A127793 Inverse of number triangle A(n,k) = 1/floor((n+2)/2) if k <= n <= 2k, 0 otherwise.

1, 0, 1, 0, -1, 2, 0, 1, -2, 2, 0, 0, 0, -2, 3, 0, -1, 2, 0, -3, 3, 0, 0, 0, 0, 0, -3, 4, 0, 1, -2, 2, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, -4, 5, 0, 0, 0, -2, 3, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 6

Offset

0,6

Comments

It is conjectured that the triangle is an integer triangle. The triangle and its inverse both appear to have row sums equal to the all 1's sequence.

The triangle is equivalent to the lower semi-matrix R = e_{1,1} + Sum_{i>=2} Sum_{p>=0} ( e_{2^p i, i} ceiling(i/2) - e_{2^p (i+1), i} ceiling(i/2) ) , where e_{i,j} is the matrix unit. The conjecture above is true, deduced from the formula of the matrix. - FUNG Cheok Yin, Sep 12 2022

Links

Table of n, a(n) for n=0..65.

FUNG Cheok Yin, the triangle with the first 61 rows

Example

Triangle begins
  1;
  0,  1;
  0, -1,  2;
  0,  1, -2,  2;
  0,  0,  0, -2,  3;
  0, -1,  2,  0, -3,  3;
  0,  0,  0,  0,  0, -3,  4;
  0,  1, -2,  2,  0,  0, -4,  4;
  0,  0,  0,  0,  0,  0,  0, -4,  5;
  0,  0,  0, -2,  3,  0,  0,  0, -5,  5;
  0,  0,  0,  0,  0,  0,  0,  0,  0, -5,  6;
  0, -1,  2,  0, -3,  3,  0,  0,  0,  0, -6,  6;
  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -6,  7;
Inverse of the triangle begins
  1;
  0,  1;
  0, 1/2, 1/2;
  0,  0,  1/2, 1/2;
  0,  0,  1/3, 1/3, 1/3;
  0,  0,   0,  1/3, 1/3, 1/3;
  0,  0,   0,  1/4, 1/4, 1/4, 1/4;
  0,  0,   0,   0,  1/4, 1/4, 1/4, 1/4;
  0,  0,   0,   0,  1/5, 1/5, 1/5, 1/5, 1/5;
  0,  0,   0,   0,   0,  1/5, 1/5, 1/5, 1/5, 1/5;
  0,  0,   0,   0,   0,  1/6, 1/6, 1/6, 1/6, 1/6, 1/6;

Mathematica

rows = 11;
A[n_, k_] := If[k <= n, If[n <= 2 k, 1/Floor[(n+2)/2] , 0], 0];
T = Table[A[n, k], {n, 0, rows-1}, {k, 0, rows-1}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Stefano Spezia, Sep 30 2018 *)

Crossrefs

Sequence in context: A036862 A285124 A094238 * A127771 A248806 A118407

Adjacent sequences: A127790 A127791 A127792 * A127794 A127795 A127796

Keyword

sign,tabl

Author

Paul Barry, Jan 29 2007

Status

approved