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A161944 a(n) = ((4+sqrt(2))*(3+sqrt(2))^n + (4-sqrt(2))*(3-sqrt(2))^n)/4.

%I A161944 #19 Sep 08 2022 08:45:45
%S A161944 2,7,28,119,518,2275,10024,44219,195146,861343,3802036,16782815,
%T A161944 74082638,327016123,1443518272,6371996771,28127352722,124160138935,
%U A161944 548069364556,2419295214791,10679285736854,47140647917587
%N A161944 a(n) = ((4+sqrt(2))*(3+sqrt(2))^n + (4-sqrt(2))*(3-sqrt(2))^n)/4.
%C A161944 Third binomial transform of A135530.
%H A161944 Vincenzo Librandi, <a href="/A161944/b161944.txt">Table of n, a(n) for n = 0..300</a>
%H A161944 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F A161944 a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 2; a(1) = 7.
%F A161944 G.f.: (2-5*x)/(1-6*x+7*x^2).
%F A161944 E.g.f.: exp(3*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - _G. C. Greubel_, Apr 03 2018
%t A161944 LinearRecurrence[{6,-7}, {2,7}, 50] (* _G. C. Greubel_, Apr 03 2018 *)
%t A161944 Table[((4+Sqrt[2])(3+Sqrt[2])^n+(4-Sqrt[2])(3-Sqrt[2])^n)/4,{n,0,30}]// Simplify (* _Harvey P. Dale_, Jun 03 2020 *)
%o A161944 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/4: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 01 2009
%o A161944 (PARI) x='x+O('x^30); Vec((2-5*x)/(1-6*x+7*x^2)) \\ _G. C. Greubel_, Apr 03 2018
%o A161944 (GAP) a:=[2,7];; for n in [3..25] do a[n]:=6*a[n-1]-7*a[n-2]; od; a; # _Muniru A Asiru_, Apr 04 2018
%Y A161944 Cf. A135530, A161941 (second binomial transform of A135530).
%K A161944 nonn,easy
%O A161944 0,1
%A A161944 Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009
%E A161944 Edited and extended beyond a(4) by _Klaus Brockhaus_, Jul 01 2009