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A161734 a(n) = ((2+sqrt(2))*(5+sqrt(2))^n+(2-sqrt(2))*(5-sqrt(2))^n)/4.

%I A161734 #36 Aug 31 2023 09:51:11
%S A161734 1,6,37,232,1469,9354,59753,382388,2449561,15700686,100666957,
%T A161734 645553792,4140197909,26554241874,170317866833,1092431105228,
%U A161734 7007000115121,44944085730966,288279854661877,1849084574806552,11860409090842349,76075145687872794
%N A161734 a(n) = ((2+sqrt(2))*(5+sqrt(2))^n+(2-sqrt(2))*(5-sqrt(2))^n)/4.
%C A161734 Fifth binomial transform of A016116. Fourth binomial transform of the sequence of the absolute values of A077985. Third binomial transform of A007052. Second binomial transform of A086351. - _R. J. Mathar_, Jun 18 2009
%H A161734 Vincenzo Librandi, <a href="/A161734/b161734.txt">Table of n, a(n) for n = 0..1000</a>
%H A161734 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-23).
%F A161734 a(n) = 10*a(n-1) - 23*a(n-2). - _R. J. Mathar_, Jun 18 2009
%F A161734 G.f.: (1-4*x)/(1-10*x+23*x^2). - _R. J. Mathar_, Jun 18 2009
%F A161734 E.g.f.: exp(5*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x))/2. - _G. C. Greubel_, Apr 03 2018
%t A161734 CoefficientList[Series[(1-4*z)/(23*z^2-10*z+1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 12 2011 *)
%t A161734 LinearRecurrence[{10,-23}, {1,6}, 50] (* _G. C. Greubel_, Apr 03 2018 *)
%o A161734 (PARI) F=nfinit(x^2-2); for(n=0, 20, print1(nfeltdiv(F, ((2+x)*(5+x)^n+(2-x)*(5-x)^n), 4)[1], ",")) \\ _Klaus Brockhaus_, Jun 19 2009
%o A161734 (Magma) [Floor(((2+Sqrt(2))*(5+Sqrt(2))^n+(2-Sqrt(2))*(5-Sqrt(2))^n)/4): n in [0..30]]; // _Vincenzo Librandi_, Aug 18 2011
%Y A161734 Cf. A016116, A077985, A000129, A007052, A086351.
%K A161734 nonn,easy
%O A161734 0,2
%A A161734 Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009
%E A161734 Extended by _R. J. Mathar_ and _Klaus Brockhaus_, Jun 18 2009
%E A161734 Edited by _Klaus Brockhaus_, Jul 05 2009