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%I #74 Sep 08 2022 08:46:04
%S 1,2,22,41,439,818,8758,16319,174721,325562,3485662,6494921,69538519,
%T 129572858,1387284718,2584962239,27676155841,51569671922,552135832102,
%U 1028808476201,11015040486199,20524599852098,219748673891878,409463188565759
%N Numbers m such that 11*m^2 + 5 is a square.
%C Corresponding squares are: 16, 49, 5329, 18496, 2119936, 7360369, 843728209, 2929407376, ... (subsequence of A016778).
%C The Diophantine equation 11*x^2+k = y^2, for |k|<11, has integer solutions with the following k values:
%C k = -10, the nonnegative x values are in A198947;
%C k = -8, " 2*A075839;
%C k = -7, " A221763;
%C k = -2, " A075839;
%C k = 1, " A001084;
%C k = 4, " A075844;
%C k = 5, " this sequence;
%C k = 9, " 3*A001084.
%C Also, the Diophantine equation h*x^2+5 = y^2 has infinitely many integer solutions for h = 5, 11, 19, 20, 29, 31, 41, 44, 55, 59, ...
%C a(n+1)/a(n) tends alternately to (1+sqrt(11))^2/10 and (4+sqrt(11))^2/5.
%C a(n+2)/a(n) tends to A176395^2/2.
%H Bruno Berselli, <a href="/A221762/b221762.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,20,0,-1).
%F G.f.: x*(1+2*x+2*x^2+x^3)/(1-20*x^2+x^4).
%F a(n) = -a(1-n) = ((-11*(-1)^n+4*t)*(10+3*t)^floor(n/2)-(11*(-1)^n+4*t)*(10-3*t)^floor(n/2))/22, where t=sqrt(11).
%F a(n) = 20*a(n-2) - a(n-4) for n>4, a(1)=1, a(2)=2, a(3)=22, a(4)=41.
%F a(n)*a(n-3)-a(n-1)*a(n-2) = -(3/2)*(9-7*(-1)^n).
%F a(n+1) + a(n-1) = A198949(n), with a(0)=-1.
%F 2*a(n-1) - a(n) = A001084(n/2-1) for even n.
%p A221762:=proc(q)
%p local n;
%p for n from 1 to q do if type(sqrt(11*n^2+5), integer) then print(n);
%p fi; od; end:
%p A221762(1000); # _Paolo P. Lava_, Feb 19 2013
%t LinearRecurrence[{0, 20, 0, -1}, {1, 2, 22, 41}, 24]
%t CoefficientList[Series[(1 + 2 x + 2 x^2 + x^3)/(1 - 20 x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o (Maxima) makelist(expand(((-11*(-1)^n+4*sqrt(11))*(10+3*sqrt(11))^floor(n/2)-(11*(-1)^n+4*sqrt(11))*(10-3*sqrt(11))^floor(n/2))/22), n, 1, 24);
%o (Magma) m:=24; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2+x^3)/(1-20*x^2+x^4)));
%o (Magma) I:=[1,2,22,41]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Aug 18 2013
%Y Cf. A001084, A075839, A075844, A198947, A198949, A221763.
%Y Cf. A049629 (numbers m such that 20*m^2 + 5 is a square), A075796 (numbers m such that 5*m^2 + 5 is a square).
%K nonn,easy
%O 1,2
%A _Bruno Berselli_, Jan 24 2013
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