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A273751 Triangle of the natural numbers written by decreasing antidiagonals.

%I A273751 #59 Sep 29 2024 07:49:37
%S A273751 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 A273751 37,43,20,24,28,33,38,44,50,57,25,29,34,39,45,51,58,65,73,30,35,40,46,
%U A273751 52,59,66,74,82,91,36,41,47,53,60,67,75,83,92,101,111
%N A273751 Triangle of the natural numbers written by decreasing antidiagonals.
%C A273751 A permutation of the natural numbers.
%C A273751 a(n) and A091995(n) are different at the ninth term.
%C A273751 Antidiagonal sums: 1, 2, 7, 11, ... = A235355(n+1). Same idea.
%C A273751 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 A273751 Indicates the order in which electrons fill the different atomic orbitals (s,p,d,f,g,h). - _Alexander Goebel_, May 12 2020
%H A273751 Wikipedia, <a href="https://en.wikipedia.org/wiki/Atomic_orbital#Orbital_energy">Atomic orbital</a>
%H A273751 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A273751 T(n, k) = (2 * (n+k)^2 + 7 + (-1)^(n-k)) / 8 - k. - _Werner Schulte_, Sep 27 2024
%F A273751 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
%e A273751 1,
%e A273751 2,   3,
%e A273751 4,   5,  7,
%e A273751 6,   8, 10, 13,
%e A273751 9,  11, 14, 17, 21,
%e A273751 12, 15, 18, 22, 26, 31,
%e A273751 16, 19, 23, 27, 32, 37, 43,
%e A273751 20, etc.
%p A273751 A273751 := proc(n,k)
%p A273751     option remember;
%p A273751     if k = n then
%p A273751         A002061(n) ;
%p A273751     elif k > n or k < 0 then
%p A273751         0;
%p A273751     elif k = n-1 then
%p A273751         procname(n-1,k)+k ;
%p A273751     else
%p A273751         procname(n-1,k+1)+1 ;
%p A273751     end if;
%p A273751 end proc: # _R. J. Mathar_, Jun 13 2016
%t A273751 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 A273751 Table[T[n, k], {n, 12}, {k, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2016 *)
%Y A273751 Cf. A002061 (right diagonal), A002620 (first column), A033638, A091995, A234305 (antidiagonals of the triangle).
%K A273751 nonn
%O A273751 1,2
%A A273751 _Paul Curtz_, May 30 2016