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A268344 a(n) = 11*a(n - 1) - 3*a(n - 2) for n>1, a(0)=0, a(1)=1.

%I A268344 #21 Sep 08 2022 08:46:15
%S A268344 0,1,11,118,1265,13561,145376,1558453,16706855,179100046,1919979941,
%T A268344 20582479213,220647331520,2365373209081,25357163305331,
%U A268344 271832676731398,2914087954129385,31239469465229041,334891900255131296,3590092494410757133
%N A268344 a(n) = 11*a(n - 1) - 3*a(n - 2) for n>1, a(0)=0, a(1)=1.
%C A268344 In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1, is x/(1 - k*x + m*x^2). This recurrence gives the closed form b(n) = (2^(-n)*((sqrt(k^2 - 4*m) + k)^n - (k - sqrt(k^2 - 4*m))^n))/sqrt(k^2 - 4*m).
%H A268344 G. C. Greubel, <a href="/A268344/b268344.txt">Table of n, a(n) for n = 0..965</a>
%H A268344 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-3).
%F A268344 G.f.: x/(1 - 11*x + 3*x^2).
%F A268344 a(n) = ( ((11 + sqrt(109))/2)^n - ((11 - sqrt(109))/2)^n )/sqrt(109).
%t A268344 LinearRecurrence[{11, -3}, {0, 1}, 20] (* or *) Table[(((11 + Sqrt[109])/2)^n - ((11 - Sqrt[109])/2)^n)/Sqrt[109], {n, 0, 20}]
%o A268344 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-11*x+3*x^2))) \\ _G. C. Greubel_, Jan 14 2018
%o A268344 (Magma) I:=[0,1]; [n le 2 select I[n] else 11*Self(n-1) - 3*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 14 2018
%Y A268344 Cf. A190872, A190980.
%K A268344 nonn,easy
%O A268344 0,3
%A A268344 _Ilya Gutkovskiy_, Feb 02 2016