A273751 Triangle of the natural numbers written by decreasing antidiagonals.

1, 2, 3, 4, 5, 7, 6, 8, 10, 13, 9, 11, 14, 17, 21, 12, 15, 18, 22, 26, 31, 16, 19, 23, 27, 32, 37, 43, 20, 24, 28, 33, 38, 44, 50, 57, 25, 29, 34, 39, 45, 51, 58, 65, 73, 30, 35, 40, 46, 52, 59, 66, 74, 82, 91, 36, 41, 47, 53, 60, 67, 75, 83, 92, 101, 111

Offset: 1

Author

Paul Curtz, May 30 2016

Keywords

• nonn

Comments

A permutation of the natural numbers.

a(n) and A091995(n) are different at the ninth term.

Antidiagonal sums: 1, 2, 7, 11, ... = A235355(n+1). Same idea.

Row sums: 1, 5, 16, 37, 72, 124, 197, 294, ... = 7*n^3/12 -n^2/8 +5*n/12 +1/16 -1/16*(-1)^n with g.f. x*(1+2*x+3*x^2+x^3) / ( (1+x)*(x-1)^4 ). The third difference is of period 2: repeat [3, 4].

Indicates the order in which electrons fill the different atomic orbitals (s,p,d,f,g,h). - Alexander Goebel, May 12 2020

Examples

			1,
2,   3,
4,   5,  7,
6,   8, 10, 13,
9,  11, 14, 17, 21,
12, 15, 18, 22, 26, 31,
16, 19, 23, 27, 32, 37, 43,
20, etc.
		

Links

• Wikipedia, Atomic orbital
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A002061 (right diagonal), A002620 (first column), A033638, A091995, A234305 (antidiagonals of the triangle).

Programs

Maple

A273751 := proc(n,k)
    option remember;
    if k = n then
        A002061(n) ;
    elif k > n or k < 0 then
        0;
    elif k = n-1 then
        procname(n-1,k)+k ;
    else
        procname(n-1,k+1)+1 ;
    end if;
end proc: # R. J. Mathar, Jun 13 2016

Mathematica

T[n_, k_] := T[n, k] = Which[k == n, n(n-1) + 1, k == n-1, (n-1)^2 + 1, k == 1, n + T[n-2, 1], 1 < k < n-1, T[n-1, k+1] + 1,True, 0];
Table[T[n, k], {n, 12}, {k, n}] // Flatten (* Jean-François Alcover, Jun 10 2016 *)

Formula

T(n, k) = (2 * (n+k)^2 + 7 + (-1)^(n-k)) / 8 - k. - Werner Schulte, Sep 27 2024

G.f.: x*y*(1 + x^4*y + x^2*(y - 1)*y + x^5*y^2 - x^3*y*(y + 2))/((1 - x)^3*(1 + x)*(1 - x*y)^3). - Stefano Spezia, Sep 28 2024