A235355 -id:A235355 - cp's OEIS frontend

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

Paul Curtz, May 30 2016

nonn

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

			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.

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

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

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 *)

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

A271647 Irregular triangle read by rows: the natural numbers from right to left.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 8, 7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, 17, 25, 24, 23, 22, 21, 30, 29, 28, 27, 26, 36, 35, 34, 33, 32, 31, 42, 41, 40, 39, 38, 37, 49, 48, 47, 46, 45, 44, 43, 56, 55, 54, 53, 52, 51, 50, 64, 63, 62, 61, 60, 59, 58, 57

Offset: 1

Paul Curtz, Apr 11 2016

A permutation of the natural numbers. Mentioned as d(n) in A269837.
Difference table:
1,  2,  4,  3,  6,  5,  9,  8,  7, 12, 11, 10, 16, 15, 14, 13, 20, 19, 18, ...
1,  2, -1,  3, -1,  4, -1, -1,  5, -1, -1,  6, -1, -1, -1,  7, -1, -1, -1, ...
1, -3,  4, -4,  5, -5,  0,  6, -6,  0,  7, -7,  0,  0,  8, -8,  0,  0,  9, ...
etc.

			Irregular triangle:
1,
2,
4,   3,
6,   5,
9,   8,  7,
12, 11, 10,
16, 15, 14, 13,
20, 19, 18, 17,
25, 24, 23, 22, 21,
30, 29, 28, 27, 26,
etc.

Cf. A000027, A002620, A024206, A033638, A075356, A235355, A269837, A271584

count:= 0:
for r from 1 to 20 do
  d:= ceil(r/2);
  for i from 0 to d-1 do A[r,i]:= count+ d-i od;
  count:= count+d;
od:
seq(seq(A[r,i],i=0..ceil(r/2)-1),r=1..20); # Robert Israel, Apr 11 2016

Table[Reverse@ Range[Floor[n/2]] + Floor[(n - 1)^2/4], {n, 16}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

With offset=0, a(n) = A271584(n) + A269837(n)

Empirical g.f. as triangle: (1-y*x^3+y^2*x^4-2*y*x^4-y^2*x^5+y*x^5+y^2*x^7)*x/((1+x)*(1-x)^3*(1-y*x^2)^3). - Robert Israel, Apr 11 2016

cp's OEIS Frontend

A273751 Triangle of the natural numbers written by decreasing antidiagonals.

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