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

%I A327853 #24 Oct 27 2019 14:15:46
%S A327853 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,
%T A327853 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,
%U A327853 2,3,0,0,0,0,8,9,10,11,0,0,0,0,4,0,0,0,8,0,0,0,12
%N A327853 Triangle read by rows, Sierpinski's gasket, A047999 * (0,1,2,3,4,...) diagonalized.
%C A327853 This is similar to A166555, the difference being that this is scaled "linearly" instead of exponentially.
%C A327853 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).
%C A327853 If instead of (0,1,2,3,4,...), we use the A000217 (triangular numbers), then the border of the scatterplot will be truly linear.
%H A327853 Matej Veselovac, <a href="/A327853/b327853.txt">Table of n, a(n) for n = 1..100000</a>
%H A327853 Math StackExchange, <a href="https://math.stackexchange.com/q/2585840/318073">Pattern in Pascal's triangle </a>.
%H A327853 Matej Veselovac, <a href="https://imgur.com/gallery/6hAL81C">Scatterplot and Logarithmic Scatterplot of triangle read by rows T(n,k) = k * (binomial(n, k) mod 2), n=1...10^5.</a>.
%F A327853 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.
%F A327853 The entries of the triangle are given by T(n, k) = k * (binomial(n, k) (mod 2)), then it is read by rows.
%e A327853 First 16 rows of the triangle:
%e A327853   0;
%e A327853   0, 1;
%e A327853   0, 0, 2;
%e A327853   0, 1, 2, 3;
%e A327853   0, 0, 0, 0, 4;
%e A327853   0, 1, 0, 0, 4, 5;
%e A327853   0, 0, 2, 0, 4, 0, 6;
%e A327853   0, 1, 2, 3, 4, 5, 6, 7;
%e A327853   0, 0, 0, 0, 0, 0, 0, 0, 8;
%e A327853   0, 1, 0, 0, 0, 0, 0, 0, 8, 9;
%e A327853   0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 10;
%e A327853   0, 1, 2, 3, 0, 0, 0, 0, 8, 9, 10, 11;
%e A327853   0, 0, 0, 0, 4, 0, 0, 0, 8, 0,  0,  0, 12;
%e A327853   0, 1, 0, 0, 4, 5, 0, 0, 8, 9,  0,  0, 12, 13;
%e A327853   0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 10,  0, 12,  0, 14;
%e A327853   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
%t A327853 r[n0_]:=Flatten[Table[(k)(Mod[Binomial[n,k],2]),{n,0,n0},{k,0,n}]]; r[20] (* _Matej Veselovac_, Sep 28 2019 *)
%Y A327853 Cf. A001317, A007318, A003056, A000217.
%Y A327853 Cf. A166555 (2^k is used instead of k).
%Y A327853 Cf. A080099 (similar scatterplot visualization).
%Y A327853 Cf. A327889 (alternating, normalized (linear) modification of the sequence, transformed by first decimal digit indicator function).
%K A327853 nonn,tabl,hear,look
%O A327853 1,6
%A A327853 _Matej Veselovac_, Sep 28 2019