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%I #13 Aug 03 2021 19:45:38
%S 0,16,1440,389295,33994015,1802660535,2968421391,157411709575,
%T 13745486642391,3714721448623416,324376465956415720,
%U 17201282202880383816,28325163305411682840,1502048325307681783960,131161685794667995415400,35446480734732882983897895
%N Numbers n such that 67*n^2 + 67*n + 1 is a square.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,9542163854,-9542163854,0,0,0,0,-1,1).
%F a(1)=0, a(2)=16, a(3)=1440, a(4)=389295, a(5)=33994015, a(6)=1802660535, a(7)=9542163854*a(1)+4771081926-a(6), a(8)=9542163854*a(2)+4771081926-a(5), a(9)=9542163854*a(3)+4771081926-a(4), a(10)=9542163854*a(4)+4771081926-a(3), a(11)=9542163854*a(5)+4771081926-a(2), a(12)=9542163854*a(6)+4771081926-a(1), then a(n)=9542163854*a(n-6)+4771081926-a(n-12).
%F G.f.: -x^2*(16*x^10 +1424*x^9 +387855*x^8 +33604720*x^7 +1768666520*x^6 +1165760856*x^5 +1768666520*x^4 +33604720*x^3 +387855*x^2 +1424*x +16) / ((x -1)*(x^6 -97684*x^3 +1)*(x^6 +97684*x^3 +1)). [_Colin Barker_, Mar 07 2013]
%t CoefficientList[Series[-x^2(16x^10+1424x^9+387855x^8+33604720x^7+1768666520x^6+1165760856x^5+1768666520x^4+33604720x^3+387855x^2+1424x+16)/((x-1)(x^6-97684x^3+1)(x^6+97684x^3+1)),{x,0,30}],x] (* or *) LinearRecurrence[{1,0,0,0,0,9542163854,-9542163854,0,0,0,0,-1,1},{0,0,16,1440,389295,33994015,1802660535,2968421391,157411709575,13745486642391,3714721448623416,324376465956415720,17201282202880383816},30] (* _Harvey P. Dale_, Aug 03 2021 *)
%Y Cf. A106175 (square roots of 67*a(n)^2+67*a(n)+1).
%K nonn,easy
%O 1,2
%A _Pierre CAMI_, Apr 24 2005
%E a(15)-a(16) from _Colin Barker_, Mar 07 2013
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