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A257781 Positive integers whose square is the sum of 50 consecutive squares.

%I A257781 #14 Aug 07 2025 13:14:53
%S A257781 245,385,495,655,795,1055,1365,2205,2855,3795,4615,6135,7945,12845,
%T A257781 16635,22115,26895,35755,46305,74865,96955,128895,156755,208395,
%U A257781 269885,436345,565095,751255,913635,1214615,1573005,2543205,3293615,4378635,5325055,7079295
%N A257781 Positive integers whose square is the sum of 50 consecutive squares.
%C A257781 Positive integers x in the solutions to 2*x^2-100*y^2-4900*y-80850 = 0.
%H A257781 Colin Barker, <a href="/A257781/b257781.txt">Table of n, a(n) for n = 1..1000</a>
%H A257781 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,6,0,0,0,0,0,-1).
%F A257781 a(n) = 6*a(n-6)-a(n-12).
%F A257781 G.f.: -5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)).
%e A257781 245 is in the sequence because 245^2 = 60025 = 7^2+8^2+...+56^2.
%t A257781 LinearRecurrence[{0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -1}, {245, 385, 495, 655, 795, 1055, 1365, 2205, 2855, 3795, 4615, 6135}, 50] (* _Vincenzo Librandi_, May 11 2015 *)
%t A257781 Select[Sqrt[Total/@Partition[Range[10^6]^2,50,1]],IntegerQ] (* _Harvey P. Dale_, Aug 07 2025 *)
%o A257781 (PARI) Vec(-5*x*(39*x^11 +31*x^10 +27*x^9 +23*x^8 +21*x^7 +21*x^6 -211*x^5 -159*x^4 -131*x^3 -99*x^2 -77*x -49) / ((x^6 -2*x^3 -1)*(x^6 +2*x^3 -1)) + O(x^100))
%o A257781 (Magma) I:=[245,385,495,655,795,1055,1365,2205,2855,3795, 4615,6135]; [n le 12 select I[n] else 6*Self(n-6)-Self(n-12): n in [1..40]]; // _Vincenzo Librandi_, May 11 2015
%Y A257781 Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257823-A257828.
%K A257781 nonn,easy
%O A257781 1,1
%A A257781 _Colin Barker_, May 08 2015