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

Offset

1,2

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

Links

Table of n, a(n) for n=1..57.

Wikipedia, Atomic orbital

Index entries for sequences that are permutations of the natural numbers

Example

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.

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 *)

Crossrefs

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

Sequence in context: A283734 A363162 A293052 * A056017 A091995 A343150

Adjacent sequences: A273748 A273749 A273750 * A273752 A273753 A273754

Keyword

nonn

Author

Paul Curtz, May 30 2016

Status

approved