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A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

%I A130646 #11 Feb 15 2020 10:52:27
%S A130646 0,56,1925,2181,2465,13056,14540,16188,77865,86513,96117,455588,
%T A130646 505992,561968,2657117,2950893,3277145,15488568,17200820,19102356,
%U A130646 90275745,100255481,111338445,526167356,584333520,648929768,3066729845
%N A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.
%C A130646 Also values x of Pythagorean triples (x, x+727, y).
%C A130646 Corresponding values y of solutions (x, y) are in A159893.
%C A130646 For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
%C A130646 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A130646 lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.
%C A130646 lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.
%H A130646 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A130646 a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.
%F A130646 G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A130646 a(3*k+1) = 727*A001652(k) for k >= 0.
%t A130646 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* _Harvey P. Dale_, Jan 16 2013 *)
%o A130646 (PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}
%Y A130646 Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).
%K A130646 nonn,easy
%O A130646 1,2
%A A130646 _Mohamed Bouhamida_, Jun 20 2007
%E A130646 Edited and one term added by _Klaus Brockhaus_, Apr 30 2009