cp's OEIS Frontend

A118676 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.

%I A118676 #19 Oct 21 2022 22:09:10
%S A118676 0,20,161,237,341,1140,1580,2184,6837,9401,12921,40040,54984,75500,
%T A118676 233561,320661,440237,1361484,1869140,2566080,7935501,10894337,
%U A118676 14956401,46251680,63497040,87172484,269574737,370088061,508078661,1571196900,2157031484,2961299640
%N A118676 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.
%C A118676 Also values x of Pythagorean triples (x, x+79, y).
%C A118676 Corresponding values y of solutions (x, y) are in A159758.
%C A118676 For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
%C A118676 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A118676 lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {1, 2}.
%C A118676 lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 0.
%H A118676 G. C. Greubel, <a href="/A118676/b118676.txt">Table of n, a(n) for n = 1..1000</a>
%H A118676 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A118676 a(n) = 6*a(n-3) -a(n-6) +158 for n > 6; a(1)=0, a(2)=20, a(3)=161, a(4)=237, a(5)=341, a(6)=1140.
%F A118676 G.f.: x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1-6*x^3+x^6)).
%F A118676 a(3*k+1) = 79*A001652(k) for k >= 0.
%t A118676 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,20,161,237,341,1140,1580},75] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)
%o A118676 (PARI) forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+158*n+6241), print1(n, ",")))
%o A118676 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1- 6*x^3+x^6)))); // _G. C. Greubel_, May 07 2018
%Y A118676 Cf. A159758, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).
%K A118676 nonn,easy
%O A118676 1,2
%A A118676 _Mohamed Bouhamida_, May 19 2006
%E A118676 Edited by _Klaus Brockhaus_, Apr 30 2009