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A159844 Positive numbers y such that y^2 is of the form x^2+(x+359)^2 with integer x.

%I A159844 #26 Sep 08 2022 08:45:44
%S A159844 325,359,401,1549,1795,2081,8969,10411,12085,52265,60671,70429,304621,
%T A159844 353615,410489,1775461,2061019,2392505,10348145,12012499,13944541,
%U A159844 60313409,70013975,81274741,351532309,408071351,473703905,2048880445
%N A159844 Positive numbers y such that y^2 is of the form x^2+(x+359)^2 with integer x.
%C A159844 (-36, a(1)) and (A130610(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.
%C A159844 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A159844 lim_{n -> infinity} a(n)/a(n-1) = (363+38*sqrt(2))/359 for n mod 3 = {0, 2}.
%C A159844 lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 1.
%C A159844 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)+2p with a(1)=0, a(2)=2m+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+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+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [_Mohamed Bouhamida_, Sep 09 2009]
%H A159844 G. C. Greubel, <a href="/A159844/b159844.txt">Table of n, a(n) for n = 1..1000</a>
%H A159844 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159844 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=359, a(3)=401, a(4)=1549, a(5)=1795, a(6)=2081.
%F A159844 G.f.: (1-x)*(325+684*x+1085*x^2+684*x^3+325*x^4) / (1-6*x^3+x^6).
%F A159844 a(3*k-1) = 359*A001653(k) for k >= 1.
%e A159844 (-36, a(1)) = (-36, 325) is a solution: (-36)^2+(-36+359)^2 = 1296+104329 = 105625 = 325^2.
%e A159844 (A130610(1), a(2)) = (0, 359) is a solution: 0^2+(0+359)^2 = 128881 = 359^2.
%e A159844 (A130610(3), a(4)) = (901, 1549) is a solution: 901^2+(901+359)^2 = 811801+1587600 = 2399401 = 1549^2.
%t A159844 t={325,359,401,1549,1795,2081}; Do[AppendTo[t, 6*t[[-3]]-t[[-6]]], {25}]; t
%t A159844 CoefficientList[Series[(325+359 x+401 x^2-401 x^3-359 x^4-325 x^5)/(1-6 x^3+x^6),{x,0,30}],x]  (* _Harvey P. Dale_, Feb 16 2011 *)
%t A159844 LinearRecurrence[{0,0,6,0,0,-1}, {325, 359, 401, 1549, 1795, 2081}, 50] (* _G. C. Greubel_, May 19 2018 *)
%o A159844 (PARI) {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+718*n+128881, &k), print1(k, ",")))}
%o A159844 (PARI) V=[]; v=[[-323,-325], [-323,325], [0,-359], [-359,359], [-399,-401], [399,401]]; for(n=1,100,u=[]; for(i=1,#v,if(v[i][2]>0, u=concat(u,v[i][2])); t=3*v[i][1]+2*v[i][2]+359; v[i][2]=4*v[i][1]+3*v[i][2]+718; v[i][1]=t); V=concat(V,u)); vecsort(V,,8) \\ _Charles R Greathouse IV_, Feb 14 2011
%o A159844 (Magma) I:=[325, 359, 401, 1549, 1795, 2081]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 19 2018
%Y A159844 Cf. A130610, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).
%K A159844 nonn,easy
%O A159844 1,1
%A A159844 _Klaus Brockhaus_, Apr 30 2009