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Eigentriangle, row sums = the Padovan sequence, A000931
1

%I #7 Oct 03 2024 07:47:42

%S 1,1,1,-1,1,2,0,-1,2,2,1,0,-2,2,3,-1,1,0,-2,3,4,0,-1,2,0,-3,4,5,1,0,

%T -2,2,0,-4,5,7,-1,1,0,-2,3,0,-5,7,9,0,-1,2,0,-3,4,0,-7,9,12,1,0,-2,2,

%U 0,-4,5,0,-9,12,16,-1,1,0,-2,3,0,-5,7,0,-12,16,21

%N Eigentriangle, row sums = the Padovan sequence, A000931

%C Right border = Padovan sequence starting with offset 6.

%C Row sums = Padovan sequence starting with offset 7.

%C Sum of n-th row terms = rightmost term of next row.

%F Triangle read by rows, T(n,k) = M * (A000931 * 0^(n-k)). M = an infinite lower triangular matrix with A106510 in every column: (1, 1, -1, 0, 1, -1, 0, 1, -1,...); and A000931 is a diagonalized infinite lower triangular matrix with the Padovan sequence starting with offset 6: (1, 1, 2, 2, 3, 4, 5, 7, 9,...) as the main diagonal and the rest zeros.

%e First few rows of the triangle =

%e 1;

%e 1, 1;

%e -1, 1, 2;

%e 0, -1, 2, 2;

%e 1, 0, -2, 2, 3;

%e -1, 1, 0, -2, 3, 4;

%e 0, -1, 2, 0, -3, 4, 5;

%e 1, 0, -2, 2, 0, -4, 5, 7;

%e -1, 1, 0, -2, 3, 0, -5, 7, 9;

%e 0, -1, 2, 0, -3, 4, 0, -7, 9, 12;

%e 1, 0, -2, 2, 0, -4, 5, 0, -9, 12, 16;

%e ...

%e Example: Row 10 = (1, 0, -2, 2, 3) with A000931(10) = 3, rightmost term. This row = the termwise products of (1, 0, -1, 1, 1) and (1, 1, 2, 2, 3); where the Padovan sequence starting with offset 6 = (1, 1, 2, 2, 3, 4, 5, 7, 9,...).

%Y A000931, A106510

%K eigen,tabl,sign

%O 6,6

%A _Gary W. Adamson_, Oct 10 2008