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A163614 a(n) = ((1 + 3*sqrt(2))*(3 + sqrt(2))^n + (1 - 3*sqrt(2))*(3 - sqrt(2))^n)/2.

%I A163614 #8 Sep 08 2022 08:45:46
%S A163614 1,9,47,219,985,4377,19367,85563,377809,1667913,7362815,32501499,
%T A163614 143469289,633305241,2795546423,12340141851,54472026145,240451163913,
%U A163614 1061402800463,4685258655387,20681732329081,91293583386777
%N A163614 a(n) = ((1 + 3*sqrt(2))*(3 + sqrt(2))^n + (1 - 3*sqrt(2))*(3 - sqrt(2))^n)/2.
%C A163614 Binomial transform of A163613. Third binomial transform of A163864. Inverse binomial transform of A163615.
%H A163614 G. C. Greubel, <a href="/A163614/b163614.txt">Table of n, a(n) for n = 0..1000</a>
%H A163614 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F A163614 a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
%F A163614 G.f.: (1+3*x)/(1-6*x+7*x^2).
%F A163614 E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) ). - _G. C. Greubel_, Jul 30 2017
%t A163614 LinearRecurrence[{6,-7},{1,9},30] (* _Harvey P. Dale_, Sep 24 2015 *)
%o A163614 (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+r)^n+(1-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 06 2009
%o A163614 (PARI) x='x+O('x^50); Vec((1+3*x)/(1-6*x+7*x^2)) \\ _G. C. Greubel_, Jul 30 2017
%Y A163614 Cf. A163613, A163864, A163615.
%K A163614 nonn
%O A163614 0,2
%A A163614 Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009
%E A163614 Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 06 2009