A157257 - OEIS

%I #11 Sep 08 2022 08:45:41

%S 29,41,85,89,205,481,505,1189,2801,2941,6929,16325,17141,40385,95149,

%T 99905,235381,554569,582289,1371901,3232265,3393829,7996025,18839021,

%U 19780685,46604249,109801861,115290281,271629469,639972145,671961001

%N Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

%C (-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^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) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.

%H G. C. Greubel, <a href="/A157257/b157257.txt">Table of n, a(n) for n = 1..1001</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)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.

%F G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).

%F a(3*k-1) = 41*A001653(k) for k >= 1.

%e (-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.

%e (A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.

%e (A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.

%t CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x,0,50}], x] (* _G. C. Greubel_, Feb 04 2018 *)

%o (PARI) {forstep(n=-20, 500000000, [3 ,1], if(issquare(n^2+(n+41)^2, &k), print1(k, ",")))}

%o (PARI) x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ _G. C. Greubel_, Feb 04 2018

%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // _G. C. Greubel_, Feb 04 2018

%Y Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Feb 26 2009