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%I #43 Jan 01 2024 11:44:58
%S 1,4,28,208,1552,11584,86464,645376,4817152,35955712,268377088,
%T 2003193856,14952042496,111603564544,833020346368,6217748512768,
%U 46409906716672,346408259682304,2585626450591744,19299378566004736
%N a(n) = 8*a(n-1) - 4*a(n-2), where a(0) = 1, a(1) = 4.
%H G. C. Greubel, <a href="/A090965/b090965.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-4).
%F a(n) = Sum_{k>=0} binomial(2*n, 2*k)*3^k = Sum_{k>=0} A086645(n, k)*3^k.
%F a(n) = 2^n*A001075(n).
%F G.f.: (1-4*x)/(1-8*x+4*x^2). - _Philippe Deléham_, Sep 07 2009
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k-4)/(x*(3*k-1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 28 2013
%F From _Peter Bala_, Feb 19 2022: (Start)
%F a(n) = Sum_{k = 0..floor(n/2)} 4^(n-2*k)*12^k*binomial(n,2*k).
%F a(n) = [x^n] (4*x + sqrt(1 + 12*x^2))^n.
%F G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = 1/sqrt(1 - 8*x + 4*x^2) is the g.f. of A069835.
%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
%t LinearRecurrence[{8,-4}, {1,4}, 20] (* _G. C. Greubel_, Feb 03 2019 *)
%o (Sage) [lucas_number2(n,8,4)/2 for n in range(0,21)] # _Zerinvary Lajos_, Jul 08 2008
%o (PARI) my(x='x+O('x^20)); Vec((1-4*x)/(1-8*x+4*x^2)) \\ _G. C. Greubel_, Feb 03 2019
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-4*x)/(1-8*x+4*x^2) )); // _G. C. Greubel_, Feb 03 2019
%o (GAP) a:=[1,4];; for n in [3..20] do a[n]:=8*a[n-1]-4*a[n-2]; od; a; # _G. C. Greubel_, Feb 03 2019
%Y Cf. A001075.
%Y Sum_{k>=0} A086645(n,k)*m^k for m = 0, 1, 2, 4 gives A000007, A081294, A001541, A083884.
%K easy,nonn
%O 0,2
%A _Philippe Deléham_, Feb 29 2004
%E Corrected by _T. D. Noe_, Nov 07 2006
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