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

0,2

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).

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). - Eric W. Weisstein, Mar 31 2017

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 *)

Prog

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

Crossrefs

Cf. A098647.

Sequence in context: A158196 A191649 A009637 * A026295 A118650 A180187

Adjacent sequences: A098645 A098646 A098647 * A098649 A098650 A098651

Keyword

nonn,easy

Author

Paul Barry, Sep 18 2004

Status

approved