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A160200 Positive numbers y such that y^2 is of the form x^2+(x+761)^2 with integer x. 3

%I #2 Mar 30 2012 17:28:01

%S 541,761,1465,1781,3805,8249,10145,22069,48029,59089,128609,279925,

%T 344389,749585,1631521,2007245,4368901,9509201,11699081,25463821,

%U 55423685,68187241,148414025,323032909,397424365,865020329,1882773769

%N Positive numbers y such that y^2 is of the form x^2+(x+761)^2 with integer x.

%C (-341, a(1)) and (A122694(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+761)^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) = (1003+462*sqrt(2))/761 for n mod 3 = {0, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 for n mod 3 = 1.

%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=541, a(2)=761, a(3)=1465, a(4)=1781, a(5)=3805, a(6)=8249.

%F G.f.: (1-x)*(541+1302*x+2767*x^2+1302*x^3+541*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 761*A001653(k) for k >= 1.

%e (-341, a(1)) = (-341, 541) is a solution: (-341)^2+(-341+761)^2 = 116281+176400 = 292681 = 541^2.

%e (A122694(1), a(2)) = (0, 761) is a solution: 0^2+(0+761)^2 = 579121 = 761^2.

%e (A122694(3), a(4)) = (820, 1781) is a solution: 820^2+(820+761)^2 = 672400+2499561 = 3171961 = 1781^2.

%o (PARI) {forstep(n=-344, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121, &k), print1(k, ",")))}

%Y Cf. A122694, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160201 (decimal expansion of (1003+462*sqrt(2))/761), A160202 (decimal expansion of (591603+85478*sqrt(2))/761^2).

%K nonn

%O 1,1

%A _Klaus Brockhaus_, May 18 2009

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