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A154516 a(n) = 9n^2 - n.

%I A154516 #29 Sep 08 2022 08:45:40
%S A154516 8,34,78,140,220,318,434,568,720,890,1078,1284,1508,1750,2010,2288,
%T A154516 2584,2898,3230,3580,3948,4334,4738,5160,5600,6058,6534,7028,7540,
%U A154516 8070,8618,9184,9768,10370,10990,11628,12284,12958,13650,14360
%N A154516 a(n) = 9n^2 - n.
%C A154516 The identity (648*n^2-72*n+1)^2-(9*n^2-n)*(216*n-12)^2=1 can be written as A154514(n)^2-a(n)*A154518(n)^2=1 (see also the second comment in A154514). - _Vincenzo Librandi_, Jan 30 2012
%C A154516 The continued fraction expansion of sqrt(a(n)) is [3n-1; {1, 4, 1, 6n-2}]. For n=1, this collapses to [2; {1, 4}]. - _Magus K. Chu_, Sep 06 2022
%H A154516 Vincenzo Librandi, <a href="/A154516/b154516.txt">Table of n, a(n) for n = 1..10000</a>
%H A154516 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A154516 a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - _Vincenzo Librandi_, Jan 30 2012
%F A154516 G.f.: x*(-8-10*x)/(x-1)^3. - _Vincenzo Librandi_, Jan 30 2012
%t A154516 LinearRecurrence[{3, -3, 1}, {8, 34, 78}, 50] (* _Vincenzo Librandi_, Jan 30 2012 *)
%o A154516 (Sage) [lucas_number1(3,3*n,n) for n in range(0, 41)] # _Zerinvary Lajos_, Nov 20 2009
%o A154516 (PARI) a(n)=9*n^2-n \\ _Charles R Greathouse IV_, Dec 27 2011
%o A154516 (Magma) I:=[8, 34, 78]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Jan 30 2012
%Y A154516 Cf. A154514, A154518.
%K A154516 nonn,easy
%O A154516 1,1
%A A154516 _Vincenzo Librandi_, Jan 11 2009