reviewed

approved

proposed

reviewed

editing

proposed

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,...).

approved

editing

_Gary W. Adamson (qntmpkt(AT)yahoo.com), _, Oct 10 2008

eigen,nonn,tabl,newsign

Eigentriangle, row sums = the Padovan sequence, A000931

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

6,6

Right border = Padovan sequence starting with offset 6.

Row sums = Padovan sequence starting with offset 7.

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

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.

First few rows of the triangle =

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, -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, 0, -4, 5, 0, -9, 12, 16;

...

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,...).

A000931, A106510

eigen,nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 10 2008

approved