Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #8 Feb 17 2014 18:11:41
%S 1,3,1,9,4,1,27,13,7,1,81,40,34,8,1,243,121,142,42,11,1,729,364,547,
%T 184,75,12,1,2187,1093,2005,731,409,87,15,1,6561,3280,7108,2736,1958,
%U 496,132,16,1,19683,9841,24604,9844,8610,2454,892,148,19,1
%N 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.
%C Companion triangle A140070 uses an analogous operation with the main diagonal [1,3,1,3,1,3,...].
%F From _Peter Bala_, Jan 17 2014: (Start)
%F 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 + ....
%F Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).
%F 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.
%F Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) + R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)
%e First few rows of the triangle are:
%e 1;
%e 3, 1;
%e 9, 4, 1;
%e 27, 13, 7, 1;
%e 81, 40, 34, 8, 1;
%e 243, 121, 142, 42, 11, 1;
%e 729, 364, 547, 184, 75, 12, 1;
%e 2187, 1093, 2005, 731, 409, 87, 15, 1;
%e 6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1;
%e ...
%Y Cf. A140070, A007070 (row sums), A157751.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, May 04 2008