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%I #6 Jun 30 2023 00:21:00
%S 17,65,373,2173,12665,73817,430237,2507605,14615393,85184753,
%T 496493125,2893773997,16866150857,98303131145,572952636013,
%U 3339412684933,19463523473585,113441728156577,661186845465877,3853679344638685
%N a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.
%C lim_{n -> infinity} a(n)/a(n-1) = 3+2*sqrt(2).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).
%F a(n) = ((74+47*sqrt(2))*(3-2*sqrt(2))^n+(74-47*sqrt(2))*(3+2*sqrt(2))^n)/4.
%F G.f.: x*(17-37*x)/(1-6*x+x^2).
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r2>:=NumberField(x^2-2); S:=[ ((74+47*r2)*(3-2*r2)^n+(74-47*r2)*(3+2*r2)^n)/4: n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ];
%o (PARI) {m=20; v=concat([17, 65], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}
%Y First trisection of A156567.
%Y Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156569.
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Feb 11 2009, Feb 16 2009
%E G.f. corrected by _Klaus Brockhaus_, Sep 22 2009
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