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A163604 a(n) = ((3+2*sqrt(2))*(4+sqrt(2))^n + (3-2*sqrt(2))*(4-sqrt(2))^n)/2.

%I A163604 #10 Sep 08 2022 08:45:46
%S A163604 3,16,86,464,2508,13568,73432,397504,2151984,11650816,63078752,
%T A163604 341518592,1849046208,10011109376,54202228096,293462293504,
%U A163604 1588867154688,8602465128448,46575580861952,252170135097344,1365302948711424,7392041698328576,40022092304668672
%N A163604 a(n) = ((3+2*sqrt(2))*(4+sqrt(2))^n + (3-2*sqrt(2))*(4-sqrt(2))^n)/2.
%C A163604 Binomial transform of A163606. Inverse binomial transform of A163605.
%H A163604 G. C. Greubel, <a href="/A163604/b163604.txt">Table of n, a(n) for n = 0..1000</a>
%H A163604 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14).
%F A163604 a(n) = 8*a(n-1)-14*a(n-2) for n > 1; a(0) = 3, a(1) = 16.
%F A163604 G.f.: (3-8*x)/(1-8*x+14*x^2).
%F A163604 E.g.f.: exp(4*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). -  _G. C. Greubel_, Jul 29 2017
%t A163604 LinearRecurrence[{8, -14}, {3, 16}, 50] (* _G. C. Greubel_, Jul 29 2017 *)
%o A163604 (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+2*r)*(4+r)^n+(3-2*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 07 2009
%o A163604 (PARI) x='x+O('x^50); Vec((3-8*x)/(1-8*x+14*x^2)) \\ _G. C. Greubel_, Jul 29 2017
%Y A163604 Cf. A163606, A163605.
%K A163604 nonn
%O A163604 0,1
%A A163604 Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009
%E A163604 Edited and extended beyond a(5) by _Klaus Brockhaus_ and _R. J. Mathar_, Aug 07 2009