cp's OEIS Frontend

A160580 Positive numbers y such that y^2 is of the form x^2+(x+457)^2 with integer x.

%I A160580 #2 Mar 30 2012 17:28:01
%S A160580 325,457,877,1073,2285,4937,6113,13253,28745,35605,77233,167533,
%T A160580 207517,450145,976453,1209497,2623637,5691185,7049465,15291677,
%U A160580 33170657,41087293,89126425,193332757,239474293,519466873,1126825885,1395758465
%N A160580 Positive numbers y such that y^2 is of the form x^2+(x+457)^2 with integer x.
%C A160580 (-204, a(1)) and (A129642(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.
%C A160580 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A160580 lim_{n -> infinity} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {0, 2}.
%C A160580 lim_{n -> infinity} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 1.
%F A160580 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=457, a(3)=877, a(4)=1073, a(5)=2285, a(6)=4937.
%F A160580 G.f.: (1-x)*(325+782*x+1659*x^2+782*x^3+325*x^4) / (1-6*x^3+x^6).
%F A160580 a(3*k-1) = 457*A001653(k) for k >= 1.
%e A160580 (-204, a(1)) = (-204, 325) is a solution: (-204)^2+(-204+457)^2 = 41616+64009 = 105625 = 325^2.
%e A160580 (A129642(1), a(2)) = (0, 457) is a solution: 0^2+(0+457)^2 = 208849 = 457^2.
%e A160580 (A129642(3), a(4)) = (495, 1073) is a solution: 495^2+(495+457)^2 = 245025+906304 = 1151329 = 1073^2.
%o A160580 (PARI) {forstep(n=-204, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849, &k), print1(k, ",")))}
%Y A160580 Cf. A129642, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).
%K A160580 nonn
%O A160580 1,1
%A A160580 _Klaus Brockhaus_, Jun 08 2009