A098648 Expansion of (1-3*x)/(1 - 6*x + 4*x^2).

1, 3, 14, 72, 376, 1968, 10304, 53952, 282496, 1479168, 7745024, 40553472, 212340736, 1111830528, 5821620224, 30482399232, 159607914496, 835717890048, 4375875682304, 22912382533632, 119970792472576, 628175224700928, 3289168178315264, 17222308171087872

Offset: 0

Author

Paul Barry, Sep 18 2004

Keywords

• nonn
• easy

Comments

Binomial transform of A001077. Second binomial transform of A084057. Third binomial transform of 1/(1-5*x^2). Let A=[1,1,1,1;3,1,-1,-3;3,-1,-1,3;1,-1,1,-1], the 4 X 4 Krawtchouk matrix. Then a(n)=trace((16(A*A`)^(-1))^n)/4.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for linear recurrences with constant coefficients, signature (6,-4).

Crossrefs

Cf. A098647.

Programs

Mathematica

a[n_]:=(MatrixPower[{{5,1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
CoefficientList[Series[(1-3x)/(1-6x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-4},{1,3},31] (* Harvey P. Dale, Jun 06 2011 *)
Table[2^(n - 1) LucasL[2 n], {n, 0, 20}] (* Eric W. Weisstein, Mar 31 2017 *)

PARI

Vec((1-3*x)/(1 - 6*x + 4*x^2) + O(x^25)) \\ Jinyuan Wang, Jul 24 2021

Formula

E.g.f.: exp(3*x)*cosh(sqrt(5)*x).

a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2.

a(n) = 2*(3*a(n-1) - 2*a(n-2)). - Lekraj Beedassy, Oct 22 2004

a(n) = A084326(n+1) - 3*A084326(n). - R. J. Mathar, Nov 10 2013

a(n) = 2^(n-1)*Lucas(2*n) = 2^(n-1)*A005248(n), n>0. - Eric W. Weisstein, Mar 31 2017