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A161947 a(n) = ((4+sqrt(2))*(5+sqrt(2))^n + (4-sqrt(2))*(5-sqrt(2))^n)/4.

%I A161947 #22 Oct 24 2023 17:36:19
%S A161947 2,11,64,387,2398,15079,95636,609543,3895802,24938531,159781864,
%T A161947 1024232427,6567341398,42116068159,270111829436,1732448726703,
%U A161947 11111915190002,71272831185851,457154262488464,2932267507610067,18808127038865998,120639117713628439,773804255242366436
%N A161947 a(n) = ((4+sqrt(2))*(5+sqrt(2))^n + (4-sqrt(2))*(5-sqrt(2))^n)/4.
%C A161947 Fifth binomial transform of A135530.
%H A161947 Muniru A Asiru, <a href="/A161947/b161947.txt">Table of n, a(n) for n = 0..252</a>
%H A161947 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-23).
%F A161947 a(n) = 10*a(n-1) - 23*a(n-2) for n>1; a(0) = 2; a(1) = 11.
%F A161947 G.f.: (2-9*x)/(1-10*x+23*x^2).
%F A161947 E.g.f.: exp(5*x)*(4*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - _Stefano Spezia_, Oct 24 2023
%p A161947 seq(simplify(((4+sqrt(2))*(5+sqrt(2))^n+(4-sqrt(2))*(5-sqrt(2))^n)*1/4), n = 0 .. 20); # _Emeric Deutsch_, Jun 28 2009
%t A161947 LinearRecurrence[{10,-23}, {2,11}, 50] (* _G. C. Greubel_, Aug 17 2018 *)
%t A161947 Table[(((4+Sqrt[2])(5+Sqrt[2])^n)+((4-Sqrt[2])(5-Sqrt[2])^n))/4,{n,0,20}]//Simplify (* _Harvey P. Dale_, Mar 07 2020 *)
%o A161947 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((4+r)*(5+r)^n+(4-r)*(5-r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 01 2009
%o A161947 (GAP) a := [2, 11];; for n in [3..10^2] do a[n] := 10*a[n-1] - 23*a[n-2]; od; a; # _Muniru A Asiru_, Feb 02 2018
%o A161947 (PARI) x='x+O('x^30); Vec((2-9*x)/(1-10*x+23*x^2)) \\ _G. C. Greubel_, Aug 17 2018
%Y A161947 Cf. A135530.
%K A161947 nonn,easy
%O A161947 0,1
%A A161947 Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009
%E A161947 Edited and extended beyond a(4) by _Klaus Brockhaus_, Jul 01 2009
%E A161947 Extended by _Emeric Deutsch_, Jun 28 2009