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A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x. 4

%I #15 Sep 08 2022 08:45:42

%S 25,31,41,109,155,221,629,899,1285,3665,5239,7489,21361,30535,43649,

%T 124501,177971,254405,725645,1037291,1482781,4229369,6045775,8642281,

%U 24650569,35237359,50370905,143674045,205378379,293583149,837393701

%N Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

%C (-7,a(1)) and (A118674(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+31)^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) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}.

%C Lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 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="/A157646/b157646.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)=25, a(2)=31, a(3)=41, a(4)=109, a(5)=155, a(6)=221.

%F G.f.: (1-x)*(25 + 56*x + 97*x^2 + 56*x^3 + 25*x^4)/(1 - 6*x^3 + x^6).

%F a(3*k-1) = 31*A001653(k) for k >= 1.

%e (-7, a(1)) = (-7, 25) is a solution: (-7)^2+(-7+31)^2 = 49+576 = 625 = 25^2.

%e (A118674(1), a(2)) = (0, 31) is a solution: 0^2+(0+31)^2 = 961 = 31^2.

%e (A118674(3), a(4)) = (60, 109) is a solution: 60^2+(60+31)^2 = 3600+8281 = 11881 = 109^2.

%t LinearRecurrence[{0,0,6,0,0,-1},{25,31,41,109,155,221},40] (* _Harvey P. Dale_, Oct 12 2017 *)

%o (PARI) {forstep(n=-8, 840000000, [1, 3], if(issquare(2*n^2+62*n+961, &k), print1(k, ",")))};

%o (Magma) I:=[25,31,41,109,155,221]; [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. A118674, A001653, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Mar 11 2009

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)