A146973 Eigentriangle, row sums = A000931 starting with offset 3.

1, -1, 1, 2, -1, 0, -2, 2, 0, 1, 3, -2, 0, -1, 1, -3, 3, 0, 2, -1, 1, 4, -3, 0, -2, 2, -1, 2, -4, 4, 0, 3, -2, 2, -2, 2, 5, -4, 0, -3, 3, -2, 4, -2, 3, -5, 5, 0, 4, -3, 3, -4, 4, -3, 4, 6, -5, 0, -4, 4, -3, 6, -4, 5, -6, 6, 0, 5, -4, 4, -6, 6, -6, 8, -5, 7

Offset: 3

Author

Gary W. Adamson, Nov 03 2008

Keywords

• eigen
• tabl
• sign

Comments

Row sums and right border = the Padovan sequence, A000931 starting with offset 3: (1, 1, 0, 1, 1, 1, 2, 2, 3,...).

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

Examples

			First few rows of the triangle =
  1;
  -1, 1;
  2, -1, 0;
  -2, 2, 0, 1;
  3, -2, 0, -1, 1;
  -3, 3, 0, 2, -1, 1;
  4, -3, 0, -2, 2, -1, 2;
  -4, 4, 0, 3, -2, 2, -2, 2;
  5, -4, 0, -3, 3, -2, 4, -2, 3;
  -5, 5, 0, 4, -3, 3, -4, 4, -3, 4;
  6, -5, 0, -4, 4, -3, 6, -4, 6, -4, 5;
  -6, 6, 0, 5, -4, 4, -6, 6, -6, 8, -5, 7;
  7, -6, 0, -5, 5, -4, 8, -6, 9, -8, 10, -7, 9
  -7, 7, 0, 6, -5, 5, -8, 8, -9, 12, -10, 14, -9, 12;
  ...
Row 6 = (-2, 2, 0, 1) = termwise products of (-2, 2, 0, 1) and (1, 1, 0, 1).
		

Crossrefs

Cf. A000931.

Formula

Triangle read by rows, T * Q, where T = an infinite lower triangular matrix with (1, -1, 2, -2, 3, -3,...) in every column and Q = an infinite lower triangular matrix with the Padovan sequence, A000931 as the main diagonal starting with offset 3: (1, 1, 0, 1, 1, 1, 2, 2, 3,...). The rest of triangle Q = all zeros. This triangle = T * Q.