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A327853
Triangle read by rows, Sierpinski's gasket, A047999 * (0,1,2,3,4,...) diagonalized.
2
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
1,6
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.
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.
EXAMPLE
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;
MATHEMATICA
r[n0_]:=Flatten[Table[(k)(Mod[Binomial[n, k], 2]), {n, 0, n0}, {k, 0, n}]]; r[20] (* Matej Veselovac, Sep 28 2019 *)
CROSSREFS
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).
Sequence in context: A319203 A321198 A128584 * A080099 A268040 A127711
KEYWORD
AUTHOR
Matej Veselovac, Sep 28 2019
STATUS
approved