A140071 Triangle read by rows: iterates of X * [1,0,0,0,...]; where X = an infinite lower bidiagonal matrix with [3,1,3,1,3,1...] in the main diagonal and [1,1,1,...] in the subdiagonal.

1, 3, 1, 9, 4, 1, 27, 13, 7, 1, 81, 40, 34, 8, 1, 243, 121, 142, 42, 11, 1, 729, 364, 547, 184, 75, 12, 1, 2187, 1093, 2005, 731, 409, 87, 15, 1, 6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1, 19683, 9841, 24604, 9844, 8610, 2454, 892, 148, 19, 1

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, May 04 2008

Keywords

• nonn
• tabl

Comments

Companion triangle A140070 uses an analogous operation with the main diagonal [1,3,1,3,1,3,...].

Examples

			First few rows of the triangle are:
1;
3, 1;
9, 4, 1;
27, 13, 7, 1;
81, 40, 34, 8, 1;
243, 121, 142, 42, 11, 1;
729, 364, 547, 184, 75, 12, 1;
2187, 1093, 2005, 731, 409, 87, 15, 1;
6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1;
...
		

Crossrefs

Cf. A140070, A007070 (row sums), A157751.

Formula

From Peter Bala, Jan 17 2014: (Start)

O.g.f. (1 + (x - 1)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 3)*z + (x^2 + 4*x + 9)*z^2 + ....

Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).

Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.

Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) + R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)