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%I #13 Sep 08 2022 08:46:12
%S 679,1545404,3675742735,81619738879,194132514608060,
%T 461744104375531831,10253011689091642135,24386783991798773338556,
%U 58003955471481693294113311,1287975802673112210113634031,3063449905150311732357259611836,7286414311424213782299531873117895
%N Positive integers whose square is the sum of 97 consecutive squares.
%C Positive integers x in the solutions to 2*x^2-194*y^2-18624*y-599072 = 0.
%H Colin Barker, <a href="/A257828/b257828.txt">Table of n, a(n) for n = 1..370</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,125619266,0,0,-1).
%F a(n) = 125619266*a(n-3)-a(n-6).
%F G.f.: -679*x*(x-1)*(x^4+2277*x^3+5415742*x^2+2277*x+1) / (x^6-125619266*x^3+1).
%e 679 is in the sequence because 679^2 = 461041 = 15^2+16^2+...+111^2.
%t LinearRecurrence[{0, 0, 125619266, 0, 0, -1}, {679, 1545404, 3675742735, 81619738879, 194132514608060, 461744104375531831}, 30] (* _Vincenzo Librandi_, May 11 2015 *)
%t Rest[CoefficientList[Series[-679x(x-1)(x^4+2277x^3+5415742x^2+ 2277x+1)/ (x^6-125619266x^3+1),{x,0,15}],x]] (* _Harvey P. Dale_, Aug 02 2021 *)
%o (PARI) Vec(-679*x*(x-1)*(x^4+2277*x^3+5415742*x^2+2277*x+1) / (x^6-125619266*x^3+1) + O(x^100))
%o (Magma) I:=[679,1545404,3675742735,81619738879, 194132514608060,461744104375531831]; [n le 6 select I[n] else 125619266*Self(n-3)-Self(n-6): n in [1..20]]; // _Vincenzo Librandi_, May 11 2015
%Y Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257826, A257827.
%K nonn,easy
%O 1,1
%A _Colin Barker_, May 10 2015
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