A128282 Regular symmetric triangle, read by rows, whose left half consists of the positive integers.

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 7, 8, 9, 8, 7, 10, 11, 12, 12, 11, 10, 13, 14, 15, 16, 15, 14, 13, 17, 18, 19, 20, 20, 19, 18, 17, 21, 22, 23, 24, 25, 24, 23, 22, 21, 26, 27, 28, 29, 30, 30, 29, 28, 27, 26, 31, 32, 33, 34, 35, 36, 35, 34, 33, 32, 31, 37, 38, 39, 40, 41, 42, 42, 41, 40, 39, 38, 37

Offset: 0

Author

Philippe Deléham, May 03 2007

Keywords

• nonn
• tabl

Comments

Left half triangle is A000027 (positive integers) (compare with example triangle):

1;

2;

3, 4;

5, 6;

7, 8, 9;

10, 11, 12;

13, 14, 15, 16;

17, 18, 19, 20;

...

Examples

			Triangle begins:
   1;
   2,  2;
   3,  4,  3;
   5,  6,  6,  5;
   7,  8,  9,  8,  7;
  10, 11, 12, 12, 11, 10;
  13, 14, 15, 16, 15, 14, 13;
  17, 18, 19, 20, 20, 19, 18, 17;
  ...
		

Links

• Jianrui Xie, On Symmetric Invertible Binary Pairing Functions, arXiv:2105.10752 [math.CO], 2021. See (6) p. 3 and p. 5

Crossrefs

Cf. A000027, A000290, A033638 (1st column and right diagonal).

Programs

Maple

A := proc(n,k) ## n >= 0 and k = 0 .. n
    1+(1/4)*n*(n+1)+min(k, n-k)+(1/2)*ceil((1/2)*n)
end proc: # Yu-Sheng Chang, May 25 2020

Mathematica

T[n_,k_]:=1+n*(n+1)/4+Min[k,n-k]+Ceiling[n/2]/2;Table[T[n,k],{n,0,11},{k,0,n}]//Flatten (* James C. McMahon, Jan 06 2025 *)

Formula

T(n,k) = T(n,n-k).

T(2*n,n) = (n+1)^2 = A000290(n+1).

T(n,0) = T(n,n) = A033638(n+1).

From Yu-Sheng Chang, May 25 2020: (Start)

O.g.f.: F(z,v) = (z/((-z+1)^3*(z+1)) - v^2*z/((-v*z+1)^3*(v*z+1)))/(1-v) + 1/((-z+1)*(-v*z+1)*(-v*z^2+1)).

T(n,k) = [v^k] (1/8)*(1-v^(n+1))*(2*(n+1)^2 - 1 - (-1)^n)/(1-v) + (v^(2+n) + (1/2*((sqrt(v)-1)^2*(-1)^n - (sqrt(v)+1)^2))*v^((1/2)*n + 1/2) + 1)/(1-v)^2.

T(n,k) = 1 + (1/4)*n*(n+1) + min(k, n-k) + (1/2)*ceiling((1/2)*n). (End)

T(n,k) = ((n+k-1)^2 - ((n+k-1) mod 2))/4 + min(n,k) for n and k >= 1, as an array. See Xie. - Michel Marcus, May 25 2021

Extensions

Name edited by Michel Marcus, May 25 2021