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Positive integers whose square is the sum of 88 consecutive squares.
4

%I #9 Sep 08 2022 08:46:12

%S 2222,2530,39358,55990,872938,994598,15506810,22059818,343935350,

%T 391869082,6109643782,8691512302,135509654962,154395423710,

%U 2407184143298,3424433787170,53390460119678,60831405072658,948424442815630,1349218220632678,21035705777498170

%N Positive integers whose square is the sum of 88 consecutive squares.

%C Positive integers x in the solutions to 2*x^2-176*y^2-15312*y-446600 = 0.

%H Colin Barker, <a href="/A257826/b257826.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,394,0,0,0,-1).

%F a(n) = 394*a(n-4)-a(n-8).

%F G.f.: -22*x*(11*x^7+11*x^6+101*x^5+115*x^4-2545*x^3-1789*x^2-115*x-101) / (x^8-394*x^4+1).

%e 2222 is in the sequence because 2222^2 = 4937284 = 192^2+193^2+...+279^2.

%t LinearRecurrence[{0, 0, 0, 394, 0, 0, 0, -1}, {2222, 2530, 39358, 55990, 872938, 994598, 15506810, 22059818}, 40] (* _Vincenzo Librandi_, May 11 2015 *)

%o (PARI) Vec(-22*x*(11*x^7+11*x^6+101*x^5+115*x^4-2545*x^3-1789*x^2-115*x-101) / (x^8-394*x^4+1) + O(x^100))

%o (Magma) I:=[2222,2530,39358,55990,872938,994598,15506810, 22059818]; [n le 8 select I[n] else 394*Self(n-4)-Self(n-8): n in [1..40]]; // _Vincenzo Librandi_, May 11 2015

%Y Cf. A001653, A180274, A218395, A257761, A257765, A257767, A257780, A257781, A257823-A257825, A257827, A257828.

%K nonn,easy

%O 1,1

%A _Colin Barker_, May 10 2015