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A156066 Numbers n with property that n^2 is a square arising in A154138.

%I A156066 #28 Sep 08 2022 08:45:41
%S A156066 2,3,9,16,52,93,303,542,1766,3159,10293,18412,59992,107313,349659,
%T A156066 625466,2037962,3645483,11878113,21247432,69230716,123839109,
%U A156066 403506183,721787222,2351806382,4206884223,13707332109,24519518116,79892186272
%N A156066 Numbers n with property that n^2 is a square arising in A154138.
%C A156066 Except for the first term, positive values of x (or y) satisfying x^2 - 6xy + y^2 + 23 = 0. - _Colin Barker_, Feb 08 2014
%H A156066 Vincenzo Librandi, <a href="/A156066/b156066.txt">Table of n, a(n) for n = 1..1000</a>
%H A156066 Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://ajc.maths.uq.edu.au/pdf/77/ajc_v77_p318.pdf">Counting families of generalized balancing numbers</a>, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
%H A156066 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A156066 a(n) = sqrt((A154138(n)^2 + A154138(n) + 6)/2).
%F A156066 a(1..4) = (2,3,9,16); a(n>4) = 6*a(n-2) - a(n-4).
%F A156066 G.f.: -x*(x-1)*(x+2)*(2*x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). - _Colin Barker_, Feb 08 2014
%F A156066 a(n) = A006452(n-1) - A006452(n) + A006452(n+1). - _Carl Najafi_, Sep 27 2018
%p A156066 seq(coeff(series(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)),x,n+1), x, n), n = 1..30); # _Muniru A Asiru_, Sep 28 2018
%t A156066 a[1]=2;a[2]=3;a[3]=9;a[4]=16;a[n_]:=a[n]=6*a[n-2]-a[n-4];A1=Table[a[n],{n,25}]
%t A156066 CoefficientList[Series[-(x - 1) (x + 2) (2 x + 1)/((x^2 - 2 x - 1) (x^2 + 2 x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 11 2014 *)
%o A156066 (PARI) Vec(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^100)) \\ _Colin Barker_, Feb 08 2014
%o A156066 (Magma) I:=[2,3,9,16]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Feb 11 2014
%o A156066 (GAP) a:=[2,3,9,16];; for n in [5..30] do a[n]:=6*a[n-2]-a[n-4]; od; a; # _Muniru A Asiru_, Sep 28 2018
%Y A156066 Cf. A154138.
%K A156066 nonn,easy
%O A156066 1,1
%A A156066 _Zak Seidov_, Oct 21 2009