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%I #12 Sep 08 2022 08:46:13
%S 105,3248,97433,2919840,87497865,2622016208,78572988473,2354567638080,
%T 70558456154025,2114399116982768,63361415053329113,
%U 1898728052482890720,56898480159433392585,1705055676730518886928,51094771821756133215353,1531138098975953477573760
%N The first of eight consecutive positive integers the sum of the squares of which is equal to the sum of the squares of seven consecutive positive integers.
%C For the first of the corresponding seven consecutive positive integers, see A262074.
%H Colin Barker, <a href="/A262075/b262075.txt">Table of n, a(n) for n = 1..676</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (31,-31,1).
%F a(n) = 31*a(n-1)-31*a(n-2)+a(n-3) for n>3.
%F G.f.: 7*x*(x-15) / ((x-1)*(x^2-30*x+1)).
%e 105 is in the sequence because 105^2 + ... + 112^2 (8 terms) = 94220 = 113^2 + ... + 119^2 (7 terms).
%t LinearRecurrence[{31, -31, 1}, {105, 3248, 97433}, 20] (* _Vincenzo Librandi_, Sep 11 2015 *)
%t Table[1/4 (-14-(15-4 Sqrt[14])^n (7+2 Sqrt[14])+(-7+2 Sqrt[14]) (15+4 Sqrt[ 14])^n),{n,2,20}]//Simplify (* _Harvey P. Dale_, Aug 01 2019 *)
%o (PARI) Vec(7*x*(x-15)/((x-1)*(x^2-30*x+1)) + O(x^20))
%o (Magma) I:=[105,3248,97433]; [n le 3 select I[n] else 31*Self(n-1)-31*Self(n-2)+Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Sep 11 2015
%Y Cf. A262074, A262076, A262077.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Sep 10 2015
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