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A164602 a(n) = ((1+4*sqrt(2))*(1+2*sqrt(2))^n + (1-4*sqrt(2))*(1-2*sqrt(2))^n)/2.

%I A164602 #13 Sep 08 2022 08:45:47
%S A164602 1,17,41,201,689,2785,10393,40281,153313,588593,2250377,8620905,
%T A164602 32994449,126335233,483631609,1851609849,7088640961,27138550865,
%U A164602 103897588457,397765032969,1522813185137,5829981601057,22319655498073
%N A164602 a(n) = ((1+4*sqrt(2))*(1+2*sqrt(2))^n + (1-4*sqrt(2))*(1-2*sqrt(2))^n)/2.
%C A164602 Binomial transform of A164703. Inverse binomial transform of A164603.
%H A164602 G. C. Greubel, <a href="/A164602/b164602.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..177 from Vincenzo Librandi)
%H A164602 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,7).
%F A164602 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 17.
%F A164602 G.f.: (1+15*x)/(1-2*x-7*x^2).
%F A164602 E.g.f.: exp(x)*(cosh(2*sqrt(2)*x) + 4*sqrt(2)*sinh(2*sqrt(2)*x)). - _G. C. Greubel_, Aug 11 2017
%t A164602 Simplify/@Table[1/2((1-4Sqrt[2])(1-2Sqrt[2])^n+(1+2Sqrt[2])^n(1+4 Sqrt[2])),{n,0,25}] (* _Harvey P. Dale_, Jul 26 2011 *)
%o A164602 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+4*r)*(1+2*r)^n+(1-4*r)*(1-2*r)^n)/2: n in [0..22] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 23 2009
%o A164602 (PARI) x='x+O('x^50); Vec((1+15*x)/(1-2*x-7*x^2)) \\ _G. C. Greubel_, Aug 11 2017
%Y A164602 Cf. A164703, A164603.
%K A164602 nonn
%O A164602 0,2
%A A164602 Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009
%E A164602 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 23 2009