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%I #20 Jun 13 2015 00:54:55
%S 0,1,4,9,16,17,20,25,33,36,41,48,49,52,57,64,65,68,73,80,81,84,89,97,
%T 100,105,112,113,116,121,128,129,132,137,144,145,148,153,161,164,169,
%U 176,177,180,185,192,193,196,201,208,209,212,217,225,228,233,240
%N Values of n for which the equation x^2 - 16*y^2 = n has integer solutions.
%C This equation is a Pellian equation of the form x^2 - D^2*y^2 = N.
%H Colin Barker, <a href="/A233457/b233457.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F G.f.: x^2*(x +1)*(7*x^3 +5*x^2 +3*x +1)*(x^4 +1)*(x^6 -x^5 +x^4 -x^3 +x^2 -x +1) / ((x -1)^2*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)*(x^8 -x^7 +x^5 -x^4 +x^3 -x +1)).
%e 33 is in the sequence because the equation x^2 - 16*y^2 = 33 has solutions (X,Y) = (7,1) and (17,4).
%t LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,4,9,16,17,20,25,33,36,41,48,49,52,57,64},60] (* _Harvey P. Dale_, Sep 06 2014 *)
%o (PARI) concat(0, Vec((7*x^15 +5*x^14 +3*x^13 +x^12 +7*x^11 +5*x^10 +3*x^9 +8*x^8 +5*x^7 +3*x^6 +x^5 +7*x^4 +5*x^3 +3*x^2 +x)/(x^16 -x^15 -x +1) + O(x^100)))
%Y Cf. A042965, A230239, A230240.
%K nonn,easy
%O 1,3
%A _Colin Barker_, Mar 18 2014
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