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Numbers m such that 10*m^2 - 6 is a square.
4

%I #36 Mar 16 2024 11:20:44

%S 1,7,31,265,1177,10063,44695,382129,1697233,14510839,64450159,

%T 551029753,2447408809,20924619775,92937084583,794584521697,

%U 3529161805345,30173287204711,134015211518527,1145790329257321,5089048875898681,43509859224573487

%N Numbers m such that 10*m^2 - 6 is a square.

%C See the first two comments on A221874.

%C For the corresponding numbers whose square is 10*m^2 - 6, see A281647. - _Jon E. Schoenfield_, Aug 05 2023

%H Vincenzo Librandi, <a href="/A221875/b221875.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,38,0,-1).

%F G.f.: x*(1-x)*(1+8*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2)).

%F a(n) = ((5-t*(-1)^n)*(3+t)^(2*floor(n/2)) + (5+t*(-1)^n)*(3-t)^(2*floor(n/2)))/10, where t=sqrt(10).

%F a(n)*a(n-3) - a(n-1)*a(n-2) = 36 + 12(-1)^n.

%t LinearRecurrence[{0, 38, 0, -1}, {1, 7, 31, 265}, 22]

%o (Magma)

%o m:=22; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1+8*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2))));

%o (Maxima)

%o makelist(expand(((5-sqrt(10)*(-1)^n)*(3+sqrt(10))^(2*floor(n/2))+(5+sqrt(10)*(-1)^n)*(3-sqrt(10))^(2*floor(n/2)))/10), n, 1, 22);

%o (Magma) I:=[1, 7, 31, 265]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Mar 25 2013

%Y Cf. A221874, A281647.

%K nonn,easy

%O 1,2

%A _Bruno Berselli_, Jan 28 2013