
%e A130517 6, 4, 2, 1, 3, 5;
%e A130517 7, 5, 3, 1, 2, 4, 6;
%e A130517 8, 6, 4, 2, 1, 3, 5, 7;
%e A130517 9, 7, 5, 3, 1, 2, 4, 6, 8;
%e A130517 10, 8, 6, 4, 2, 1, 3, 5, 7, 9;
%e A130517 ...
%e A130517 Also:
%e A130517 1;
%e A130517 2, 1;
%e A130517 3, 1, 2;
%e A130517 4, 2, 1, 3;
%e A130517 5, 3, 1, 2, 4;
%e A130517 6, 4, 2, 1, 3, 5;
%e A130517 7, 5, 3, 1, 2, 4, 6;
%e A130517 8, 6, 4, 2, 1, 3, 5, 7;
%e A130517 9, 7, 5, 3, 1, 2, 4, 6, 8;
%e A130517 10, 8, 6, 4, 2, 1, 3, 5, 7, 9;
%e A130517 ...
%e A130517 In this view each column contains the same numbers.
%e A130517 From _Miquel A. Fiol_, Apr 07 2024: (Start)
%e A130517 Eigenvalues of the odd graphs O(n) for n=2..10:
%e A130517 2, -1;
%e A130517 3, 1, -2;
%e A130517 4, 2, -1, -3;
%e A130517 5, 3, 1, -2, -4;
%e A130517 6, 4, 2, -1, -3, -5;
%e A130517 7, 5, 3, 1, -2, -4, -6;
%e A130517 8, 6, 4, 2, -1, -3, -5, -7;
%e A130517 9, 7, 5, 3, 1, -2, -4, -6, -8;
%e A130517 10, 8, 6, 4, 2, -1, -3, -5, -7, -9;
%e A130517 ... (End)
%p A130517 A130517 := proc(n,k)
%p A130517 if k <= (n+1)/2 then
%p A130517 n-2*(k-1) ;
%p A130517 else
%p A130517 1-n+2*(k-1) ;
%p A130517 end if;
%p A130517 end proc: # _R. J. Mathar_, Jul 21 2012
%t A130517 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 A130517 (Haskell)
%o A130517 a130517 n k = a130517_tabl !! (n-1) !! (k-1)
%o A130517 a130517_row n = a130517_tabl !! (n-1)
%o A130517 a130517_tabl = iterate (\row -> (head row + 1) : reverse row) [1]
%o A130517 -- _Reinhard Zumkeller_, Dec 03 2012
%o A130517 (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 A130517 Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.
%Y A130517 Cf. A130556, A130598, A130602.
%Y A130517 Other versions are A004736, A212121, A213361, A213371.
%Y A130517 Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.
%K A130517 nonn,tabl,easy
%O A130517 1,2
%A A130517 _Omar E. Pol_, Aug 08 2007