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A221874 Numbers m such that 10*m^2 + 6 is a square.

%I A221874 #40 Aug 05 2023 21:19:14
%S A221874 1,5,43,191,1633,7253,62011,275423,2354785,10458821,89419819,
%T A221874 397159775,3395598337,15081612629,128943316987,572704120127,
%U A221874 4896450447169,21747674952197,185936173675435,825838944063359,7060678149219361,31360132199455445
%N A221874 Numbers m such that 10*m^2 + 6 is a square.
%C A221874 The Diophantine equation 10*x^2 + k = y^2, for |k| < 10, has integer solutions with the following k values:
%C A221874 k =  1, the nonnegative x values are in A084070;
%C A221874 k = -1,               "                 A097315;
%C A221874 k =  4,               "                 2*A084070;
%C A221874 k = -4,               "                 2*A097315;
%C A221874 k =  6,               "                 this sequence;
%C A221874 k = -6,               "                 A221875;
%C A221874 k =  9,               "                 A075836;
%C A221874 k = -9,               "                 A052454.
%C A221874 a(n+1)/a(n) tends alternately to (sqrt(2)+sqrt(5))^2/3 and (2*sqrt(2)+sqrt(5))^2/3; a(n+2)/a(n) tends to A176398^2.
%H A221874 Vincenzo Librandi, <a href="/A221874/b221874.txt">Table of n, a(n) for n = 1..1000</a>
%H A221874 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,38,0,-1).
%F A221874 G.f.: x*(1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2)).
%F A221874 a(n) = ((-5*(-1)^n+2*t)*(3+t)^(2*floor(n/2)) - (5*(-1)^n+2*t)*(3-t)^(2*floor(n/2)))/10, where t=sqrt(10).
%F A221874 a(n) = 2*A129556(n) + 1.
%F A221874 a(n)*a(n-3) - a(n-1)*a(n-2) = -36 + 12(-1)^n.
%p A221874 A221874:=proc(q)
%p A221874 local n;
%p A221874 for n from 1 to q do if type(sqrt(10*n^2+6),integer) then print(n);
%p A221874 fi; od; end:
%p A221874 A221874(100000000000000000); # _Paolo P. Lava_, Feb 11 2013
%t A221874 LinearRecurrence[{0, 38, 0, -1}, {1, 5, 43, 191}, 22]
%o A221874 (Magma) m:=22; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2))));
%o A221874 (Maxima) makelist(expand(((-5*(-1)^n+2*sqrt(10))*(3+sqrt(10))^(2*floor(n/2))-(5*(-1)^n+2*sqrt(10))*(3-sqrt(10))^(2*floor(n/2)))/10), n, 1, 22);
%Y A221874 Cf. A097315, A084070, A075836, A052454, A129556, A221875.
%Y A221874 Subsequence of A031150.
%K A221874 nonn,easy
%O A221874 1,2
%A A221874 _Bruno Berselli_, Jan 28 2013