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Triangle read by rows, (Sierpinski's gasket, A047999) * A166966 (diagonalized as a lower triangular matrix)
2

%I #9 May 01 2013 04:06:26

%S 1,1,1,1,0,2,1,1,2,3,1,0,0,0,7,1,1,0,0,7,8,1,0,2,0,7,0,17,1,1,2,3,7,8,

%T 17,27,1,0,0,0,0,0,0,0,66,1,1,0,0,0,0,0,0,66,67,1,0,2,0,0,0,0,0,66,0,

%U 135,1,1,2,3,0,0,0,0,66,67,135,204

%N Triangle read by rows, (Sierpinski's gasket, A047999) * A166966 (diagonalized as a lower triangular matrix)

%C An eigentriangle (a given triangle * its own eigensequence); in this case A047999 * A166966.

%C Triangle A166967 has the properties of: row sums = the eigensequence, A166966 and sum of n-th row terms = rightmost term of next row.

%F Let Sierpinski's gasket, A047999 = S; and Q = the eigensequence of A047999 prefaced with a 1: (1, 1, 2, 3, 7, 8, 17,...) then diagonalized as an infinite lower triangular matrix: [1; 0,1; 0,0,2; 0,0,0,3; 0,0,0,0,7,...].

%F Triangle A166967 = S * Q.

%e First few rows of the triangle =

%e 1;

%e 1, 1;

%e 1, 1, 2, 3;

%e 1, 0, 0, 0, 7;

%e 1, 1, 0, 0, 7, 8;

%e 1, 0, 2, 0, 7, 0, 17;

%e 1, 1, 2, 3, 7, 8, 17, 27;

%e 1, 0, 0, 0, 0, 0,..0,..0, 66;

%e 1, 1, 0, 0, 0, 0,..0,..0, 66, 67;

%e 1, 0, 2, 0, 0, 0,..0,..0, 66,..0, 135;

%e 1, 1, 2, 3, 0, 0,..0,..0, 66, 67, 135, 204;

%e 1, 0, 0, 0, 7, 0,..0,..0, 66,..0,...0,...0, 479;

%e 1, 1, 0, 0, 7, 8,..0,..0, 66, 67,...0,...0, 479, 553

%e 1, 0, 2, 0, 7, 0, 17,..0, 66,..0, 135,...0, 479,...0, 1182;

%e 1, 1, 2, 3, 7, 8, 17, 27, 66, 67, 135, 204, 479, 553, 1182, 1189;

%e ...

%Y Cf. A047999, A166966.

%K nonn,tabl

%O 0,6

%A _Gary W. Adamson_, Oct 25 2009