A177878 Triangle in which row n is generated from (1,2,3,...,n) dot (n, n-1,...,1) with subtractive carryovers.

1, 2, 0, 3, 1, 2, 4, 2, 4, 0, 5, 3, 6, 2, 3, 6, 4, 8, 4, 6, 0, 7, 5, 10, 6, 9, 3, 4, 8, 6, 12, 8, 12, 6, 8, 0, 9, 7, 14, 10, 15, 9, 12, 4, 5, 10, 8, 16, 12, 18, 12, 16, 8, 10, 0, 11, 9, 18, 14, 21, 15, 20, 12, 15, 5, 6, 12, 10, 20, 16, 24, 18, 24, 16, 20, 10, 12, 0

Offset: 0

Author

Gary W. Adamson, Dec 13 2010

Keywords

• nonn
• tabl

Comments

The subtractive carryover dot product of two vectors (a(1),a(2),...,a(n)) dot (b(1),b(2),...,b(n)) = (c(1),...,c(n)) is defined by c(1) = a(1)*b(1) and c(i) = a(i)*b(i)-c(i-1), i>1.

A177877 = analogous triangle with additive carryovers.

A160770 = the analogous triangle using the triangular series as the generating vector.

Examples

			Row 3 = (4, 2, 4, 0) = (1, 2, 3, 4) dot (4, 3, 2, 1) with subtractive carryovers = (4), then (2*3 - 4 = 2), (3*2 - 2 = 4), and (4*1 - 4 = 0).
First few rows of the triangle:
  1;
  2, 0;
  3, 1, 2;
  4, 2, 4, 0;
  5, 3, 6, 2, 3;
  6, 4, 8, 4, 6, 0;
  7, 5, 10, 6, 9, 3, 4;
  8, 6, 12, 8, 12, 6, 8, 0;
  9, 7, 14, 10, 15, 9, 12, 4, 5;
  10, 8, 16, 12, 18, 12, 16, 8, 10, 0;
  11, 9, 18, 14, 21, 15, 20, 12, 15, 5, 6;
  12, 10, 20, 16, 24, 18, 24, 16, 20, 10, 12, 0;
  ...
		

Crossrefs

Cf. A005993 (row sums), A177877, A160770

Formula

By rows, dot product of (1,2,3,...) and (...3,2,1) with subtractive carryovers; such that current row product subtracts previous product.