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%I #16 Sep 08 2022 08:45:47
%S 3,14,50,172,588,2008,6856,23408,79920,272864,931616,3180736,10859712,
%T 37077376,126590080,432205568,1475642112,5038157312,17201345024,
%U 58729065472,200513571840,684596156416,2337357481984,7980237615104
%N a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 3, a(1) = 14.
%C Binomial transform of A164303. Second binomial transform of A164654. Inverse binomial transform of A164305.
%H G. C. Greubel, <a href="/A164304/b164304.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..185 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 3, a(1) = 14.
%F a(n) = ((3+4*sqrt(2))*(2+sqrt(2))^n + (3-4*sqrt(2))*(2-sqrt(2))^n)/2.
%F G.f.: (3+2*x)/(1-4*x+2*x^2).
%F E.g.f.: (3*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x). - _G. C. Greubel_, Sep 13 2017
%t (3 + 2*x)/(1 - 4*x + 2*x^2) + O[x]^24 // CoefficientList[#, x]& (* _Jean-François Alcover_, Jun 22 2017 *)
%t LinearRecurrence[{4,-2}, {3,14}, 50] (* _G. C. Greubel_, Sep 13 2017 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+4*r)*(2+r)^n+(3-4*r)*(2-r)^n)/2: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 20 2009
%o (PARI) x='x+O('x^50); Vec((3+2*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Sep 13 2017
%Y Cf. A164303, A164654, A164305.
%K nonn,easy
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 20 2009
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