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%I #12 Sep 08 2022 08:46:12
%S 652,724,788,1012,1828,2372,2596,2908,6164,6908,7564,9836,17996,23404,
%T 25628,28724,60988,68356,74852,97348,178132,231668,253684,284332,
%U 603716,676652,740956,963644,1763324,2293276,2511212,2814596,5976172,6698164,7334708
%N Positive integers whose square is the sum of 96 consecutive squares.
%C Positive integers x in the solutions to 2*x^2-192*y^2-18240*y-580640 = 0.
%H Colin Barker, <a href="/A257827/b257827.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,-1).
%F a(n) = 10*a(n-8) -a(n-16).
%F G.f.: -4*x*(89*x^15 +83*x^14 +79*x^13 +71*x^12 +71*x^11 +79*x^10 +83*x^9 +89*x^8 -727*x^7 -649*x^6 -593*x^5 -457*x^4 -253*x^3 -197*x^2 -181*x-163) / (x^16 -10*x^8 +1).
%e 652 is in the sequence because 652^2 = 425104 = 13^2+14^2+...+108^2.
%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, -1}, {652, 724, 788, 1012, 1828, 2372, 2596, 2908, 6164, 6908, 7564, 9836, 17996, 23404, 25628, 28724}, 40] (* _Vincenzo Librandi_, May 11 2015 *)
%o (PARI)
%o Vec(-4*x*(89*x^15 +83*x^14 +79*x^13 +71*x^12 +71*x^11 +79*x^10 +83*x^9 +89*x^8 -727*x^7 -649*x^6 -593*x^5 -457*x^4 -253*x^3 -197*x^2 -181*x -163) / (x^16-10*x^8+1) + O(x^100))
%o (Magma) I:=[652,724,788,1012,1828,2372,2596,2908,6164,6908, 7564,9836,17996,23404,25628,28724]; [n le 16 select I[n] else 10*Self(n-8)-Self(n-16): n in [1..40]]; // _Vincenzo Librandi_, May 11 2015
%Y Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257826, A257828.
%K nonn,easy
%O 1,1
%A _Colin Barker_, May 10 2015
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