|
%I #4 Feb 18 2024 11:40:37
%S 845,953,1093,3977,4765,5713,23017,27637,33185,134125,161057,193397,
%T 781733,938705,1127197,4556273,5471173,6569785,26555905,31888333,
%U 38291513,154779157,185858825,223179293,902119037,1083264617,1300784245
%N Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x.
%C (-116, a(1)) and (A129975(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 1.
%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)=845, a(2)=953, a(3)=1093, a(4)=3977, a(5)=4765, a(6)=5713.
%F G.f.: (1-x)*(845+1798*x+2891*x^2+1798*x^3+845*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 953*A001653(k) for k >= 1.
%e (-116, a(1)) = (-116, 845) is a solution: (-116)^2+(-116+953)^2 = 13456+700569 = 714025 = 845^2.
%e (A129975(1), a(2)) = (0, 953) is a solution: 0^2+(0+953)^2 = 908209 = 953^2.
%e (A129975(3), a(4)) = (2295, 3977) is a solution: 2295^2+(2295+953)^2 = 5267025+10549504 = 15816529 = 3977^2.
%t LinearRecurrence[{0,0,6,0,0,-1},{845,953,1093,3977,4765,5713},30] (* _Harvey P. Dale_, Feb 18 2024 *)
%o (PARI) {forstep(n=-116, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209, &k), print1(k, ",")))}
%Y Cf. A129975, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, May 18 2009
|
