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%I #42 Dec 21 2023 11:57:49
%S 1,8,44,232,1216,6368,33344,174592,914176,4786688,25063424,131233792,
%T 687149056,3597959168,18839158784,98643116032,516502061056,
%U 2704439902208,14160631169024,74146027405312,388233639755776,2032817728913408,10643971814457344
%N Trinomial transform of Lucas numbers (A000032).
%H G. C. Greubel, <a href="/A082762/b082762.txt">Table of n, a(n) for n = 0..1000</a>
%H Yuhan Jiang, <a href="https://arxiv.org/abs/2312.09427">The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model</a>, arXiv:2312.09427 [math.CO], 2023.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).
%F a(n) = Sum_{k=0..2*n} Trinomial(n,k)*Lucas(k+1), where Trinomial(n,k) = trinomial coefficients (A027907).
%F a(n) = 2^n*Lucas(2*n+1), where Lucas = A000032.
%F From _Philippe Deléham_, Mar 01 2004: (Start)
%F a(n) = 2^n*A002878(n) = 2^(-n)*Sum_{k>=0} C(2*n+1,2*k)*5^k; see A091042.
%F a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1). (End)
%F From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009: (Start)
%F a(n) = ((1+sqrt(5))*(3+sqrt(5))^n + (1-sqrt(5))*(3-sqrt(5))^n)/2.
%F Third binomial transform of 1, 5, 5, 25, 25, 125. (End)
%F G.f.: (1 + 2*x)/(1 - 6*x + 4*x^2). - _Colin Barker_, Mar 23 2012
%t a[n_]:=(MatrixPower[{{2,2},{2,4}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* _Robert G. Wilson v_, Mar 07 2011 *)
%t LinearRecurrence[{6,-4}, {1, 8}, 30] (* _G. C. Greubel_, Dec 21 2017 *)
%o (PARI) x='x+O('x^30); Vec((1 + 2*x)/(1 - 6*x + 4*x^2)) \\ _G. C. Greubel_, Dec 21 2017
%o (Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1)-4*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 21 2017
%Y Cf. A000032, A027907, A091042, A292277.
%K nonn,easy
%O 0,2
%A _Emanuele Munarini_, May 21 2003
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