%I #10 Apr 18 2024 06:11:16
%S 545,577,613,2657,2885,3133,15397,16733,18185,89725,97513,105977,
%T 522953,568345,617677,3047993,3312557,3600085,17765005,19306997,
%U 20982833,103542037,112529425,122296913,603487217,655869553,712798645,3517381265
%N Positive numbers y such that y^2 is of the form x^2+(x+577)^2 with integer x.
%C (-33,a(1)) and (A130005(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+577)^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)=545, a(2)=577, a(3)=613, a(4)=2657, a(5)=2885, a(6)=3133.
%F G.f.: (1-x)*(545+1122*x+1735*x^2+1122*x^3+545*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 577*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) = (579+34*sqrt(2))/577 for n mod 3 = {0, 2}.
%F Limit_{n -> oo} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 1.
%e (-33, a(1)) = (-33, 545) is a solution: (-33)^2+(-33+577)^2 = 1089+295936 = 297025 = 545^2.
%e (A130005(1), a(2)) = (0, 577) is a solution: 0^2+(0+577)^2 = 332929 = 577^2.
%e (A130005(3), a(4)) = (1568, 2657) is a solution: 1568^2+(1568+577)^2 = 2458624+4601025 = 7059649 = 2657^2.
%o (PARI) {forstep(n=-36, 50000000, [3, 1], if(issquare(2*n^2+1154*n+332929, &k), print1(k, ",")))}
%Y Cf. A130005, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, Apr 21 2009