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%I #15 Sep 08 2022 08:45:47
%S 3,17,81,367,1635,7241,32001,141319,623907,2754209,12157905,53667967,
%T 236902467,1045739033,4616116929,20376528343,89946351555,397042410929,
%U 1752630004689,7736483151631,34150488876963,150747551200361,665431885063425,2937358451978023
%N a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 3, a(1) = 17.
%C Binomial transform of A164304. Third binomial transform of A164654. Inverse binomial transform of A164535.
%H G. C. Greubel, <a href="/A164305/b164305.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..169 from Vincenzo Librandi)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 3, a(1) = 17.
%F G.f.: (3-x)/(1-6*x+7*x^2).
%F a(n) = ((3+4*sqrt(2))*(3+sqrt(2))^n + (3-4*sqrt(2))*(3-sqrt(2))^n)/2.
%F E.g.f.: (3*cosh(sqrt(2)*x) + 4*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - _G. C. Greubel_, Sep 13 2017
%t LinearRecurrence[{6,-7},{3,17},30] (* _Harvey P. Dale_, Jun 03 2015 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+4*r)*(3+r)^n+(3-4*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Aug 20 2009
%o (PARI) x='x+O('x^50); Vec((3-x)/(1-6*x+7*x^2)) \\ _G. C. Greubel_, Sep 13 2017
%Y Cf. A164304, A164654, A164535.
%K nonn,easy
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Aug 20 2009