cp's OEIS Frontend

A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.

%I A118630 #12 Mar 16 2024 14:25:53
%S A118630 0,539,924,1220,1715,2744,3503,4095,5096,7203,9996,12075,13703,16464,
%T A118630 22295,26640,30044,35819,48020,64239,76328,85800,101871,135828,161139,
%U A118630 180971,214620,285719,380240,450695,505899,599564,797475,944996,1060584
%N A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.
%C A118630 Also values x of Pythagorean triples (x, x+2401, y); 2401=7^4.
%C A118630 Corresponding values y of solutions (x, y) are in A157247.
%C A118630 Limit_{n -> oo} a(n)/a(n-9) = 3+2*sqrt(2).
%C A118630 Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 9 = {1, 2, 6}.
%C A118630 Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 3, 5, 7}.
%C A118630 Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {4, 8}.
%H A118630 <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1).
%F A118630 a(n) = 6*a(n-9)-a(n-18)+4802 for n > 18; a(1)=0, a(2)=539, a(3)=924, a(4)=1220, a(5)=1715, a(6)=2744, a(7)=3503, a(8)=4095, a(9)=5096, a(10)=7203, a(11)=9996, a(12)=12075, a(13)=13703, a(14)=16464,a (15)=22295, a(16)=26640, a(17)=30044, a(18)=35819.
%F A118630 G.f.: x*(539+385*x+296*x^2+495*x^3+1029*x^4+759*x^5+592*x^6 +1001*x^7+2107*x^8-441*x^9-231*x^10-148*x^11-209*x^12-343*x^13 -209*x^14-148*x^15-231*x^16-441*x^17) / ((1-x)*(1-6*x^9+x^18)).
%F A118630 a(9*k+1) = 2401*A001652(k) for k >= 0.
%e A118630 924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.
%o A118630 (PARI) {forstep(n=0, 1100000, [3 ,1], if(issquare(n^2+(n+2401)^2), print1(n, ",")))}
%Y A118630 Cf. A157247, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).
%K A118630 nonn,easy
%O A118630 1,2
%A A118630 _Mohamed Bouhamida_, May 09 2006
%E A118630 Edited by _Klaus Brockhaus_, Feb 25 2009