%I #22 Feb 16 2024 06:38:21

%S 0,36,39,123,319,336,820,1960,2059,4879,11523,12100,28536,67260,70623,

%T 166419,392119,411720,970060,2285536,2399779,5654023,13321179,

%U 13987036,32954160,77641620,81522519,192071019,452528623,475148160,1119472036

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+41, y).

%C Corresponding values y of solutions (x, y) are in A157257.

%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 = {1, 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 = 0.

%H G. C. Greubel, <a href="/A129288/b129288.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).

%F a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.

%F G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

%F a(3*k + 1) = 41*A001652(k) for k >= 0.

%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,36,39,123,319,336,820},40] (* _Harvey P. Dale_, Jan 18 2015 *)

%o (PARI) forstep(n=0, 1200000000, [3 ,1], if(issquare(2*n^2+82*n+1681), print1(n, ",")))

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // _G. C. Greubel_, May 07 2018

%Y Cf. A157257, A001652, 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 1,2

%A _Mohamed Bouhamida_, May 26 2007

%E Edited and extended by _Klaus Brockhaus_, Feb 26 2009