%I #25 Jun 30 2024 18:15:24

%S 2,6,28,144,752,3936,20608,107904,564992,2958336,15490048,81106944,

%T 424681472,2223661056,11643240448,60964798464,319215828992,

%U 1671435780096,8751751364608,45824765067264,239941584945152,1256350449401856,6578336356630528,34444616342175744

%N Binomial transform of A014448.

%C The binomial transform of this sequence is 2, 8, 42, 248,... = 2*A108404(n).

%D C. Smith, A Treatise on Algebra, Macmillan, London, 5th ed., 1950, p. 360, Example 44.

%H Colin Barker, <a href="/A228842/b228842.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Bhadouria, D. Jhala, and B. Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence B_4.

%H Takao Komatsu, <a href="https://arxiv.org/abs/2105.08277">Asymmetric Circular Graph with Hosoya Index and Negative Continued Fractions</a>, arXiv:2105.08277 [math.CO], 2021.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).

%F G.f.: 2*( 1-3*x ) / ( 1-6*x+4*x^2 ).

%F a(n) = 2*A098648(n).

%F From _Colin Barker_, Sep 21 2017: (Start)

%F a(n) = (3-sqrt(5))^n + (3+sqrt(5))^n.

%F a(n) = 6*a(n-1) - 4*a(n-2) for n>1.

%F (End)

%t CoefficientList[Series[2*(1 - 3 x)/(1 - 6 x + 4 x^2), {x, 0, 23}], x] (* _Michael De Vlieger_, Aug 26 2021 *)

%t LinearRecurrence[{6,-4},{2,6},30] (* _Harvey P. Dale_, Jun 30 2024 *)

%o (PARI) Vec(2*(1 - 3*x) / (1 - 6*x + 4*x^2) + O(x^30)) \\ _Colin Barker_, Sep 21 2017

%Y Cf. A014448, A108404, A098648.

%Y When divided by 2^n this becomes(essentially) A005248.

%K nonn,easy

%O 0,1

%A _R. J. Mathar_, Nov 10 2013