%I #55 Sep 27 2024 16:14:09

%S 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,

%T 37,43,20,24,28,33,38,44,50,57,25,29,34,39,45,51,58,65,73,30,35,40,46,

%U 52,59,66,74,82,91,36,41,47,53,60,67,75,83,92,101,111

%N Triangle of the natural numbers written by decreasing antidiagonals.

%C A permutation of the natural numbers.

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

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

%C 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].

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Atomic_orbital#Orbital_energy">Atomic orbital</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F T(n, k) = (2 * (n+k)^2 + 7 + (-1)^(n-k)) / 8 - k. - _Werner Schulte_, Sep 27 2024.

%e 1,

%e 2, 3,

%e 4, 5, 7,

%e 6, 8, 10, 13,

%e 9, 11, 14, 17, 21,

%e 12, 15, 18, 22, 26, 31,

%e 16, 19, 23, 27, 32, 37, 43,

%e 20, etc.

%p A273751 := proc(n,k)

%p option remember;

%p if k = n then

%p A002061(n) ;

%p elif k > n or k < 0 then

%p 0;

%p elif k = n-1 then

%p procname(n-1,k)+k ;

%p else

%p procname(n-1,k+1)+1 ;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 13 2016

%t 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];

%t Table[T[n, k], {n, 12}, {k, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2016 *)

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

%K nonn,changed

%O 1,2

%A _Paul Curtz_, May 30 2016