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%I #15 Apr 18 2024 12:51:43
%S 221,241,265,1061,1205,1369,6145,6989,7949,35809,40729,46325,208709,
%T 237385,270001,1216445,1383581,1573681,7089961,8064101,9172085,
%U 41323321,47001025,53458829,240849965,273942049,311580889,1403776469,1596651269
%N Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x.
%C (-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.
%F G.f.: x*(1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 241*A001653(k) for k >= 1.
%F Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
%F Limit_{n -> oo} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.
%F Limit_{n -> oo} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 1.
%e (-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.
%e (A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.
%e (A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.
%t LinearRecurrence[{0,0,6,0,0,-1},{221,241,265,1061,1205,1369},30] (* _Harvey P. Dale_, Nov 21 2011 *)
%o (PARI) {forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ",")))}
%Y Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).
%K nonn,easy,changed
%O 1,1
%A _Klaus Brockhaus_, Apr 16 2009
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