%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