A327853 Triangle read by rows, Sierpinski's gasket, A047999 * (0,1,2,3,4,...) diagonalized.

0, 0, 1, 0, 0, 2, 0, 1, 2, 3, 0, 0, 0, 0, 4, 0, 1, 0, 0, 4, 5, 0, 0, 2, 0, 4, 0, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 1, 0, 0, 0, 0, 0, 0, 8, 9, 0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 10, 0, 1, 2, 3, 0, 0, 0, 0, 8, 9, 10, 11, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 12

Offset: 1

Author

Matej Veselovac, Sep 28 2019

Keywords

• nonn
• tabl
• hear
• look

Comments

This is similar to A166555, the difference being that this is scaled "linearly" instead of exponentially.

The scatterplot of the sequence resembles Sierpinski's gasket (triangle), with a square root border (the "linear" scaling is not normalized and actually resembles the scale of the function of the positive inverse of triangular numbers: A003056).

If instead of (0,1,2,3,4,...), we use the A000217 (triangular numbers), then the border of the scatterplot will be truly linear.

Examples

			First 16 rows of the triangle:
  0;
  0, 1;
  0, 0, 2;
  0, 1, 2, 3;
  0, 0, 0, 0, 4;
  0, 1, 0, 0, 4, 5;
  0, 0, 2, 0, 4, 0, 6;
  0, 1, 2, 3, 4, 5, 6, 7;
  0, 0, 0, 0, 0, 0, 0, 0, 8;
  0, 1, 0, 0, 0, 0, 0, 0, 8, 9;
  0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 10;
  0, 1, 2, 3, 0, 0, 0, 0, 8, 9, 10, 11;
  0, 0, 0, 0, 4, 0, 0, 0, 8, 0,  0,  0, 12;
  0, 1, 0, 0, 4, 5, 0, 0, 8, 9,  0,  0, 12, 13;
  0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 10,  0, 12,  0, 14;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
		

Links

• Matej Veselovac, Table of n, a(n) for n = 1..100000
• Math StackExchange, Pattern in Pascal's triangle .
• Matej Veselovac, Scatterplot and Logarithmic Scatterplot of triangle read by rows T(n,k) = k * (binomial(n, k) mod 2), n=1...10^5..

Crossrefs

Cf. A001317, A007318, A003056, A000217.

Cf. A166555 (2^k is used instead of k).

Cf. A080099 (similar scatterplot visualization).

Cf. A327889 (alternating, normalized (linear) modification of the sequence, transformed by first decimal digit indicator function).

Programs

Mathematica

r[n0_]:=Flatten[Table[(k)(Mod[Binomial[n,k],2]),{n,0,n0},{k,0,n}]]; r[20] (* Matej Veselovac, Sep 28 2019 *)

Formula

Triangle read by rows, A047999 * Q. A047999 = Sierpinski's gasket, Q = an infinite lower triangular matrix with (0,1,2,3,...) as the main diagonal and the rest zeros.

The entries of the triangle are given by T(n, k) = k * (binomial(n, k) (mod 2)), then it is read by rows.