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%I #19 Sep 08 2022 08:45:42
%S 85,97,113,397,485,593,2297,2813,3445,13385,16393,20077,78013,95545,
%T 117017,454693,556877,682025,2650145,3245717,3975133,15446177,
%U 18917425,23168773,90026917,110258833,135037505,524715325,642635573,787056257
%N Positive numbers y such that y^2 is of the form x^2 + (x+97)^2 with integer x.
%C (-13,a(1)) and (A129836(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+97)^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) = (99+14*sqrt(2))/97 for n mod 3 = {0, 2}.
%C Lim_{n -> infinity} a(n)/a(n-1) = (19491+12070*sqrt(2))/97^2 for n mod 3 = 1.
%C For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, the x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - _Mohamed Bouhamida_, Sep 09 2009
%H G. C. Greubel, <a href="/A157469/b157469.txt">Table of n, a(n) for n = 1..1000</a>
%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)=85, a(2)=97, a(3)=113, a(4)=397, a(5)=485, a(6)=593.
%F G.f.: (1-x)*(85 + 182*x + 295*x^2 + 182*x^3 + 85*x^4)/(1-6*x^3+x^6).
%F a(3*k-1) = 97*A001653(k) for k >= 1.
%e (-13, a(1)) = (-13, 85) is a solution: (-13)^2+(-13+97)^2 = 169+7056 = 7225 = 85^2.
%e (A129836(1), a(2)) = (0, 97) is a solution: 0^2+(0+97)^2 = 9409 = 97^2.
%e (A129836(3), a(4)) = (228, 397) is a solution: 228^2+(228+97)^2 = 51984+105625 = 157609 = 397^2.
%t LinearRecurrence[{0,0,6,0,0,-1},{85,97,113,397,485,593},30] (* _Harvey P. Dale_, Apr 04 2013 *)
%o (PARI) {forstep(n=-20, 800000000, [3, 1], if(issquare(2*n^2+194*n+9409, &k), print1(k, ",")))};
%o (Magma) I:=[85,97,113,397,485,593]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // _G. C. Greubel_, Mar 31 2018
%Y Cf. A129836, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, Mar 12 2009