A176333 Expansion of (1-3*x)/(1-4*x+9*x^2).

1, 1, -5, -29, -71, -23, 547, 2395, 4657, -2927, -53621, -188141, -269975, 613369, 4883251, 14012683, 12101473, -77708255, -419746277, -979610813, -140726759, 8253590281, 34280901955, 62841295291, -57162936431, -794223403343, -2662427185493, -3501698111885

Offset: 0

Author

Paul Barry, Apr 15 2010

Keywords

• easy
• sign

Comments

Hankel transform of A176332.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2097
• Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
• Index entries for linear recurrences with constant coefficients, signature (4,-9).

Crossrefs

Cf. A176332, A190958, A190967.

Programs

GAP

a:=[1,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Dec 07 2019

Magma

I:=[1,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 07 2019

Maple

seq(coeff(series((1-3*x)/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 07 2019

Mathematica

CoefficientList[Series[(1-3x)/(1-4x+9x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,-9},{1,1},30] (* Harvey P. Dale, Sep 17 2012 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)/(1-4*x+9*x^2)) \\ G. C. Greubel, Dec 07 2019

Sage

def A176333_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-4*x+9*x^2) ).list()
A176333_list(30) # G. C. Greubel, Dec 07 2019

Formula

a(n) = 3^n*( cos(2*n*atan(1/sqrt(5))) - sin(2*n*atan(1/sqrt(5)))/sqrt(5) ).

a(0)=1, a(1)=1, a(n) = 4*a(n-1) - 9*a(n-2). - Harvey P. Dale, Sep 17 2012

a(n) = -3*A190967(n) + A190967(n+1). - R. J. Mathar, May 04 2013