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A159641 Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x.

%I A159641 #12 Apr 18 2024 06:11:48
%S A159641 613,647,685,2993,3235,3497,17345,18763,20297,101077,109343,118285,
%T A159641 589117,637295,689413,3433625,3714427,4018193,20012633,21649267,
%U A159641 23419745,116642173,126181175,136500277,679840405,735437783,795581917
%N A159641 Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x.
%C A159641 (-35,a(1)) and (A130013(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.
%H A159641 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159641 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=613, a(2)=647, a(3)=685, a(4)=2993, a(5)=3235, a(6)=3497.
%F A159641 G.f.: (1-x)*(613+1260*x+1945*x^2+1260*x^3+613*x^4) / (1-6*x^3+x^6).
%F A159641 a(3*k-1) = 647*A001653(k) for k >= 1.
%F A159641 Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
%F A159641 Limit_{n -> oo} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {0, 2}.
%F A159641 Limit_{n -> oo} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 1.
%e A159641 (-35, a(1)) = (-35, 613) is a solution: (-35)^2+(-35+647)^2 = 1225+374544 = 375769 = 613^2.
%e A159641 (A130013(1), a(2)) = (0, 647) is a solution: 0^2+(0+647)^2 = 418609 = 647^2.
%e A159641 (A130013(3), a(4)) = (1768, 2993) is a solution: 1768^2+(1768+647)^2 = 3125824+5832225 = 8958049 = 2993^2.
%t A159641 LinearRecurrence[{0,0,6,0,0,-1},{613,647,685,2993,3235,3497},30] (* _Harvey P. Dale_, Jun 22 2022 *)
%o A159641 (PARI) {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609, &k), print1(k, ",")))}
%Y A159641 Cf. A130013, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).
%K A159641 nonn,easy
%O A159641 1,1
%A A159641 _Klaus Brockhaus_, Apr 21 2009