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

Offset

0,2

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;

...

Links

Table of n, a(n) for n=0..73.

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

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

Example

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;
  ...

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

Crossrefs

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

Sequence in context: A355029 A290735 A165634 * A146985 A132993 A106408

Adjacent sequences: A128279 A128280 A128281 * A128283 A128284 A128285

Keyword

nonn,tabl

Author

Philippe Deléham, May 03 2007

Extensions

Name edited by Michel Marcus, May 25 2021

Status

approved