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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..55.

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)

EXAMPLE

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.

Sequence in context: A162852 A054448 A106516 * A285280 A285266 A067417

Adjacent sequences:  A140068 A140069 A140070 * A140072 A140073 A140074

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)