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%I #13 Sep 08 2022 08:45:44
%S 145,167,197,673,835,1037,3893,4843,6025,22685,28223,35113,132217,
%T 164495,204653,770617,958747,1192805,4491485,5587987,6952177,26178293,
%U 32569175,40520257,152578273,189827063,236169365,889291345,1106393203
%N Positive numbers y such that y^2 is of the form x^2+(x+167)^2 with integer x.
%C (-24, a(1)) and (A130608(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+167)^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) = (171+26*sqrt(2))/167 for n mod 3 = {0, 2}.
%C Lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 1.
%C For the generic case x^2+(x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by: a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by: b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4) = 5*p, b(5) = 5*m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - _Mohamed Bouhamida_, Sep 09 2009
%H G. C. Greubel, <a href="/A159777/b159777.txt">Table of n, a(n) for n = 1..3900</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)=145, a(2)=167, a(3)=197, a(4)=673, a(5)=835, a(6)=1037.
%F G.f.: (1-x)*(145+312*x+509*x^2+312*x^3+145*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 167*A001653(k) for k >= 1.
%e (-24, a(1)) = (-24, 145) is a solution: (-24)^2 + (-24+167)^2 = 576 + 20449 = 21025 = 145^2.
%e (A130608(1), a(2)) = (0, 167) is a solution: 0^2 + (0+167)^2 = 27889 = 167^2.
%e (A130608(3), a(4)) = (385, 673) is a solution: 385^2 + (385+167)^2 = 148225 + 304704 = 452929 = 673^2.
%t LinearRecurrence[{0,0,6,0,0,-1}, {145,167,197,673,835,1037}, 50] (* _G. C. Greubel_, May 21 2018 *)
%o (PARI) {forstep(n=-24, 10000000, [1, 3], if(issquare(2*n^2+334*n+27889, &k), print1(k, ",")))};
%o (Magma) I:=[145,167,197,673,835,1037]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 21 2018
%Y Cf. A130608, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, Apr 30 2009
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